Research ArticleTemperature Dependent Viscosity of a Third Order Thin FilmFluid Layer on a Lubricating Vertical Belt

T Gul1 S Islam1 R A Shah2 I Khan3 and L C C Dennis4

1Department of Mathematics Abdul Wali Khan University Mardan Khyber Pakhtunkhwa Pakistan2Department of Mathematics UET Peshawar Khyber Pakhtunkhwa Pakistan3College of Engineering Majmaah University Saudi Arabia4Department of Fundamental and Applied Sciences Universiti Teknologi PETRONAS 31750 Perak Malaysia

Correspondence should be addressed to L C C Dennis dennislingpetronascommy

Received 23 July 2014 Revised 28 September 2014 Accepted 30 September 2014

Academic Editor Yasir Khan

Copyright copy 2015 T Gul et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper aims to study the influence of heat transfer on thin film flow of a reactive third order fluid with variable viscosityand slip boundary condition The problem is formulated in the form of coupled nonlinear equations governing the flow togetherwith appropriate boundary conditions Approximate analytical solutions for velocity and temperature are obtained using AdomianDecomposition Method (ADM) Such solutions are also obtained by using Optimal Homotopy Asymptotic Method (OHAM) andare compared with ADM solutions Both of these solutions are found identical as shown in graphs and tables The graphical resultsfor embedded flow parameters are also shown

1 Introduction

Thin film fluid flows exist in various aspects of daily life Inengineering we see their usage in condensers distillationunits and heat exchangers In geophysical events we see thinfluid films in the forms of drillingmud heat pipes and debrisflow In biological sciences we can see thin fluid films coatingthe airways in the lungs and thin tear films covering theeye These are a few examples of common use of thin fluidfilms All these occurrences can be modeled using mathe-matical principles The material properties such as viscosityand density change accordingly when a fluid is subjectedto a temperature change This phenomenon mostly occursin heat exchangers chemical reactors or processes wherecomponents are cooled

Properties of isothermal flows (viscosity at constant tem-perature) are important in many situations There have alsobeen many studies done on variety of nonisothermal (tem-perature dependent viscosity) flows of fluids particularlyGoussis andKelly [1 2] andHwang andWeng [3] investigatedthe stability of a fluid film with variable viscosity flowing

down a heated plan while Reisfeld and Bankoff [4] and Wuand Hwang [5] independently considered the progressionand final substrate of a fluid film with variable viscosity on aheated horizontal substrate subjected to surface tension andvan derWaals forces Sansom et al [6] considered the scatter-ing of a thin film flow with temperature dependent viscosityon a horizontal substrate for different viscosity models forboth heated and cooled substrate without internal heatingwithin the film and for a substrate at the ambient temperaturewith constant internal heating within the film

Many researchers have contributed relevant work in thisfield Amongst them Khaled et al [7] studied heat transferinside thin film flow with variable pressure whereas Nadeemand Awais [8] investigated the thin film unsteady flow withvariable viscosityThe freemicropolar convection for the caseof a symmetric boundary conditions or asymmetric heatingwas investigated by Chamkha et al [9] later on proceededby Saleh et al [10] Few other relevant studies may be foundin [11 12] and the references therein One of the well-knownmodels amongst non-Newtonian fluids for thin film is a thirdgrade fluid which has its constitutive equations based on

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2 Abstract and Applied Analysis

strong theoretical foundations where the relation betweenstress and strain is nonlinear Therefore in the present workwe have chosen third grade fluid to study its thin film flowover a vertical belt

On the other hand several numerical and approximatemethods are used by many authors to study real worldproblems appearing in mathematics fluid mechanics andengineering sciences Few of these methods include Keller-box [13] shooting [14] Homotopy Perturbation Method(HPM) [15] Homotopy Analysis Method (HAM) and Opti-mal Homotopy Asymptotic Method (OHAM) [16 17] Appli-cations of OHAM for solving nonlinear equations arising inheat transfer have been investigated byMarinca andHerisanu[18] Marinca et al [19] analysed steady flow of a fourth gradefluid past a porous plate using OHAM Micropolar flow in aporous channel with highmass transfer has been investigatedby Joneidi [20] These methods deal with the nonlinearproblems effectively The work under various configurationson the thin film flows has been discussed by Siddiqui et al[21] In [22] Siddiqui et al examined the thin film flows ofSisko and Oldroyd-6 constant fluids on a moving belt Usingthe same idea Gul et al [23 24] investigated MHD thin filmflow of a third grade fluid for lifting and drainage problemsConstant and variable viscosity fluid has been used in allthese studies In [24] they studied variable viscosity with no-slip boundary conditions However in the present problemwe have chosen Reynolds model (exponential expression)which is based on variable viscosity in the presence of slipboundary conditions More exactly we have shown the effectof variable viscosity with heat transfer in a thin film fluid flowsuch as silicate melts and polymers past a lubricating beltIn these fluids viscous friction generates a local increase intemperature near the belt with a resultant viscosity decreaseand frequent rise of the flow velocity This velocity increasemay produce an additional growth of the local temperaturediscussed by Costa et al [25] Analytical solution for MHDflow in a third grade fluid with variable viscosity has beendiscussed by Ellahi and Riaz [26] Approximate analyticalsolution for flow of a third grade fluid through a parallel platechannel filled with a porousmedium has been investigated byAksoy et al [27]

Based on the above motivation the present paper aimsto study the effect of heat on viscosity into a thin film flowof a third grade fluid over a vertical belt with slip boundaryconditions using ADM and OHAM In 1992 Adomian [2829] introduced the ADM for the approximate solutionsfor linear and nonlinear problems Wazwaz [30 31] usedADM for reliable treatment of Bratu-type and Emden-Fowlerequations Alam et al investigated the thin film flow ofJohnson-Segalman fluids for lifting and drainage problems[32]

2 Basic Equations

Continuity momentum and energy equations of an incom-pressible isothermal third grade fluid are

nabla sdot v = 0 (1)

120588

119863v119863119905

= nabla sdot T + 120588g (2)

120588119888119901

119863Θ

119863119905

= 120581nabla

2Θ + tr (T sdot L) (3)

where L = nablav and119863119863119905 = 120597120597119905+(vnabla) denotesmaterial timederivative

Cauchy stress tensor T is given by

T = minus119901I + S (4)

where minus119901I denotes spherical stress and shear stress S isdefined as

S = 120583A1 + 1205721A2 + 1205722A2

1+ 1205731A3 + 1205732 (A1A2 + A2A1)

+ 1205733 (trA2

2)A1

(5)

Here A1 A2 and A3 are the kinematical tensors given by

A0 = I

A1 = (nablav) + (nablav)119879

A119899 =119863A119899minus1119863119905

+ A119899minus1 (nablav) + (nablav)119879A119899minus1 119899 ge 2

(6)

3 Formulation of the Lift Problem

Consider a wide flat belt moves vertically upward at constantspeed 119881 through a large bath of a third grade liquid Thebelt carries with it a layer of liquid of constant thickness120575 Cartesian coordinate system is chosen for analysis inwhich the 119910-axis is taken parallel to the belt and 119909-axis isperpendicular to the belt Assume that the flow is steadyand laminar after a small distance above the liquid surfacelayer while external pressure is atmospheric everywhere Thegeometry of the problems is shown in Figures 1(a) and 1(b)

Velocity and temperature fields are

v = (0 V (119909) 0)

Θ = Θ (119909)

(7)

Modeled slip boundary conditions are

v = 119881 minus 120574119879119909119910 Θ = Θ0 at 119909 = 0

119889v119889119909

= 0 Θ = Θ1 at 119909 = 120575(8)

Inserting the velocity field from (7) in continuity equation (1)and in Cauchy stress equation (4) the continuity equation (1)satisfies identically while (4) gives the components of stresstensor as

119879119910119910 = minus119875 + (21205721 + 1205722) (119889v119889119909

)

2

119879119909119910 = 120583119889v119889119909

+ 2 (1205732 + 1205733) (119889v119889119909

)

3

Abstract and Applied Analysis 3

Moving belt

Thin film

Heat

x

y

g

(a)

Stationary belt

Thin film

Heat

x

y

g

(b)

Figure 1 (a) Geometry of lift problem (b) Geometry of drainage problem

119879119909119909 = minus119875 + 1205722 (119889v119889119909

)

119879119911119911 = minus119875

119879119909119911 = 119879119910119911 = 0

(9)

Inserting (9) in (2) and (3) the momentum and energy equa-tions reduce to

0 = 120583

119889

2v119889119909

2+

119889V119889119909

119889120583

119889119909

+ 6 (1205732 + 1205733) (119889V119889119909

)

2

(

119889

2V119889119909

2)

minus 120588119892

0 = 120581

119889

2Θ

119889119909

2+ 120583(

119889v119889119909

)

2

+ 2 (1205732 + 1205733) (119889v119889119909

)

4

(10)

In addition we introduce the dimensionless parameters

V =v119881

119909 =

119909

120575

Θ =Θ minus Θ0

Θ1 minus Θ0

120583 =

120583

1205830

120574 =

120574

1205740

119861119903 =

12058301198812

119896 (Θ1 minus Θ0)

119878119905 =

120575

2120588119892

1205830119881

120573 =

(1205732 + 1205733) 1198812

12058301205752

Λ = 1205741205830

(11)

For Reynoldrsquos model the dimensionless viscosity is

120583 = exp (minus119872Θ) (12)

By making use of Taylor series expansion one may representviscosity and its derivative as

120583 cong 1 minus119872Θ

119889120583

119889119909

cong minus119872

119889Θ

119889119909

(13)

Inserting dimensionless variables from (11) in modeledboundary conditions (8) and in momentum as well as inenergy equations (10) and dropping out the bar notations weobtain

119889

2V119889119909

2minus119872(

119889V119889119909

119889Θ

119889119909

+ Θ (119909)

119889

2V119889119909

2)

+ 6120573(

119889V119889119909

)

2

(

119889

2V119889119909

2) minus 119878119905 = 0

(14)

119889

2Θ

119889119909

2+ 119861119903 ((

119889V119889119909

)

2

minus119872Θ(

119889V119889119909

)

2

+ 2120573(

119889V119889119909

)

4

) = 0 (15)

4 Abstract and Applied Analysis

According to Gul et al [24] the boundary conditions for zerocomponent solution are

V119899 = 1 minus Λ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

Θ119899 = 0 at 119909 = 0

(16)

119889V119899119889119909

= 0 Θ119899 = 1 at 119909 = 1 when 119899 = 0 (17)

Similarly the boundary conditions for first second and thirdcomponents solutions are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

Θ119899 = 0 at 119909 = 0

(18)

119889V119899119889119909

= 0 Θ119899 = 0 at 119909 = 1 when 119899 = 1 2 3 (19)

4 Solution of Lifting Problem

41 The ADM Solution The inverse operator 119871minus1 = ∬119889V1015840of

the Adomian Decomposition Method is used in the secondorder coupled equations (14) and (15)

V = 1198911 minus 6120573119871minus1[(

119889V119889119909

)

2119889

2V119889

2119909

]

+119872[

119889V119889119909

119889Θ

119889119909

+ Θ

119889

2V119889119909

2]

Θ = 1198912 minus 119861119903119871minus1(

119889V119889119909

)

2

+ 119861119903119872119871minus1Θ(

119889V119889119909

)

2

minus 2119861119903120573119871minus1(

119889V119889119909

)

4

(20)

where11989111198912 are the terms whichwe get by double integratingsource terms 119878119905 0 andusing the boundary (initial) conditionsAs ADM is a series solution method according to Aksoy etal [27] the solutions Θ and the nonlinear terms 1198731V 1198732V1198733V 1198734Θ 1198735Θ and 1198736Θ can be expressed respectively inthe following series

V =infin

sum

119899=0

V119899

Θ =

infin

sum

119899=0

Θ119899

1198731V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119860 119899 = (119889V119889119909

)

2

(

119889

2V119889119909

2)

1198732V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119861 119899 = (119889V119889119909

)(

119889Θ

119889119909

)

1198733V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119862 119899 = Θ(119889

2V119889119909

2)

1198734Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119863 119899 = (119889V119889119909

)

2

1198735Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119864 119899 = Θ(119889V119889119909

)

2

1198736Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119865 119899 = (119889V119889119909

)

4

(21)

where ⏞⏞⏞⏞⏞⏞⏞119860 119899⏞⏞⏞⏞⏞⏞⏞

119861 119899⏞⏞⏞⏞⏞⏞⏞

119862 119899⏞⏞⏞⏞⏞⏞⏞

119863 119899⏞⏞⏞⏞⏞⏞⏞

119864 119899 and⏞⏞⏞⏞⏞⏞⏞

119865 119899 are calledthe Adomian polynomials For every nonlinear term thesepolynomials are obtained from the following formula

⏞⏞⏞⏞⏞⏞⏞

119860 0 = 1198911 (V0)

1198630 = 1198912 (Θ0)

⏞⏞⏞⏞⏞⏞⏞

119860 119894 =⏞⏞⏞⏞⏞⏞⏞

119861 119894 =⏞⏞⏞⏞⏞⏞⏞

119862 119894 =

119894

sum

120582=1

119862 (120582 119899) 1198911

(120582)(V0)

⏞⏞⏞⏞⏞⏞⏞

119863 119894 =⏞⏞⏞⏞⏞⏞⏞

119864 119894 =⏞⏞⏞⏞⏞⏞⏞

119865 119894 =

119894

sum

120582=1

119862 (120582 119899) 1198912

(120582)(Θ0)

(22)

Here 1198911(V) and 1198912(Θ) are the nonlinear functions and 1198911(120582)

1198912

(120582) are the 120582th derivative of 1198911(V) and 1198912(Θ) evaluated atV = V0 andΘ = Θ0The (120582 119899) is a well-situated representationof the series coefficients and is formed by forming productsor sumof the products of 120582 components of Vwhose subscriptsadd up to 119894 divided by the factorial of the number of repeatedsubscripts [26] The convergence of this method has beendiscussed in [28ndash30]

The series solutions of (20) are as follows

infin

sum

119899=0

V119899 = 1198911 minus 6120573119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119860 119899] +119872119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119861 119899]

+119872119871

minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119862 119899]

infin

sum

119899=0

Θ119899 = 1198912 minus 119861119903119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119863 119899]

+ 119861119903119872119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119864 119899]

minus 2119861119903120573119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119865 119899]

(23)

Abstract and Applied Analysis 5

The first few Adomian polynomials are derived in (24) and(27) when 119899 ge 0

⏞⏞⏞⏞⏞⏞⏞

119860 0 = (119889V0119889119909

)

2119889

2V0119889119909

2

⏞⏞⏞⏞⏞⏞⏞

119861 0 =119889V0119889119909

119889Θ0

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 0 = Θ0 (119909)

119889

2V0119889119909

2

(24)

⏞⏞⏞⏞⏞⏞⏞

119863 0 = (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119864 0 = Θ0 (119909) (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119865 0 = (119889V0119889119909

)

4

(25)

⏞⏞⏞⏞⏞⏞⏞

119860 1 = (119889V0119889119909

)

2119889

2V1119889119909

2+ 2

119889V0119889119909

119889V119889119909

⏞⏞⏞⏞⏞⏞⏞

119861 1 =119889V1119889119909

119889Θ0

119889119909

+

119889V0119889119909

119889Θ1

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 1 = Θ1 (119909)

119889

2V0119889119909

2+ Θ0 (119909)

119889

2V1119889119909

2

(26)

⏞⏞⏞⏞⏞⏞⏞

119863 1 = 2119889V0119889119909

119889V1119889119909

⏞⏞⏞⏞⏞⏞⏞

119864 1 = Θ1 (119909) (119889V0119889119909

)

2

+ 2Θ0 (119909)

119889V1119889119909

119889V0119889119909

⏞⏞⏞⏞⏞⏞⏞

119865 1 = 4(119889V0119889119909

)

3119889V1119889119909

(27)

Solution of (23) in series form is derived as

V0 + V1 + sdot sdot sdot = 1198911 minus 6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0 +⏞⏞⏞⏞⏞⏞⏞

119860 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0 +⏞⏞⏞⏞⏞⏞⏞

119861 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0 +⏞⏞⏞⏞⏞⏞⏞

119862 1 + sdot sdot sdot ]

Θ0 + Θ1 + sdot sdot sdot = 1198912 minus 119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0 +⏞⏞⏞⏞⏞⏞⏞

119863 1 + sdot sdot sdot ]

+ 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0 +⏞⏞⏞⏞⏞⏞⏞

119864 1 + sdot sdot sdot ]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0 +⏞⏞⏞⏞⏞⏞⏞

119865 1 + sdot sdot sdot ]

(28)

The velocity and heat components up to second componentare obtained by comparing both sides of (28)

Components of the lift problem are

V0 (119909) = 1198911 (29)

Θ0 (119909) = 1198912 (30)

V1 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0]

(31)

Θ1 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0] + 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0]

(32)

V2 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 1] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 1]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 1]

(33)

Θ2 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 1] + Λ119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 1]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 1]

(34)

Zero component slip boundary conditions from (16) and (19)are

V0 = 1 minus Λ(119889V0119889119909

minus119872Θ0

119889V0119889119909

+ 2120573(

119889V0119889119909

)

3

)

Θ0 = 0 at 119909 = 0

(35)

119889V0119889119909

= 0 Θ0 = 1 at 119909 = 1 (36)

First component slip boundary conditions from (17) and (19)are

V1 = minusΛ(119889V1119889119909

minus119872Θ0

119889V1119889119909

minus119872Θ1

119889V0119889119909

+ 6120573(

119889V0119889119909

)

2119889V1119889119909

)

Θ1 = 0 at 119909 = 0(37)

119889V1119889119909

= 0 Θ1 = 0 at 119909 = 1 (38)

Second component slip boundary conditions from (17) and(19) are

V2 = minusΛ[119889V2119889119909

minus119872(Θ0

119889V2119889119909

+ Θ1

119889V1119889119909

+ Θ2

119889V0119889119909

)

+ 6120573((

119889V0119889119909

)

2119889V2119889119909

+ (

119889V1119889119909

)

2119889V0119889119909

)]

Θ2 = 0 at 119909 = 0

(39)

119889V2119889119909

= 0 Θ2 = 0 at 119909 = 1 (40)

Inserting slip boundary conditions from (35) and (36) into(29) and (30) V0(119909) and Θ0(119909) are

V0 (119909) = (1 + Λ119878119905 + 2120573Λ1198783

119905) minus (119878119905) 119909 + (

119878119905

2

) 119909

2

Θ0 (119909) = 119909

(41)

6 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 10

080

085

090

095

100

ADMOHAM

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 2 Comparison of ADM and OHAMmethods for lift velocity profile (a) and temperature distribution (b) 119878119905 = 05119872 = 01 120573 = 03Λ = 001 119861119903 = 4 1198621 = 0212619 119878119905 = 01 119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702416 1198622 = minus024397 1198623 = 0016111198624 = minus0016823 1198622 = minus003461 1198623 = minus449647 and 1198624 = minus1642906

minus001

x

u(x)

ADMOHAM

00 02 04 06 08 10

000

001

002

003

004

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 3 Comparison of ADM and OHAMmethods for drainage velocity profile (a) and temperature distribution (b) 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus0000019 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus00733881198622 = 00000691198623 = 17284476081198624 = 152167755 1198622 = 0049829 1198623 = minus1123666 and 1198624 = minus00230394

Inserting slip boundary conditions from (37) and (38) into(31) and (32) V1(119909) and Θ1(119909) are

V1 (119909) = (2120573Λ1198783

119905+ 12120573

2Λ119878

5

119905) + (2120573119878

3

119905) 119909

minus (

119872119878119905

2

+ 3120573119878

3

119905)119909

2+ (

119872119878119905

3

+ 2120573119878

3

119905)119909

3

minus (

1

2

120573119878

3

119905)119909

4

(42)

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

1198721198611199031198782

119905+

1

3

1205731198611199031198784

119905)119909

minus (

1

2

1198611199031198782

119905+ 120573119861119903119878

4

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+

4

3

1205731198611199031198784

119905)119909

3

minus (

1

12

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+ 120573119861119903119878

4

119905)119909

4

+ (

1

20

1198721198611199031198782

119905+

2

5

1205731198611199031198784

119905)119909

5

minus (

1

15

1205731198611199031198784

119905)119909

6

(43)

Abstract and Applied Analysis 7

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1007

08

09

10

11

12

13

(a)

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 4 Brinkman number on the lift velocity (a) and temperature distribution (b) when 119878119905 = 1Λ = 04119872 = 01 and 120573 = 03 and 119878119905 = 05Λ = 04119872 = 01 and 120573 = 03

x

u(x)

St = 06

St = 02

St = 03

St = 04

St = 05

00 02 04 06 08 10080

085

090

095

100

105

(a)

x

St = 02

St = 04

St = 05

St = 01

St = 03

00 02 04 06 08 1000

02

04

06

08

10

12

120579(x)

(b)

Figure 5 Effect of the stock number on lift velocity (a) and temperature distribution (b) when119872 = 001 120573 = 03 Λ = 01 and 119861119903 = 4 and120573 = 03119872 = 001 and 119861119903 = 100

The terms V2(119909) and Θ2(119909) are too large to be written abovetherefore their expression is mentioned graphically

5 Formulation of Drainage Problem

The geometry and assumptions of the problem are the sameas in previous problem Consider a film of non-Newtonianliquid draining at volume flow rate 119876 down the vertical beltThe belt is stationary and the fluid drains down the belt dueto gravity The gravity in this case is opposite to the previouscase Therefore the stock number is positively mentioned in(14)The coordinate system is selected the same as in previouscase Assume the flow is steady and laminar while externalpressure is neglected Consider fluid shear forces keep thegravity balanced and the film thickness remains constant

Modeled slip boundary conditions for drainage problemare

V = minus120574119879119909119910 at 119909 = 0

119889V119889119909

= 0 at 119909 = 1(44)

Making use of nondimensional variables the slip boundaryconditions in (42) are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

at 119909 = 0

119889V119899119889119909

= 0 at 119909 = 1 when 119899 = 0 1 2 3

(45)

8 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 1004

06

08

10

12

14

16

Λ = 00

Λ = 02

Λ = 04

Λ = 06

Λ = 08

Figure 6 Effect of slip parameter on the lift velocity profile when119878119905 = 1119872 = 01 120573 = 03 and 119861119903 = 50

For heat the boundary conditions are identical as given in(36) (38) and (40) regarding both the problems

6 Solution of Drainage Problem

61 The ADM Solution By making use of ADM in (14)and (15) the Adomian polynomials in (24)ndash(27) for bothproblems are the sameThe different velocity components areobtained as follows

Components of ProblemThe boundary conditions of first andsecond components for drainage velocity profile are the sameas given in (37) The boundary conditions for temperaturedistribution are also the same as given in (38) but the solutionof these components is different dependent on the differentvelocity profile of drainage and lift problemsThe terms V2(119909)and Θ2(119909) are too large to be written above therefore theirexpression is mentioned graphically

Inserting slip boundary conditions from (45) and from(18) and (19) in (29)ndash(34) the different components solutionis obtained as

V0 (119909) =1

2

(2119878119905119909 minus 1198781199051199092)

Θ0 (119909 119905) = 119909

V1 (119909) = (minus1

3

119872

2119878119905 minus 120573119878

3

119905)119909

+ (

Λ119878119905

2

+

3120573119878

3

119905

2

)119909

2

+ (

119872

2119878119905

6

minus

Λ119878119905

3

minus 120573119878

3

119905)119909

3

+ (minus

1

24

119872

2119878119905 +

120573119878

3

119905

4

)119909

4

Table 1 Comparison of present work and existing work [23] for liftvelocity profile when119898 = 119878119905 = 05119872 = 0 120573 = 06 Λ = 0 120572 = 1

119909 Present work Existing work Absolute error00 1 1 001 095485367 095485367 002 091553824 091553824 003 088114335 088114335 004 085118976 085118976 0148 times 10

minus14

05 082550781 082550781 006 080413216 080413216 111 times 10

minus29

07 078721265 0787212652 008 077494144 077494144 minus111 times 10

minus29

09 076749627 0767496272 010 0765 0765 minus111 times 10

minus29

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

Λ1198611199031198782

119905+

1

6

1205731198611199031198784

119905)119909

+ (minus

1

2

1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

Λ1198611199031198782

119905+

2

3

1205731198611199031198784

119905)119909

3

+ (minus

1

12

1198611199031198782

119905minus

1

6

Λ1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

4

+ (

1

20

Λ1198611199031198782

119905+

1

5

1205731198611199031198784

119905)119909

5

+ (minus

1

30

1205731198611199031198784

119905)119909

6

(46)

In the existence work [23]119898 is used as gravitational parame-ter instead of stock number 119878119905 in the present work Magneticparameter is defined as 119872 in the existing work and in thepresent work119872 is embedding parameter If we put magneticparameter = 0 slip parameter Λ = 0 in the existing workand embedding parameter119872 = 0 slip parameter Λ = 0 inpresent work then comparison of existing and present workis as shown in Table 1

OHAM solution of the present work has been obtainedand used only for comparison through Tables 1 2 3 4 and 5

7 Results and Discussion

In this study the heat transfer of a third grade fluid onthin film flow has been examined The geometry of lift anddrainage problems has been shown in Figures 1(a) and 1(b)respectively The velocity and heat transfer analysis for bothlift and drainage problems have been investigated under theinfluence of stock number 119878119905 slip parameterΛ the Brinkmannumber 119861119903 non-Newtonian parameter 120573 and viscosityparameter 119872 The effects of these physical parameters havebeen discussed in Figures 2ndash13 The OHAM solution of thepresent work has been shown graphically and numerically

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

strong theoretical foundations where the relation betweenstress and strain is nonlinear Therefore in the present workwe have chosen third grade fluid to study its thin film flowover a vertical belt

On the other hand several numerical and approximatemethods are used by many authors to study real worldproblems appearing in mathematics fluid mechanics andengineering sciences Few of these methods include Keller-box [13] shooting [14] Homotopy Perturbation Method(HPM) [15] Homotopy Analysis Method (HAM) and Opti-mal Homotopy Asymptotic Method (OHAM) [16 17] Appli-cations of OHAM for solving nonlinear equations arising inheat transfer have been investigated byMarinca andHerisanu[18] Marinca et al [19] analysed steady flow of a fourth gradefluid past a porous plate using OHAM Micropolar flow in aporous channel with highmass transfer has been investigatedby Joneidi [20] These methods deal with the nonlinearproblems effectively The work under various configurationson the thin film flows has been discussed by Siddiqui et al[21] In [22] Siddiqui et al examined the thin film flows ofSisko and Oldroyd-6 constant fluids on a moving belt Usingthe same idea Gul et al [23 24] investigated MHD thin filmflow of a third grade fluid for lifting and drainage problemsConstant and variable viscosity fluid has been used in allthese studies In [24] they studied variable viscosity with no-slip boundary conditions However in the present problemwe have chosen Reynolds model (exponential expression)which is based on variable viscosity in the presence of slipboundary conditions More exactly we have shown the effectof variable viscosity with heat transfer in a thin film fluid flowsuch as silicate melts and polymers past a lubricating beltIn these fluids viscous friction generates a local increase intemperature near the belt with a resultant viscosity decreaseand frequent rise of the flow velocity This velocity increasemay produce an additional growth of the local temperaturediscussed by Costa et al [25] Analytical solution for MHDflow in a third grade fluid with variable viscosity has beendiscussed by Ellahi and Riaz [26] Approximate analyticalsolution for flow of a third grade fluid through a parallel platechannel filled with a porousmedium has been investigated byAksoy et al [27]

Based on the above motivation the present paper aimsto study the effect of heat on viscosity into a thin film flowof a third grade fluid over a vertical belt with slip boundaryconditions using ADM and OHAM In 1992 Adomian [2829] introduced the ADM for the approximate solutionsfor linear and nonlinear problems Wazwaz [30 31] usedADM for reliable treatment of Bratu-type and Emden-Fowlerequations Alam et al investigated the thin film flow ofJohnson-Segalman fluids for lifting and drainage problems[32]

2 Basic Equations

Continuity momentum and energy equations of an incom-pressible isothermal third grade fluid are

nabla sdot v = 0 (1)

120588

119863v119863119905

= nabla sdot T + 120588g (2)

120588119888119901

119863Θ

119863119905

= 120581nabla

2Θ + tr (T sdot L) (3)

where L = nablav and119863119863119905 = 120597120597119905+(vnabla) denotesmaterial timederivative

Cauchy stress tensor T is given by

T = minus119901I + S (4)

where minus119901I denotes spherical stress and shear stress S isdefined as

S = 120583A1 + 1205721A2 + 1205722A2

1+ 1205731A3 + 1205732 (A1A2 + A2A1)

+ 1205733 (trA2

2)A1

(5)

Here A1 A2 and A3 are the kinematical tensors given by

A0 = I

A1 = (nablav) + (nablav)119879

A119899 =119863A119899minus1119863119905

+ A119899minus1 (nablav) + (nablav)119879A119899minus1 119899 ge 2

(6)

3 Formulation of the Lift Problem

Consider a wide flat belt moves vertically upward at constantspeed 119881 through a large bath of a third grade liquid Thebelt carries with it a layer of liquid of constant thickness120575 Cartesian coordinate system is chosen for analysis inwhich the 119910-axis is taken parallel to the belt and 119909-axis isperpendicular to the belt Assume that the flow is steadyand laminar after a small distance above the liquid surfacelayer while external pressure is atmospheric everywhere Thegeometry of the problems is shown in Figures 1(a) and 1(b)

Velocity and temperature fields are

v = (0 V (119909) 0)

Θ = Θ (119909)

(7)

Modeled slip boundary conditions are

v = 119881 minus 120574119879119909119910 Θ = Θ0 at 119909 = 0

119889v119889119909

= 0 Θ = Θ1 at 119909 = 120575(8)

Inserting the velocity field from (7) in continuity equation (1)and in Cauchy stress equation (4) the continuity equation (1)satisfies identically while (4) gives the components of stresstensor as

119879119910119910 = minus119875 + (21205721 + 1205722) (119889v119889119909

)

2

119879119909119910 = 120583119889v119889119909

+ 2 (1205732 + 1205733) (119889v119889119909

)

3

Abstract and Applied Analysis 3

Moving belt

Thin film

Heat

x

y

g

(a)

Stationary belt

Thin film

Heat

x

y

g

(b)

Figure 1 (a) Geometry of lift problem (b) Geometry of drainage problem

119879119909119909 = minus119875 + 1205722 (119889v119889119909

)

119879119911119911 = minus119875

119879119909119911 = 119879119910119911 = 0

(9)

Inserting (9) in (2) and (3) the momentum and energy equa-tions reduce to

0 = 120583

119889

2v119889119909

2+

119889V119889119909

119889120583

119889119909

+ 6 (1205732 + 1205733) (119889V119889119909

)

2

(

119889

2V119889119909

2)

minus 120588119892

0 = 120581

119889

2Θ

119889119909

2+ 120583(

119889v119889119909

)

2

+ 2 (1205732 + 1205733) (119889v119889119909

)

4

(10)

In addition we introduce the dimensionless parameters

V =v119881

119909 =

119909

120575

Θ =Θ minus Θ0

Θ1 minus Θ0

120583 =

120583

1205830

120574 =

120574

1205740

119861119903 =

12058301198812

119896 (Θ1 minus Θ0)

119878119905 =

120575

2120588119892

1205830119881

120573 =

(1205732 + 1205733) 1198812

12058301205752

Λ = 1205741205830

(11)

For Reynoldrsquos model the dimensionless viscosity is

120583 = exp (minus119872Θ) (12)

By making use of Taylor series expansion one may representviscosity and its derivative as

120583 cong 1 minus119872Θ

119889120583

119889119909

cong minus119872

119889Θ

119889119909

(13)

Inserting dimensionless variables from (11) in modeledboundary conditions (8) and in momentum as well as inenergy equations (10) and dropping out the bar notations weobtain

119889

2V119889119909

2minus119872(

119889V119889119909

119889Θ

119889119909

+ Θ (119909)

119889

2V119889119909

2)

+ 6120573(

119889V119889119909

)

2

(

119889

2V119889119909

2) minus 119878119905 = 0

(14)

119889

2Θ

119889119909

2+ 119861119903 ((

119889V119889119909

)

2

minus119872Θ(

119889V119889119909

)

2

+ 2120573(

119889V119889119909

)

4

) = 0 (15)

4 Abstract and Applied Analysis

According to Gul et al [24] the boundary conditions for zerocomponent solution are

V119899 = 1 minus Λ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

Θ119899 = 0 at 119909 = 0

(16)

119889V119899119889119909

= 0 Θ119899 = 1 at 119909 = 1 when 119899 = 0 (17)

Similarly the boundary conditions for first second and thirdcomponents solutions are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

Θ119899 = 0 at 119909 = 0

(18)

119889V119899119889119909

= 0 Θ119899 = 0 at 119909 = 1 when 119899 = 1 2 3 (19)

4 Solution of Lifting Problem

41 The ADM Solution The inverse operator 119871minus1 = ∬119889V1015840of

the Adomian Decomposition Method is used in the secondorder coupled equations (14) and (15)

V = 1198911 minus 6120573119871minus1[(

119889V119889119909

)

2119889

2V119889

2119909

]

+119872[

119889V119889119909

119889Θ

119889119909

+ Θ

119889

2V119889119909

2]

Θ = 1198912 minus 119861119903119871minus1(

119889V119889119909

)

2

+ 119861119903119872119871minus1Θ(

119889V119889119909

)

2

minus 2119861119903120573119871minus1(

119889V119889119909

)

4

(20)

where11989111198912 are the terms whichwe get by double integratingsource terms 119878119905 0 andusing the boundary (initial) conditionsAs ADM is a series solution method according to Aksoy etal [27] the solutions Θ and the nonlinear terms 1198731V 1198732V1198733V 1198734Θ 1198735Θ and 1198736Θ can be expressed respectively inthe following series

V =infin

sum

119899=0

V119899

Θ =

infin

sum

119899=0

Θ119899

1198731V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119860 119899 = (119889V119889119909

)

2

(

119889

2V119889119909

2)

1198732V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119861 119899 = (119889V119889119909

)(

119889Θ

119889119909

)

1198733V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119862 119899 = Θ(119889

2V119889119909

2)

1198734Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119863 119899 = (119889V119889119909

)

2

1198735Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119864 119899 = Θ(119889V119889119909

)

2

1198736Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119865 119899 = (119889V119889119909

)

4

(21)

where ⏞⏞⏞⏞⏞⏞⏞119860 119899⏞⏞⏞⏞⏞⏞⏞

119861 119899⏞⏞⏞⏞⏞⏞⏞

119862 119899⏞⏞⏞⏞⏞⏞⏞

119863 119899⏞⏞⏞⏞⏞⏞⏞

119864 119899 and⏞⏞⏞⏞⏞⏞⏞

119865 119899 are calledthe Adomian polynomials For every nonlinear term thesepolynomials are obtained from the following formula

⏞⏞⏞⏞⏞⏞⏞

119860 0 = 1198911 (V0)

1198630 = 1198912 (Θ0)

⏞⏞⏞⏞⏞⏞⏞

119860 119894 =⏞⏞⏞⏞⏞⏞⏞

119861 119894 =⏞⏞⏞⏞⏞⏞⏞

119862 119894 =

119894

sum

120582=1

119862 (120582 119899) 1198911

(120582)(V0)

⏞⏞⏞⏞⏞⏞⏞

119863 119894 =⏞⏞⏞⏞⏞⏞⏞

119864 119894 =⏞⏞⏞⏞⏞⏞⏞

119865 119894 =

119894

sum

120582=1

119862 (120582 119899) 1198912

(120582)(Θ0)

(22)

Here 1198911(V) and 1198912(Θ) are the nonlinear functions and 1198911(120582)

1198912

(120582) are the 120582th derivative of 1198911(V) and 1198912(Θ) evaluated atV = V0 andΘ = Θ0The (120582 119899) is a well-situated representationof the series coefficients and is formed by forming productsor sumof the products of 120582 components of Vwhose subscriptsadd up to 119894 divided by the factorial of the number of repeatedsubscripts [26] The convergence of this method has beendiscussed in [28ndash30]

The series solutions of (20) are as follows

infin

sum

119899=0

V119899 = 1198911 minus 6120573119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119860 119899] +119872119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119861 119899]

+119872119871

minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119862 119899]

infin

sum

119899=0

Θ119899 = 1198912 minus 119861119903119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119863 119899]

+ 119861119903119872119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119864 119899]

minus 2119861119903120573119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119865 119899]

(23)

Abstract and Applied Analysis 5

The first few Adomian polynomials are derived in (24) and(27) when 119899 ge 0

⏞⏞⏞⏞⏞⏞⏞

119860 0 = (119889V0119889119909

)

2119889

2V0119889119909

2

⏞⏞⏞⏞⏞⏞⏞

119861 0 =119889V0119889119909

119889Θ0

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 0 = Θ0 (119909)

119889

2V0119889119909

2

(24)

⏞⏞⏞⏞⏞⏞⏞

119863 0 = (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119864 0 = Θ0 (119909) (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119865 0 = (119889V0119889119909

)

4

(25)

⏞⏞⏞⏞⏞⏞⏞

119860 1 = (119889V0119889119909

)

2119889

2V1119889119909

2+ 2

119889V0119889119909

119889V119889119909

⏞⏞⏞⏞⏞⏞⏞

119861 1 =119889V1119889119909

119889Θ0

119889119909

+

119889V0119889119909

119889Θ1

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 1 = Θ1 (119909)

119889

2V0119889119909

2+ Θ0 (119909)

119889

2V1119889119909

2

(26)

⏞⏞⏞⏞⏞⏞⏞

119863 1 = 2119889V0119889119909

119889V1119889119909

⏞⏞⏞⏞⏞⏞⏞

119864 1 = Θ1 (119909) (119889V0119889119909

)

2

+ 2Θ0 (119909)

119889V1119889119909

119889V0119889119909

⏞⏞⏞⏞⏞⏞⏞

119865 1 = 4(119889V0119889119909

)

3119889V1119889119909

(27)

Solution of (23) in series form is derived as

V0 + V1 + sdot sdot sdot = 1198911 minus 6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0 +⏞⏞⏞⏞⏞⏞⏞

119860 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0 +⏞⏞⏞⏞⏞⏞⏞

119861 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0 +⏞⏞⏞⏞⏞⏞⏞

119862 1 + sdot sdot sdot ]

Θ0 + Θ1 + sdot sdot sdot = 1198912 minus 119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0 +⏞⏞⏞⏞⏞⏞⏞

119863 1 + sdot sdot sdot ]

+ 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0 +⏞⏞⏞⏞⏞⏞⏞

119864 1 + sdot sdot sdot ]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0 +⏞⏞⏞⏞⏞⏞⏞

119865 1 + sdot sdot sdot ]

(28)

The velocity and heat components up to second componentare obtained by comparing both sides of (28)

Components of the lift problem are

V0 (119909) = 1198911 (29)

Θ0 (119909) = 1198912 (30)

V1 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0]

(31)

Θ1 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0] + 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0]

(32)

V2 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 1] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 1]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 1]

(33)

Θ2 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 1] + Λ119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 1]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 1]

(34)

Zero component slip boundary conditions from (16) and (19)are

V0 = 1 minus Λ(119889V0119889119909

minus119872Θ0

119889V0119889119909

+ 2120573(

119889V0119889119909

)

3

)

Θ0 = 0 at 119909 = 0

(35)

119889V0119889119909

= 0 Θ0 = 1 at 119909 = 1 (36)

First component slip boundary conditions from (17) and (19)are

V1 = minusΛ(119889V1119889119909

minus119872Θ0

119889V1119889119909

minus119872Θ1

119889V0119889119909

+ 6120573(

119889V0119889119909

)

2119889V1119889119909

)

Θ1 = 0 at 119909 = 0(37)

119889V1119889119909

= 0 Θ1 = 0 at 119909 = 1 (38)

Second component slip boundary conditions from (17) and(19) are

V2 = minusΛ[119889V2119889119909

minus119872(Θ0

119889V2119889119909

+ Θ1

119889V1119889119909

+ Θ2

119889V0119889119909

)

+ 6120573((

119889V0119889119909

)

2119889V2119889119909

+ (

119889V1119889119909

)

2119889V0119889119909

)]

Θ2 = 0 at 119909 = 0

(39)

119889V2119889119909

= 0 Θ2 = 0 at 119909 = 1 (40)

Inserting slip boundary conditions from (35) and (36) into(29) and (30) V0(119909) and Θ0(119909) are

V0 (119909) = (1 + Λ119878119905 + 2120573Λ1198783

119905) minus (119878119905) 119909 + (

119878119905

2

) 119909

2

Θ0 (119909) = 119909

(41)

6 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 10

080

085

090

095

100

ADMOHAM

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 2 Comparison of ADM and OHAMmethods for lift velocity profile (a) and temperature distribution (b) 119878119905 = 05119872 = 01 120573 = 03Λ = 001 119861119903 = 4 1198621 = 0212619 119878119905 = 01 119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702416 1198622 = minus024397 1198623 = 0016111198624 = minus0016823 1198622 = minus003461 1198623 = minus449647 and 1198624 = minus1642906

minus001

x

u(x)

ADMOHAM

00 02 04 06 08 10

000

001

002

003

004

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 3 Comparison of ADM and OHAMmethods for drainage velocity profile (a) and temperature distribution (b) 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus0000019 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus00733881198622 = 00000691198623 = 17284476081198624 = 152167755 1198622 = 0049829 1198623 = minus1123666 and 1198624 = minus00230394

Inserting slip boundary conditions from (37) and (38) into(31) and (32) V1(119909) and Θ1(119909) are

V1 (119909) = (2120573Λ1198783

119905+ 12120573

2Λ119878

5

119905) + (2120573119878

3

119905) 119909

minus (

119872119878119905

2

+ 3120573119878

3

119905)119909

2+ (

119872119878119905

3

+ 2120573119878

3

119905)119909

3

minus (

1

2

120573119878

3

119905)119909

4

(42)

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

1198721198611199031198782

119905+

1

3

1205731198611199031198784

119905)119909

minus (

1

2

1198611199031198782

119905+ 120573119861119903119878

4

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+

4

3

1205731198611199031198784

119905)119909

3

minus (

1

12

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+ 120573119861119903119878

4

119905)119909

4

+ (

1

20

1198721198611199031198782

119905+

2

5

1205731198611199031198784

119905)119909

5

minus (

1

15

1205731198611199031198784

119905)119909

6

(43)

Abstract and Applied Analysis 7

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1007

08

09

10

11

12

13

(a)

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 4 Brinkman number on the lift velocity (a) and temperature distribution (b) when 119878119905 = 1Λ = 04119872 = 01 and 120573 = 03 and 119878119905 = 05Λ = 04119872 = 01 and 120573 = 03

x

u(x)

St = 06

St = 02

St = 03

St = 04

St = 05

00 02 04 06 08 10080

085

090

095

100

105

(a)

x

St = 02

St = 04

St = 05

St = 01

St = 03

00 02 04 06 08 1000

02

04

06

08

10

12

120579(x)

(b)

Figure 5 Effect of the stock number on lift velocity (a) and temperature distribution (b) when119872 = 001 120573 = 03 Λ = 01 and 119861119903 = 4 and120573 = 03119872 = 001 and 119861119903 = 100

The terms V2(119909) and Θ2(119909) are too large to be written abovetherefore their expression is mentioned graphically

5 Formulation of Drainage Problem

The geometry and assumptions of the problem are the sameas in previous problem Consider a film of non-Newtonianliquid draining at volume flow rate 119876 down the vertical beltThe belt is stationary and the fluid drains down the belt dueto gravity The gravity in this case is opposite to the previouscase Therefore the stock number is positively mentioned in(14)The coordinate system is selected the same as in previouscase Assume the flow is steady and laminar while externalpressure is neglected Consider fluid shear forces keep thegravity balanced and the film thickness remains constant

Modeled slip boundary conditions for drainage problemare

V = minus120574119879119909119910 at 119909 = 0

119889V119889119909

= 0 at 119909 = 1(44)

Making use of nondimensional variables the slip boundaryconditions in (42) are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

at 119909 = 0

119889V119899119889119909

= 0 at 119909 = 1 when 119899 = 0 1 2 3

(45)

8 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 1004

06

08

10

12

14

16

Λ = 00

Λ = 02

Λ = 04

Λ = 06

Λ = 08

Figure 6 Effect of slip parameter on the lift velocity profile when119878119905 = 1119872 = 01 120573 = 03 and 119861119903 = 50

For heat the boundary conditions are identical as given in(36) (38) and (40) regarding both the problems

6 Solution of Drainage Problem

61 The ADM Solution By making use of ADM in (14)and (15) the Adomian polynomials in (24)ndash(27) for bothproblems are the sameThe different velocity components areobtained as follows

Components of ProblemThe boundary conditions of first andsecond components for drainage velocity profile are the sameas given in (37) The boundary conditions for temperaturedistribution are also the same as given in (38) but the solutionof these components is different dependent on the differentvelocity profile of drainage and lift problemsThe terms V2(119909)and Θ2(119909) are too large to be written above therefore theirexpression is mentioned graphically

Inserting slip boundary conditions from (45) and from(18) and (19) in (29)ndash(34) the different components solutionis obtained as

V0 (119909) =1

2

(2119878119905119909 minus 1198781199051199092)

Θ0 (119909 119905) = 119909

V1 (119909) = (minus1

3

119872

2119878119905 minus 120573119878

3

119905)119909

+ (

Λ119878119905

2

+

3120573119878

3

119905

2

)119909

2

+ (

119872

2119878119905

6

minus

Λ119878119905

3

minus 120573119878

3

119905)119909

3

+ (minus

1

24

119872

2119878119905 +

120573119878

3

119905

4

)119909

4

Table 1 Comparison of present work and existing work [23] for liftvelocity profile when119898 = 119878119905 = 05119872 = 0 120573 = 06 Λ = 0 120572 = 1

119909 Present work Existing work Absolute error00 1 1 001 095485367 095485367 002 091553824 091553824 003 088114335 088114335 004 085118976 085118976 0148 times 10

minus14

05 082550781 082550781 006 080413216 080413216 111 times 10

minus29

07 078721265 0787212652 008 077494144 077494144 minus111 times 10

minus29

09 076749627 0767496272 010 0765 0765 minus111 times 10

minus29

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

Λ1198611199031198782

119905+

1

6

1205731198611199031198784

119905)119909

+ (minus

1

2

1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

Λ1198611199031198782

119905+

2

3

1205731198611199031198784

119905)119909

3

+ (minus

1

12

1198611199031198782

119905minus

1

6

Λ1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

4

+ (

1

20

Λ1198611199031198782

119905+

1

5

1205731198611199031198784

119905)119909

5

+ (minus

1

30

1205731198611199031198784

119905)119909

6

(46)

In the existence work [23]119898 is used as gravitational parame-ter instead of stock number 119878119905 in the present work Magneticparameter is defined as 119872 in the existing work and in thepresent work119872 is embedding parameter If we put magneticparameter = 0 slip parameter Λ = 0 in the existing workand embedding parameter119872 = 0 slip parameter Λ = 0 inpresent work then comparison of existing and present workis as shown in Table 1

OHAM solution of the present work has been obtainedand used only for comparison through Tables 1 2 3 4 and 5

7 Results and Discussion

In this study the heat transfer of a third grade fluid onthin film flow has been examined The geometry of lift anddrainage problems has been shown in Figures 1(a) and 1(b)respectively The velocity and heat transfer analysis for bothlift and drainage problems have been investigated under theinfluence of stock number 119878119905 slip parameterΛ the Brinkmannumber 119861119903 non-Newtonian parameter 120573 and viscosityparameter 119872 The effects of these physical parameters havebeen discussed in Figures 2ndash13 The OHAM solution of thepresent work has been shown graphically and numerically

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Moving belt

Thin film

Heat

x

y

g

(a)

Stationary belt

Thin film

Heat

x

y

g

(b)

Figure 1 (a) Geometry of lift problem (b) Geometry of drainage problem

119879119909119909 = minus119875 + 1205722 (119889v119889119909

)

119879119911119911 = minus119875

119879119909119911 = 119879119910119911 = 0

(9)

Inserting (9) in (2) and (3) the momentum and energy equa-tions reduce to

0 = 120583

119889

2v119889119909

2+

119889V119889119909

119889120583

119889119909

+ 6 (1205732 + 1205733) (119889V119889119909

)

2

(

119889

2V119889119909

2)

minus 120588119892

0 = 120581

119889

2Θ

119889119909

2+ 120583(

119889v119889119909

)

2

+ 2 (1205732 + 1205733) (119889v119889119909

)

4

(10)

In addition we introduce the dimensionless parameters

V =v119881

119909 =

119909

120575

Θ =Θ minus Θ0

Θ1 minus Θ0

120583 =

120583

1205830

120574 =

120574

1205740

119861119903 =

12058301198812

119896 (Θ1 minus Θ0)

119878119905 =

120575

2120588119892

1205830119881

120573 =

(1205732 + 1205733) 1198812

12058301205752

Λ = 1205741205830

(11)

For Reynoldrsquos model the dimensionless viscosity is

120583 = exp (minus119872Θ) (12)

By making use of Taylor series expansion one may representviscosity and its derivative as

120583 cong 1 minus119872Θ

119889120583

119889119909

cong minus119872

119889Θ

119889119909

(13)

Inserting dimensionless variables from (11) in modeledboundary conditions (8) and in momentum as well as inenergy equations (10) and dropping out the bar notations weobtain

119889

2V119889119909

2minus119872(

119889V119889119909

119889Θ

119889119909

+ Θ (119909)

119889

2V119889119909

2)

+ 6120573(

119889V119889119909

)

2

(

119889

2V119889119909

2) minus 119878119905 = 0

(14)

119889

2Θ

119889119909

2+ 119861119903 ((

119889V119889119909

)

2

minus119872Θ(

119889V119889119909

)

2

+ 2120573(

119889V119889119909

)

4

) = 0 (15)

4 Abstract and Applied Analysis

According to Gul et al [24] the boundary conditions for zerocomponent solution are

V119899 = 1 minus Λ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

Θ119899 = 0 at 119909 = 0

(16)

119889V119899119889119909

= 0 Θ119899 = 1 at 119909 = 1 when 119899 = 0 (17)

Similarly the boundary conditions for first second and thirdcomponents solutions are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

Θ119899 = 0 at 119909 = 0

(18)

119889V119899119889119909

= 0 Θ119899 = 0 at 119909 = 1 when 119899 = 1 2 3 (19)

4 Solution of Lifting Problem

41 The ADM Solution The inverse operator 119871minus1 = ∬119889V1015840of

the Adomian Decomposition Method is used in the secondorder coupled equations (14) and (15)

V = 1198911 minus 6120573119871minus1[(

119889V119889119909

)

2119889

2V119889

2119909

]

+119872[

119889V119889119909

119889Θ

119889119909

+ Θ

119889

2V119889119909

2]

Θ = 1198912 minus 119861119903119871minus1(

119889V119889119909

)

2

+ 119861119903119872119871minus1Θ(

119889V119889119909

)

2

minus 2119861119903120573119871minus1(

119889V119889119909

)

4

(20)

where11989111198912 are the terms whichwe get by double integratingsource terms 119878119905 0 andusing the boundary (initial) conditionsAs ADM is a series solution method according to Aksoy etal [27] the solutions Θ and the nonlinear terms 1198731V 1198732V1198733V 1198734Θ 1198735Θ and 1198736Θ can be expressed respectively inthe following series

V =infin

sum

119899=0

V119899

Θ =

infin

sum

119899=0

Θ119899

1198731V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119860 119899 = (119889V119889119909

)

2

(

119889

2V119889119909

2)

1198732V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119861 119899 = (119889V119889119909

)(

119889Θ

119889119909

)

1198733V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119862 119899 = Θ(119889

2V119889119909

2)

1198734Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119863 119899 = (119889V119889119909

)

2

1198735Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119864 119899 = Θ(119889V119889119909

)

2

1198736Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119865 119899 = (119889V119889119909

)

4

(21)

where ⏞⏞⏞⏞⏞⏞⏞119860 119899⏞⏞⏞⏞⏞⏞⏞

119861 119899⏞⏞⏞⏞⏞⏞⏞

119862 119899⏞⏞⏞⏞⏞⏞⏞

119863 119899⏞⏞⏞⏞⏞⏞⏞

119864 119899 and⏞⏞⏞⏞⏞⏞⏞

119865 119899 are calledthe Adomian polynomials For every nonlinear term thesepolynomials are obtained from the following formula

⏞⏞⏞⏞⏞⏞⏞

119860 0 = 1198911 (V0)

1198630 = 1198912 (Θ0)

⏞⏞⏞⏞⏞⏞⏞

119860 119894 =⏞⏞⏞⏞⏞⏞⏞

119861 119894 =⏞⏞⏞⏞⏞⏞⏞

119862 119894 =

119894

sum

120582=1

119862 (120582 119899) 1198911

(120582)(V0)

⏞⏞⏞⏞⏞⏞⏞

119863 119894 =⏞⏞⏞⏞⏞⏞⏞

119864 119894 =⏞⏞⏞⏞⏞⏞⏞

119865 119894 =

119894

sum

120582=1

119862 (120582 119899) 1198912

(120582)(Θ0)

(22)

Here 1198911(V) and 1198912(Θ) are the nonlinear functions and 1198911(120582)

1198912

(120582) are the 120582th derivative of 1198911(V) and 1198912(Θ) evaluated atV = V0 andΘ = Θ0The (120582 119899) is a well-situated representationof the series coefficients and is formed by forming productsor sumof the products of 120582 components of Vwhose subscriptsadd up to 119894 divided by the factorial of the number of repeatedsubscripts [26] The convergence of this method has beendiscussed in [28ndash30]

The series solutions of (20) are as follows

infin

sum

119899=0

V119899 = 1198911 minus 6120573119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119860 119899] +119872119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119861 119899]

+119872119871

minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119862 119899]

infin

sum

119899=0

Θ119899 = 1198912 minus 119861119903119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119863 119899]

+ 119861119903119872119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119864 119899]

minus 2119861119903120573119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119865 119899]

(23)

Abstract and Applied Analysis 5

The first few Adomian polynomials are derived in (24) and(27) when 119899 ge 0

⏞⏞⏞⏞⏞⏞⏞

119860 0 = (119889V0119889119909

)

2119889

2V0119889119909

2

⏞⏞⏞⏞⏞⏞⏞

119861 0 =119889V0119889119909

119889Θ0

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 0 = Θ0 (119909)

119889

2V0119889119909

2

(24)

⏞⏞⏞⏞⏞⏞⏞

119863 0 = (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119864 0 = Θ0 (119909) (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119865 0 = (119889V0119889119909

)

4

(25)

⏞⏞⏞⏞⏞⏞⏞

119860 1 = (119889V0119889119909

)

2119889

2V1119889119909

2+ 2

119889V0119889119909

119889V119889119909

⏞⏞⏞⏞⏞⏞⏞

119861 1 =119889V1119889119909

119889Θ0

119889119909

+

119889V0119889119909

119889Θ1

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 1 = Θ1 (119909)

119889

2V0119889119909

2+ Θ0 (119909)

119889

2V1119889119909

2

(26)

⏞⏞⏞⏞⏞⏞⏞

119863 1 = 2119889V0119889119909

119889V1119889119909

⏞⏞⏞⏞⏞⏞⏞

119864 1 = Θ1 (119909) (119889V0119889119909

)

2

+ 2Θ0 (119909)

119889V1119889119909

119889V0119889119909

⏞⏞⏞⏞⏞⏞⏞

119865 1 = 4(119889V0119889119909

)

3119889V1119889119909

(27)

Solution of (23) in series form is derived as

V0 + V1 + sdot sdot sdot = 1198911 minus 6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0 +⏞⏞⏞⏞⏞⏞⏞

119860 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0 +⏞⏞⏞⏞⏞⏞⏞

119861 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0 +⏞⏞⏞⏞⏞⏞⏞

119862 1 + sdot sdot sdot ]

Θ0 + Θ1 + sdot sdot sdot = 1198912 minus 119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0 +⏞⏞⏞⏞⏞⏞⏞

119863 1 + sdot sdot sdot ]

+ 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0 +⏞⏞⏞⏞⏞⏞⏞

119864 1 + sdot sdot sdot ]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0 +⏞⏞⏞⏞⏞⏞⏞

119865 1 + sdot sdot sdot ]

(28)

The velocity and heat components up to second componentare obtained by comparing both sides of (28)

Components of the lift problem are

V0 (119909) = 1198911 (29)

Θ0 (119909) = 1198912 (30)

V1 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0]

(31)

Θ1 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0] + 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0]

(32)

V2 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 1] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 1]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 1]

(33)

Θ2 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 1] + Λ119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 1]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 1]

(34)

Zero component slip boundary conditions from (16) and (19)are

V0 = 1 minus Λ(119889V0119889119909

minus119872Θ0

119889V0119889119909

+ 2120573(

119889V0119889119909

)

3

)

Θ0 = 0 at 119909 = 0

(35)

119889V0119889119909

= 0 Θ0 = 1 at 119909 = 1 (36)

First component slip boundary conditions from (17) and (19)are

V1 = minusΛ(119889V1119889119909

minus119872Θ0

119889V1119889119909

minus119872Θ1

119889V0119889119909

+ 6120573(

119889V0119889119909

)

2119889V1119889119909

)

Θ1 = 0 at 119909 = 0(37)

119889V1119889119909

= 0 Θ1 = 0 at 119909 = 1 (38)

Second component slip boundary conditions from (17) and(19) are

V2 = minusΛ[119889V2119889119909

minus119872(Θ0

119889V2119889119909

+ Θ1

119889V1119889119909

+ Θ2

119889V0119889119909

)

+ 6120573((

119889V0119889119909

)

2119889V2119889119909

+ (

119889V1119889119909

)

2119889V0119889119909

)]

Θ2 = 0 at 119909 = 0

(39)

119889V2119889119909

= 0 Θ2 = 0 at 119909 = 1 (40)

Inserting slip boundary conditions from (35) and (36) into(29) and (30) V0(119909) and Θ0(119909) are

V0 (119909) = (1 + Λ119878119905 + 2120573Λ1198783

119905) minus (119878119905) 119909 + (

119878119905

2

) 119909

2

Θ0 (119909) = 119909

(41)

6 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 10

080

085

090

095

100

ADMOHAM

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 2 Comparison of ADM and OHAMmethods for lift velocity profile (a) and temperature distribution (b) 119878119905 = 05119872 = 01 120573 = 03Λ = 001 119861119903 = 4 1198621 = 0212619 119878119905 = 01 119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702416 1198622 = minus024397 1198623 = 0016111198624 = minus0016823 1198622 = minus003461 1198623 = minus449647 and 1198624 = minus1642906

minus001

x

u(x)

ADMOHAM

00 02 04 06 08 10

000

001

002

003

004

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 3 Comparison of ADM and OHAMmethods for drainage velocity profile (a) and temperature distribution (b) 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus0000019 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus00733881198622 = 00000691198623 = 17284476081198624 = 152167755 1198622 = 0049829 1198623 = minus1123666 and 1198624 = minus00230394

Inserting slip boundary conditions from (37) and (38) into(31) and (32) V1(119909) and Θ1(119909) are

V1 (119909) = (2120573Λ1198783

119905+ 12120573

2Λ119878

5

119905) + (2120573119878

3

119905) 119909

minus (

119872119878119905

2

+ 3120573119878

3

119905)119909

2+ (

119872119878119905

3

+ 2120573119878

3

119905)119909

3

minus (

1

2

120573119878

3

119905)119909

4

(42)

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

1198721198611199031198782

119905+

1

3

1205731198611199031198784

119905)119909

minus (

1

2

1198611199031198782

119905+ 120573119861119903119878

4

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+

4

3

1205731198611199031198784

119905)119909

3

minus (

1

12

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+ 120573119861119903119878

4

119905)119909

4

+ (

1

20

1198721198611199031198782

119905+

2

5

1205731198611199031198784

119905)119909

5

minus (

1

15

1205731198611199031198784

119905)119909

6

(43)

Abstract and Applied Analysis 7

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1007

08

09

10

11

12

13

(a)

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 4 Brinkman number on the lift velocity (a) and temperature distribution (b) when 119878119905 = 1Λ = 04119872 = 01 and 120573 = 03 and 119878119905 = 05Λ = 04119872 = 01 and 120573 = 03

x

u(x)

St = 06

St = 02

St = 03

St = 04

St = 05

00 02 04 06 08 10080

085

090

095

100

105

(a)

x

St = 02

St = 04

St = 05

St = 01

St = 03

00 02 04 06 08 1000

02

04

06

08

10

12

120579(x)

(b)

Figure 5 Effect of the stock number on lift velocity (a) and temperature distribution (b) when119872 = 001 120573 = 03 Λ = 01 and 119861119903 = 4 and120573 = 03119872 = 001 and 119861119903 = 100

The terms V2(119909) and Θ2(119909) are too large to be written abovetherefore their expression is mentioned graphically

5 Formulation of Drainage Problem

The geometry and assumptions of the problem are the sameas in previous problem Consider a film of non-Newtonianliquid draining at volume flow rate 119876 down the vertical beltThe belt is stationary and the fluid drains down the belt dueto gravity The gravity in this case is opposite to the previouscase Therefore the stock number is positively mentioned in(14)The coordinate system is selected the same as in previouscase Assume the flow is steady and laminar while externalpressure is neglected Consider fluid shear forces keep thegravity balanced and the film thickness remains constant

Modeled slip boundary conditions for drainage problemare

V = minus120574119879119909119910 at 119909 = 0

119889V119889119909

= 0 at 119909 = 1(44)

Making use of nondimensional variables the slip boundaryconditions in (42) are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

at 119909 = 0

119889V119899119889119909

= 0 at 119909 = 1 when 119899 = 0 1 2 3

(45)

8 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 1004

06

08

10

12

14

16

Λ = 00

Λ = 02

Λ = 04

Λ = 06

Λ = 08

Figure 6 Effect of slip parameter on the lift velocity profile when119878119905 = 1119872 = 01 120573 = 03 and 119861119903 = 50

For heat the boundary conditions are identical as given in(36) (38) and (40) regarding both the problems

6 Solution of Drainage Problem

61 The ADM Solution By making use of ADM in (14)and (15) the Adomian polynomials in (24)ndash(27) for bothproblems are the sameThe different velocity components areobtained as follows

Components of ProblemThe boundary conditions of first andsecond components for drainage velocity profile are the sameas given in (37) The boundary conditions for temperaturedistribution are also the same as given in (38) but the solutionof these components is different dependent on the differentvelocity profile of drainage and lift problemsThe terms V2(119909)and Θ2(119909) are too large to be written above therefore theirexpression is mentioned graphically

Inserting slip boundary conditions from (45) and from(18) and (19) in (29)ndash(34) the different components solutionis obtained as

V0 (119909) =1

2

(2119878119905119909 minus 1198781199051199092)

Θ0 (119909 119905) = 119909

V1 (119909) = (minus1

3

119872

2119878119905 minus 120573119878

3

119905)119909

+ (

Λ119878119905

2

+

3120573119878

3

119905

2

)119909

2

+ (

119872

2119878119905

6

minus

Λ119878119905

3

minus 120573119878

3

119905)119909

3

+ (minus

1

24

119872

2119878119905 +

120573119878

3

119905

4

)119909

4

Table 1 Comparison of present work and existing work [23] for liftvelocity profile when119898 = 119878119905 = 05119872 = 0 120573 = 06 Λ = 0 120572 = 1

119909 Present work Existing work Absolute error00 1 1 001 095485367 095485367 002 091553824 091553824 003 088114335 088114335 004 085118976 085118976 0148 times 10

minus14

05 082550781 082550781 006 080413216 080413216 111 times 10

minus29

07 078721265 0787212652 008 077494144 077494144 minus111 times 10

minus29

09 076749627 0767496272 010 0765 0765 minus111 times 10

minus29

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

Λ1198611199031198782

119905+

1

6

1205731198611199031198784

119905)119909

+ (minus

1

2

1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

Λ1198611199031198782

119905+

2

3

1205731198611199031198784

119905)119909

3

+ (minus

1

12

1198611199031198782

119905minus

1

6

Λ1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

4

+ (

1

20

Λ1198611199031198782

119905+

1

5

1205731198611199031198784

119905)119909

5

+ (minus

1

30

1205731198611199031198784

119905)119909

6

(46)

In the existence work [23]119898 is used as gravitational parame-ter instead of stock number 119878119905 in the present work Magneticparameter is defined as 119872 in the existing work and in thepresent work119872 is embedding parameter If we put magneticparameter = 0 slip parameter Λ = 0 in the existing workand embedding parameter119872 = 0 slip parameter Λ = 0 inpresent work then comparison of existing and present workis as shown in Table 1

OHAM solution of the present work has been obtainedand used only for comparison through Tables 1 2 3 4 and 5

7 Results and Discussion

In this study the heat transfer of a third grade fluid onthin film flow has been examined The geometry of lift anddrainage problems has been shown in Figures 1(a) and 1(b)respectively The velocity and heat transfer analysis for bothlift and drainage problems have been investigated under theinfluence of stock number 119878119905 slip parameterΛ the Brinkmannumber 119861119903 non-Newtonian parameter 120573 and viscosityparameter 119872 The effects of these physical parameters havebeen discussed in Figures 2ndash13 The OHAM solution of thepresent work has been shown graphically and numerically

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

According to Gul et al [24] the boundary conditions for zerocomponent solution are

V119899 = 1 minus Λ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

Θ119899 = 0 at 119909 = 0

(16)

119889V119899119889119909

= 0 Θ119899 = 1 at 119909 = 1 when 119899 = 0 (17)

Similarly the boundary conditions for first second and thirdcomponents solutions are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

Θ119899 = 0 at 119909 = 0

(18)

119889V119899119889119909

= 0 Θ119899 = 0 at 119909 = 1 when 119899 = 1 2 3 (19)

4 Solution of Lifting Problem

41 The ADM Solution The inverse operator 119871minus1 = ∬119889V1015840of

the Adomian Decomposition Method is used in the secondorder coupled equations (14) and (15)

V = 1198911 minus 6120573119871minus1[(

119889V119889119909

)

2119889

2V119889

2119909

]

+119872[

119889V119889119909

119889Θ

119889119909

+ Θ

119889

2V119889119909

2]

Θ = 1198912 minus 119861119903119871minus1(

119889V119889119909

)

2

+ 119861119903119872119871minus1Θ(

119889V119889119909

)

2

minus 2119861119903120573119871minus1(

119889V119889119909

)

4

(20)

where11989111198912 are the terms whichwe get by double integratingsource terms 119878119905 0 andusing the boundary (initial) conditionsAs ADM is a series solution method according to Aksoy etal [27] the solutions Θ and the nonlinear terms 1198731V 1198732V1198733V 1198734Θ 1198735Θ and 1198736Θ can be expressed respectively inthe following series

V =infin

sum

119899=0

V119899

Θ =

infin

sum

119899=0

Θ119899

1198731V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119860 119899 = (119889V119889119909

)

2

(

119889

2V119889119909

2)

1198732V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119861 119899 = (119889V119889119909

)(

119889Θ

119889119909

)

1198733V =infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119862 119899 = Θ(119889

2V119889119909

2)

1198734Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119863 119899 = (119889V119889119909

)

2

1198735Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119864 119899 = Θ(119889V119889119909

)

2

1198736Θ =

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119865 119899 = (119889V119889119909

)

4

(21)

where ⏞⏞⏞⏞⏞⏞⏞119860 119899⏞⏞⏞⏞⏞⏞⏞

119861 119899⏞⏞⏞⏞⏞⏞⏞

119862 119899⏞⏞⏞⏞⏞⏞⏞

119863 119899⏞⏞⏞⏞⏞⏞⏞

119864 119899 and⏞⏞⏞⏞⏞⏞⏞

119865 119899 are calledthe Adomian polynomials For every nonlinear term thesepolynomials are obtained from the following formula

⏞⏞⏞⏞⏞⏞⏞

119860 0 = 1198911 (V0)

1198630 = 1198912 (Θ0)

⏞⏞⏞⏞⏞⏞⏞

119860 119894 =⏞⏞⏞⏞⏞⏞⏞

119861 119894 =⏞⏞⏞⏞⏞⏞⏞

119862 119894 =

119894

sum

120582=1

119862 (120582 119899) 1198911

(120582)(V0)

⏞⏞⏞⏞⏞⏞⏞

119863 119894 =⏞⏞⏞⏞⏞⏞⏞

119864 119894 =⏞⏞⏞⏞⏞⏞⏞

119865 119894 =

119894

sum

120582=1

119862 (120582 119899) 1198912

(120582)(Θ0)

(22)

Here 1198911(V) and 1198912(Θ) are the nonlinear functions and 1198911(120582)

1198912

(120582) are the 120582th derivative of 1198911(V) and 1198912(Θ) evaluated atV = V0 andΘ = Θ0The (120582 119899) is a well-situated representationof the series coefficients and is formed by forming productsor sumof the products of 120582 components of Vwhose subscriptsadd up to 119894 divided by the factorial of the number of repeatedsubscripts [26] The convergence of this method has beendiscussed in [28ndash30]

The series solutions of (20) are as follows

infin

sum

119899=0

V119899 = 1198911 minus 6120573119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119860 119899] +119872119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119861 119899]

+119872119871

minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119862 119899]

infin

sum

119899=0

Θ119899 = 1198912 minus 119861119903119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119863 119899]

+ 119861119903119872119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119864 119899]

minus 2119861119903120573119871minus1[

infin

sum

119899=0

⏞⏞⏞⏞⏞⏞⏞

119865 119899]

(23)

Abstract and Applied Analysis 5

The first few Adomian polynomials are derived in (24) and(27) when 119899 ge 0

⏞⏞⏞⏞⏞⏞⏞

119860 0 = (119889V0119889119909

)

2119889

2V0119889119909

2

⏞⏞⏞⏞⏞⏞⏞

119861 0 =119889V0119889119909

119889Θ0

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 0 = Θ0 (119909)

119889

2V0119889119909

2

(24)

⏞⏞⏞⏞⏞⏞⏞

119863 0 = (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119864 0 = Θ0 (119909) (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119865 0 = (119889V0119889119909

)

4

(25)

⏞⏞⏞⏞⏞⏞⏞

119860 1 = (119889V0119889119909

)

2119889

2V1119889119909

2+ 2

119889V0119889119909

119889V119889119909

⏞⏞⏞⏞⏞⏞⏞

119861 1 =119889V1119889119909

119889Θ0

119889119909

+

119889V0119889119909

119889Θ1

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 1 = Θ1 (119909)

119889

2V0119889119909

2+ Θ0 (119909)

119889

2V1119889119909

2

(26)

⏞⏞⏞⏞⏞⏞⏞

119863 1 = 2119889V0119889119909

119889V1119889119909

⏞⏞⏞⏞⏞⏞⏞

119864 1 = Θ1 (119909) (119889V0119889119909

)

2

+ 2Θ0 (119909)

119889V1119889119909

119889V0119889119909

⏞⏞⏞⏞⏞⏞⏞

119865 1 = 4(119889V0119889119909

)

3119889V1119889119909

(27)

Solution of (23) in series form is derived as

V0 + V1 + sdot sdot sdot = 1198911 minus 6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0 +⏞⏞⏞⏞⏞⏞⏞

119860 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0 +⏞⏞⏞⏞⏞⏞⏞

119861 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0 +⏞⏞⏞⏞⏞⏞⏞

119862 1 + sdot sdot sdot ]

Θ0 + Θ1 + sdot sdot sdot = 1198912 minus 119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0 +⏞⏞⏞⏞⏞⏞⏞

119863 1 + sdot sdot sdot ]

+ 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0 +⏞⏞⏞⏞⏞⏞⏞

119864 1 + sdot sdot sdot ]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0 +⏞⏞⏞⏞⏞⏞⏞

119865 1 + sdot sdot sdot ]

(28)

The velocity and heat components up to second componentare obtained by comparing both sides of (28)

Components of the lift problem are

V0 (119909) = 1198911 (29)

Θ0 (119909) = 1198912 (30)

V1 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0]

(31)

Θ1 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0] + 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0]

(32)

V2 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 1] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 1]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 1]

(33)

Θ2 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 1] + Λ119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 1]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 1]

(34)

Zero component slip boundary conditions from (16) and (19)are

V0 = 1 minus Λ(119889V0119889119909

minus119872Θ0

119889V0119889119909

+ 2120573(

119889V0119889119909

)

3

)

Θ0 = 0 at 119909 = 0

(35)

119889V0119889119909

= 0 Θ0 = 1 at 119909 = 1 (36)

First component slip boundary conditions from (17) and (19)are

V1 = minusΛ(119889V1119889119909

minus119872Θ0

119889V1119889119909

minus119872Θ1

119889V0119889119909

+ 6120573(

119889V0119889119909

)

2119889V1119889119909

)

Θ1 = 0 at 119909 = 0(37)

119889V1119889119909

= 0 Θ1 = 0 at 119909 = 1 (38)

Second component slip boundary conditions from (17) and(19) are

V2 = minusΛ[119889V2119889119909

minus119872(Θ0

119889V2119889119909

+ Θ1

119889V1119889119909

+ Θ2

119889V0119889119909

)

+ 6120573((

119889V0119889119909

)

2119889V2119889119909

+ (

119889V1119889119909

)

2119889V0119889119909

)]

Θ2 = 0 at 119909 = 0

(39)

119889V2119889119909

= 0 Θ2 = 0 at 119909 = 1 (40)

Inserting slip boundary conditions from (35) and (36) into(29) and (30) V0(119909) and Θ0(119909) are

V0 (119909) = (1 + Λ119878119905 + 2120573Λ1198783

119905) minus (119878119905) 119909 + (

119878119905

2

) 119909

2

Θ0 (119909) = 119909

(41)

6 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 10

080

085

090

095

100

ADMOHAM

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 2 Comparison of ADM and OHAMmethods for lift velocity profile (a) and temperature distribution (b) 119878119905 = 05119872 = 01 120573 = 03Λ = 001 119861119903 = 4 1198621 = 0212619 119878119905 = 01 119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702416 1198622 = minus024397 1198623 = 0016111198624 = minus0016823 1198622 = minus003461 1198623 = minus449647 and 1198624 = minus1642906

minus001

x

u(x)

ADMOHAM

00 02 04 06 08 10

000

001

002

003

004

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 3 Comparison of ADM and OHAMmethods for drainage velocity profile (a) and temperature distribution (b) 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus0000019 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus00733881198622 = 00000691198623 = 17284476081198624 = 152167755 1198622 = 0049829 1198623 = minus1123666 and 1198624 = minus00230394

Inserting slip boundary conditions from (37) and (38) into(31) and (32) V1(119909) and Θ1(119909) are

V1 (119909) = (2120573Λ1198783

119905+ 12120573

2Λ119878

5

119905) + (2120573119878

3

119905) 119909

minus (

119872119878119905

2

+ 3120573119878

3

119905)119909

2+ (

119872119878119905

3

+ 2120573119878

3

119905)119909

3

minus (

1

2

120573119878

3

119905)119909

4

(42)

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

1198721198611199031198782

119905+

1

3

1205731198611199031198784

119905)119909

minus (

1

2

1198611199031198782

119905+ 120573119861119903119878

4

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+

4

3

1205731198611199031198784

119905)119909

3

minus (

1

12

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+ 120573119861119903119878

4

119905)119909

4

+ (

1

20

1198721198611199031198782

119905+

2

5

1205731198611199031198784

119905)119909

5

minus (

1

15

1205731198611199031198784

119905)119909

6

(43)

Abstract and Applied Analysis 7

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1007

08

09

10

11

12

13

(a)

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 4 Brinkman number on the lift velocity (a) and temperature distribution (b) when 119878119905 = 1Λ = 04119872 = 01 and 120573 = 03 and 119878119905 = 05Λ = 04119872 = 01 and 120573 = 03

x

u(x)

St = 06

St = 02

St = 03

St = 04

St = 05

00 02 04 06 08 10080

085

090

095

100

105

(a)

x

St = 02

St = 04

St = 05

St = 01

St = 03

00 02 04 06 08 1000

02

04

06

08

10

12

120579(x)

(b)

Figure 5 Effect of the stock number on lift velocity (a) and temperature distribution (b) when119872 = 001 120573 = 03 Λ = 01 and 119861119903 = 4 and120573 = 03119872 = 001 and 119861119903 = 100

The terms V2(119909) and Θ2(119909) are too large to be written abovetherefore their expression is mentioned graphically

5 Formulation of Drainage Problem

The geometry and assumptions of the problem are the sameas in previous problem Consider a film of non-Newtonianliquid draining at volume flow rate 119876 down the vertical beltThe belt is stationary and the fluid drains down the belt dueto gravity The gravity in this case is opposite to the previouscase Therefore the stock number is positively mentioned in(14)The coordinate system is selected the same as in previouscase Assume the flow is steady and laminar while externalpressure is neglected Consider fluid shear forces keep thegravity balanced and the film thickness remains constant

Modeled slip boundary conditions for drainage problemare

V = minus120574119879119909119910 at 119909 = 0

119889V119889119909

= 0 at 119909 = 1(44)

Making use of nondimensional variables the slip boundaryconditions in (42) are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

at 119909 = 0

119889V119899119889119909

= 0 at 119909 = 1 when 119899 = 0 1 2 3

(45)

8 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 1004

06

08

10

12

14

16

Λ = 00

Λ = 02

Λ = 04

Λ = 06

Λ = 08

Figure 6 Effect of slip parameter on the lift velocity profile when119878119905 = 1119872 = 01 120573 = 03 and 119861119903 = 50

For heat the boundary conditions are identical as given in(36) (38) and (40) regarding both the problems

6 Solution of Drainage Problem

61 The ADM Solution By making use of ADM in (14)and (15) the Adomian polynomials in (24)ndash(27) for bothproblems are the sameThe different velocity components areobtained as follows

Components of ProblemThe boundary conditions of first andsecond components for drainage velocity profile are the sameas given in (37) The boundary conditions for temperaturedistribution are also the same as given in (38) but the solutionof these components is different dependent on the differentvelocity profile of drainage and lift problemsThe terms V2(119909)and Θ2(119909) are too large to be written above therefore theirexpression is mentioned graphically

Inserting slip boundary conditions from (45) and from(18) and (19) in (29)ndash(34) the different components solutionis obtained as

V0 (119909) =1

2

(2119878119905119909 minus 1198781199051199092)

Θ0 (119909 119905) = 119909

V1 (119909) = (minus1

3

119872

2119878119905 minus 120573119878

3

119905)119909

+ (

Λ119878119905

2

+

3120573119878

3

119905

2

)119909

2

+ (

119872

2119878119905

6

minus

Λ119878119905

3

minus 120573119878

3

119905)119909

3

+ (minus

1

24

119872

2119878119905 +

120573119878

3

119905

4

)119909

4

Table 1 Comparison of present work and existing work [23] for liftvelocity profile when119898 = 119878119905 = 05119872 = 0 120573 = 06 Λ = 0 120572 = 1

119909 Present work Existing work Absolute error00 1 1 001 095485367 095485367 002 091553824 091553824 003 088114335 088114335 004 085118976 085118976 0148 times 10

minus14

05 082550781 082550781 006 080413216 080413216 111 times 10

minus29

07 078721265 0787212652 008 077494144 077494144 minus111 times 10

minus29

09 076749627 0767496272 010 0765 0765 minus111 times 10

minus29

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

Λ1198611199031198782

119905+

1

6

1205731198611199031198784

119905)119909

+ (minus

1

2

1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

Λ1198611199031198782

119905+

2

3

1205731198611199031198784

119905)119909

3

+ (minus

1

12

1198611199031198782

119905minus

1

6

Λ1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

4

+ (

1

20

Λ1198611199031198782

119905+

1

5

1205731198611199031198784

119905)119909

5

+ (minus

1

30

1205731198611199031198784

119905)119909

6

(46)

In the existence work [23]119898 is used as gravitational parame-ter instead of stock number 119878119905 in the present work Magneticparameter is defined as 119872 in the existing work and in thepresent work119872 is embedding parameter If we put magneticparameter = 0 slip parameter Λ = 0 in the existing workand embedding parameter119872 = 0 slip parameter Λ = 0 inpresent work then comparison of existing and present workis as shown in Table 1

OHAM solution of the present work has been obtainedand used only for comparison through Tables 1 2 3 4 and 5

7 Results and Discussion

In this study the heat transfer of a third grade fluid onthin film flow has been examined The geometry of lift anddrainage problems has been shown in Figures 1(a) and 1(b)respectively The velocity and heat transfer analysis for bothlift and drainage problems have been investigated under theinfluence of stock number 119878119905 slip parameterΛ the Brinkmannumber 119861119903 non-Newtonian parameter 120573 and viscosityparameter 119872 The effects of these physical parameters havebeen discussed in Figures 2ndash13 The OHAM solution of thepresent work has been shown graphically and numerically

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

The first few Adomian polynomials are derived in (24) and(27) when 119899 ge 0

⏞⏞⏞⏞⏞⏞⏞

119860 0 = (119889V0119889119909

)

2119889

2V0119889119909

2

⏞⏞⏞⏞⏞⏞⏞

119861 0 =119889V0119889119909

119889Θ0

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 0 = Θ0 (119909)

119889

2V0119889119909

2

(24)

⏞⏞⏞⏞⏞⏞⏞

119863 0 = (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119864 0 = Θ0 (119909) (119889V0119889119909

)

2

⏞⏞⏞⏞⏞⏞⏞

119865 0 = (119889V0119889119909

)

4

(25)

⏞⏞⏞⏞⏞⏞⏞

119860 1 = (119889V0119889119909

)

2119889

2V1119889119909

2+ 2

119889V0119889119909

119889V119889119909

⏞⏞⏞⏞⏞⏞⏞

119861 1 =119889V1119889119909

119889Θ0

119889119909

+

119889V0119889119909

119889Θ1

119889119909

⏞⏞⏞⏞⏞⏞⏞

119862 1 = Θ1 (119909)

119889

2V0119889119909

2+ Θ0 (119909)

119889

2V1119889119909

2

(26)

⏞⏞⏞⏞⏞⏞⏞

119863 1 = 2119889V0119889119909

119889V1119889119909

⏞⏞⏞⏞⏞⏞⏞

119864 1 = Θ1 (119909) (119889V0119889119909

)

2

+ 2Θ0 (119909)

119889V1119889119909

119889V0119889119909

⏞⏞⏞⏞⏞⏞⏞

119865 1 = 4(119889V0119889119909

)

3119889V1119889119909

(27)

Solution of (23) in series form is derived as

V0 + V1 + sdot sdot sdot = 1198911 minus 6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0 +⏞⏞⏞⏞⏞⏞⏞

119860 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0 +⏞⏞⏞⏞⏞⏞⏞

119861 1 + sdot sdot sdot ]

+ 119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0 +⏞⏞⏞⏞⏞⏞⏞

119862 1 + sdot sdot sdot ]

Θ0 + Θ1 + sdot sdot sdot = 1198912 minus 119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0 +⏞⏞⏞⏞⏞⏞⏞

119863 1 + sdot sdot sdot ]

+ 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0 +⏞⏞⏞⏞⏞⏞⏞

119864 1 + sdot sdot sdot ]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0 +⏞⏞⏞⏞⏞⏞⏞

119865 1 + sdot sdot sdot ]

(28)

The velocity and heat components up to second componentare obtained by comparing both sides of (28)

Components of the lift problem are

V0 (119909) = 1198911 (29)

Θ0 (119909) = 1198912 (30)

V1 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 0] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 0]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 0]

(31)

Θ1 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 0] + 119872119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 0]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 0]

(32)

V2 (119909) = minus6120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119860 1] + 119872119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119861 1]

+119872119871

minus1[

⏞⏞⏞⏞⏞⏞⏞

119862 1]

(33)

Θ2 (119909) = minus119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119863 1] + Λ119861119903119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119864 1]

minus 2119861119903120573119871minus1[

⏞⏞⏞⏞⏞⏞⏞

119865 1]

(34)

Zero component slip boundary conditions from (16) and (19)are

V0 = 1 minus Λ(119889V0119889119909

minus119872Θ0

119889V0119889119909

+ 2120573(

119889V0119889119909

)

3

)

Θ0 = 0 at 119909 = 0

(35)

119889V0119889119909

= 0 Θ0 = 1 at 119909 = 1 (36)

First component slip boundary conditions from (17) and (19)are

V1 = minusΛ(119889V1119889119909

minus119872Θ0

119889V1119889119909

minus119872Θ1

119889V0119889119909

+ 6120573(

119889V0119889119909

)

2119889V1119889119909

)

Θ1 = 0 at 119909 = 0(37)

119889V1119889119909

= 0 Θ1 = 0 at 119909 = 1 (38)

Second component slip boundary conditions from (17) and(19) are

V2 = minusΛ[119889V2119889119909

minus119872(Θ0

119889V2119889119909

+ Θ1

119889V1119889119909

+ Θ2

119889V0119889119909

)

+ 6120573((

119889V0119889119909

)

2119889V2119889119909

+ (

119889V1119889119909

)

2119889V0119889119909

)]

Θ2 = 0 at 119909 = 0

(39)

119889V2119889119909

= 0 Θ2 = 0 at 119909 = 1 (40)

Inserting slip boundary conditions from (35) and (36) into(29) and (30) V0(119909) and Θ0(119909) are

V0 (119909) = (1 + Λ119878119905 + 2120573Λ1198783

119905) minus (119878119905) 119909 + (

119878119905

2

) 119909

2

Θ0 (119909) = 119909

(41)

6 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 10

080

085

090

095

100

ADMOHAM

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 2 Comparison of ADM and OHAMmethods for lift velocity profile (a) and temperature distribution (b) 119878119905 = 05119872 = 01 120573 = 03Λ = 001 119861119903 = 4 1198621 = 0212619 119878119905 = 01 119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702416 1198622 = minus024397 1198623 = 0016111198624 = minus0016823 1198622 = minus003461 1198623 = minus449647 and 1198624 = minus1642906

minus001

x

u(x)

ADMOHAM

00 02 04 06 08 10

000

001

002

003

004

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 3 Comparison of ADM and OHAMmethods for drainage velocity profile (a) and temperature distribution (b) 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus0000019 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus00733881198622 = 00000691198623 = 17284476081198624 = 152167755 1198622 = 0049829 1198623 = minus1123666 and 1198624 = minus00230394

Inserting slip boundary conditions from (37) and (38) into(31) and (32) V1(119909) and Θ1(119909) are

V1 (119909) = (2120573Λ1198783

119905+ 12120573

2Λ119878

5

119905) + (2120573119878

3

119905) 119909

minus (

119872119878119905

2

+ 3120573119878

3

119905)119909

2+ (

119872119878119905

3

+ 2120573119878

3

119905)119909

3

minus (

1

2

120573119878

3

119905)119909

4

(42)

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

1198721198611199031198782

119905+

1

3

1205731198611199031198784

119905)119909

minus (

1

2

1198611199031198782

119905+ 120573119861119903119878

4

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+

4

3

1205731198611199031198784

119905)119909

3

minus (

1

12

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+ 120573119861119903119878

4

119905)119909

4

+ (

1

20

1198721198611199031198782

119905+

2

5

1205731198611199031198784

119905)119909

5

minus (

1

15

1205731198611199031198784

119905)119909

6

(43)

Abstract and Applied Analysis 7

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1007

08

09

10

11

12

13

(a)

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 4 Brinkman number on the lift velocity (a) and temperature distribution (b) when 119878119905 = 1Λ = 04119872 = 01 and 120573 = 03 and 119878119905 = 05Λ = 04119872 = 01 and 120573 = 03

x

u(x)

St = 06

St = 02

St = 03

St = 04

St = 05

00 02 04 06 08 10080

085

090

095

100

105

(a)

x

St = 02

St = 04

St = 05

St = 01

St = 03

00 02 04 06 08 1000

02

04

06

08

10

12

120579(x)

(b)

Figure 5 Effect of the stock number on lift velocity (a) and temperature distribution (b) when119872 = 001 120573 = 03 Λ = 01 and 119861119903 = 4 and120573 = 03119872 = 001 and 119861119903 = 100

The terms V2(119909) and Θ2(119909) are too large to be written abovetherefore their expression is mentioned graphically

5 Formulation of Drainage Problem

The geometry and assumptions of the problem are the sameas in previous problem Consider a film of non-Newtonianliquid draining at volume flow rate 119876 down the vertical beltThe belt is stationary and the fluid drains down the belt dueto gravity The gravity in this case is opposite to the previouscase Therefore the stock number is positively mentioned in(14)The coordinate system is selected the same as in previouscase Assume the flow is steady and laminar while externalpressure is neglected Consider fluid shear forces keep thegravity balanced and the film thickness remains constant

Modeled slip boundary conditions for drainage problemare

V = minus120574119879119909119910 at 119909 = 0

119889V119889119909

= 0 at 119909 = 1(44)

Making use of nondimensional variables the slip boundaryconditions in (42) are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

at 119909 = 0

119889V119899119889119909

= 0 at 119909 = 1 when 119899 = 0 1 2 3

(45)

8 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 1004

06

08

10

12

14

16

Λ = 00

Λ = 02

Λ = 04

Λ = 06

Λ = 08

Figure 6 Effect of slip parameter on the lift velocity profile when119878119905 = 1119872 = 01 120573 = 03 and 119861119903 = 50

For heat the boundary conditions are identical as given in(36) (38) and (40) regarding both the problems

6 Solution of Drainage Problem

61 The ADM Solution By making use of ADM in (14)and (15) the Adomian polynomials in (24)ndash(27) for bothproblems are the sameThe different velocity components areobtained as follows

Components of ProblemThe boundary conditions of first andsecond components for drainage velocity profile are the sameas given in (37) The boundary conditions for temperaturedistribution are also the same as given in (38) but the solutionof these components is different dependent on the differentvelocity profile of drainage and lift problemsThe terms V2(119909)and Θ2(119909) are too large to be written above therefore theirexpression is mentioned graphically

Inserting slip boundary conditions from (45) and from(18) and (19) in (29)ndash(34) the different components solutionis obtained as

V0 (119909) =1

2

(2119878119905119909 minus 1198781199051199092)

Θ0 (119909 119905) = 119909

V1 (119909) = (minus1

3

119872

2119878119905 minus 120573119878

3

119905)119909

+ (

Λ119878119905

2

+

3120573119878

3

119905

2

)119909

2

+ (

119872

2119878119905

6

minus

Λ119878119905

3

minus 120573119878

3

119905)119909

3

+ (minus

1

24

119872

2119878119905 +

120573119878

3

119905

4

)119909

4

Table 1 Comparison of present work and existing work [23] for liftvelocity profile when119898 = 119878119905 = 05119872 = 0 120573 = 06 Λ = 0 120572 = 1

119909 Present work Existing work Absolute error00 1 1 001 095485367 095485367 002 091553824 091553824 003 088114335 088114335 004 085118976 085118976 0148 times 10

minus14

05 082550781 082550781 006 080413216 080413216 111 times 10

minus29

07 078721265 0787212652 008 077494144 077494144 minus111 times 10

minus29

09 076749627 0767496272 010 0765 0765 minus111 times 10

minus29

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

Λ1198611199031198782

119905+

1

6

1205731198611199031198784

119905)119909

+ (minus

1

2

1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

Λ1198611199031198782

119905+

2

3

1205731198611199031198784

119905)119909

3

+ (minus

1

12

1198611199031198782

119905minus

1

6

Λ1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

4

+ (

1

20

Λ1198611199031198782

119905+

1

5

1205731198611199031198784

119905)119909

5

+ (minus

1

30

1205731198611199031198784

119905)119909

6

(46)

In the existence work [23]119898 is used as gravitational parame-ter instead of stock number 119878119905 in the present work Magneticparameter is defined as 119872 in the existing work and in thepresent work119872 is embedding parameter If we put magneticparameter = 0 slip parameter Λ = 0 in the existing workand embedding parameter119872 = 0 slip parameter Λ = 0 inpresent work then comparison of existing and present workis as shown in Table 1

OHAM solution of the present work has been obtainedand used only for comparison through Tables 1 2 3 4 and 5

7 Results and Discussion

In this study the heat transfer of a third grade fluid onthin film flow has been examined The geometry of lift anddrainage problems has been shown in Figures 1(a) and 1(b)respectively The velocity and heat transfer analysis for bothlift and drainage problems have been investigated under theinfluence of stock number 119878119905 slip parameterΛ the Brinkmannumber 119861119903 non-Newtonian parameter 120573 and viscosityparameter 119872 The effects of these physical parameters havebeen discussed in Figures 2ndash13 The OHAM solution of thepresent work has been shown graphically and numerically

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 10

080

085

090

095

100

ADMOHAM

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 2 Comparison of ADM and OHAMmethods for lift velocity profile (a) and temperature distribution (b) 119878119905 = 05119872 = 01 120573 = 03Λ = 001 119861119903 = 4 1198621 = 0212619 119878119905 = 01 119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702416 1198622 = minus024397 1198623 = 0016111198624 = minus0016823 1198622 = minus003461 1198623 = minus449647 and 1198624 = minus1642906

minus001

x

u(x)

ADMOHAM

00 02 04 06 08 10

000

001

002

003

004

(a)

x

ADMOHAM

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 3 Comparison of ADM and OHAMmethods for drainage velocity profile (a) and temperature distribution (b) 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus0000019 119878119905 = 01119872 = 001120573 = 03Λ = 01119861119903 = 041198621 = minus00733881198622 = 00000691198623 = 17284476081198624 = 152167755 1198622 = 0049829 1198623 = minus1123666 and 1198624 = minus00230394

Inserting slip boundary conditions from (37) and (38) into(31) and (32) V1(119909) and Θ1(119909) are

V1 (119909) = (2120573Λ1198783

119905+ 12120573

2Λ119878

5

119905) + (2120573119878

3

119905) 119909

minus (

119872119878119905

2

+ 3120573119878

3

119905)119909

2+ (

119872119878119905

3

+ 2120573119878

3

119905)119909

3

minus (

1

2

120573119878

3

119905)119909

4

(42)

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

1198721198611199031198782

119905+

1

3

1205731198611199031198784

119905)119909

minus (

1

2

1198611199031198782

119905+ 120573119861119903119878

4

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+

4

3

1205731198611199031198784

119905)119909

3

minus (

1

12

1198611199031198782

119905+

1

6

1198721198611199031198782

119905+ 120573119861119903119878

4

119905)119909

4

+ (

1

20

1198721198611199031198782

119905+

2

5

1205731198611199031198784

119905)119909

5

minus (

1

15

1205731198611199031198784

119905)119909

6

(43)

Abstract and Applied Analysis 7

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1007

08

09

10

11

12

13

(a)

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 4 Brinkman number on the lift velocity (a) and temperature distribution (b) when 119878119905 = 1Λ = 04119872 = 01 and 120573 = 03 and 119878119905 = 05Λ = 04119872 = 01 and 120573 = 03

x

u(x)

St = 06

St = 02

St = 03

St = 04

St = 05

00 02 04 06 08 10080

085

090

095

100

105

(a)

x

St = 02

St = 04

St = 05

St = 01

St = 03

00 02 04 06 08 1000

02

04

06

08

10

12

120579(x)

(b)

Figure 5 Effect of the stock number on lift velocity (a) and temperature distribution (b) when119872 = 001 120573 = 03 Λ = 01 and 119861119903 = 4 and120573 = 03119872 = 001 and 119861119903 = 100

The terms V2(119909) and Θ2(119909) are too large to be written abovetherefore their expression is mentioned graphically

5 Formulation of Drainage Problem

The geometry and assumptions of the problem are the sameas in previous problem Consider a film of non-Newtonianliquid draining at volume flow rate 119876 down the vertical beltThe belt is stationary and the fluid drains down the belt dueto gravity The gravity in this case is opposite to the previouscase Therefore the stock number is positively mentioned in(14)The coordinate system is selected the same as in previouscase Assume the flow is steady and laminar while externalpressure is neglected Consider fluid shear forces keep thegravity balanced and the film thickness remains constant

Modeled slip boundary conditions for drainage problemare

V = minus120574119879119909119910 at 119909 = 0

119889V119889119909

= 0 at 119909 = 1(44)

Making use of nondimensional variables the slip boundaryconditions in (42) are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

at 119909 = 0

119889V119899119889119909

= 0 at 119909 = 1 when 119899 = 0 1 2 3

(45)

8 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 1004

06

08

10

12

14

16

Λ = 00

Λ = 02

Λ = 04

Λ = 06

Λ = 08

Figure 6 Effect of slip parameter on the lift velocity profile when119878119905 = 1119872 = 01 120573 = 03 and 119861119903 = 50

For heat the boundary conditions are identical as given in(36) (38) and (40) regarding both the problems

6 Solution of Drainage Problem

61 The ADM Solution By making use of ADM in (14)and (15) the Adomian polynomials in (24)ndash(27) for bothproblems are the sameThe different velocity components areobtained as follows

Components of ProblemThe boundary conditions of first andsecond components for drainage velocity profile are the sameas given in (37) The boundary conditions for temperaturedistribution are also the same as given in (38) but the solutionof these components is different dependent on the differentvelocity profile of drainage and lift problemsThe terms V2(119909)and Θ2(119909) are too large to be written above therefore theirexpression is mentioned graphically

Inserting slip boundary conditions from (45) and from(18) and (19) in (29)ndash(34) the different components solutionis obtained as

V0 (119909) =1

2

(2119878119905119909 minus 1198781199051199092)

Θ0 (119909 119905) = 119909

V1 (119909) = (minus1

3

119872

2119878119905 minus 120573119878

3

119905)119909

+ (

Λ119878119905

2

+

3120573119878

3

119905

2

)119909

2

+ (

119872

2119878119905

6

minus

Λ119878119905

3

minus 120573119878

3

119905)119909

3

+ (minus

1

24

119872

2119878119905 +

120573119878

3

119905

4

)119909

4

Table 1 Comparison of present work and existing work [23] for liftvelocity profile when119898 = 119878119905 = 05119872 = 0 120573 = 06 Λ = 0 120572 = 1

119909 Present work Existing work Absolute error00 1 1 001 095485367 095485367 002 091553824 091553824 003 088114335 088114335 004 085118976 085118976 0148 times 10

minus14

05 082550781 082550781 006 080413216 080413216 111 times 10

minus29

07 078721265 0787212652 008 077494144 077494144 minus111 times 10

minus29

09 076749627 0767496272 010 0765 0765 minus111 times 10

minus29

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

Λ1198611199031198782

119905+

1

6

1205731198611199031198784

119905)119909

+ (minus

1

2

1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

Λ1198611199031198782

119905+

2

3

1205731198611199031198784

119905)119909

3

+ (minus

1

12

1198611199031198782

119905minus

1

6

Λ1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

4

+ (

1

20

Λ1198611199031198782

119905+

1

5

1205731198611199031198784

119905)119909

5

+ (minus

1

30

1205731198611199031198784

119905)119909

6

(46)

In the existence work [23]119898 is used as gravitational parame-ter instead of stock number 119878119905 in the present work Magneticparameter is defined as 119872 in the existing work and in thepresent work119872 is embedding parameter If we put magneticparameter = 0 slip parameter Λ = 0 in the existing workand embedding parameter119872 = 0 slip parameter Λ = 0 inpresent work then comparison of existing and present workis as shown in Table 1

OHAM solution of the present work has been obtainedand used only for comparison through Tables 1 2 3 4 and 5

7 Results and Discussion

In this study the heat transfer of a third grade fluid onthin film flow has been examined The geometry of lift anddrainage problems has been shown in Figures 1(a) and 1(b)respectively The velocity and heat transfer analysis for bothlift and drainage problems have been investigated under theinfluence of stock number 119878119905 slip parameterΛ the Brinkmannumber 119861119903 non-Newtonian parameter 120573 and viscosityparameter 119872 The effects of these physical parameters havebeen discussed in Figures 2ndash13 The OHAM solution of thepresent work has been shown graphically and numerically

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1007

08

09

10

11

12

13

(a)

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

00 02 04 06 08 1000

02

04

06

08

10

120579(x)

(b)

Figure 4 Brinkman number on the lift velocity (a) and temperature distribution (b) when 119878119905 = 1Λ = 04119872 = 01 and 120573 = 03 and 119878119905 = 05Λ = 04119872 = 01 and 120573 = 03

x

u(x)

St = 06

St = 02

St = 03

St = 04

St = 05

00 02 04 06 08 10080

085

090

095

100

105

(a)

x

St = 02

St = 04

St = 05

St = 01

St = 03

00 02 04 06 08 1000

02

04

06

08

10

12

120579(x)

(b)

Figure 5 Effect of the stock number on lift velocity (a) and temperature distribution (b) when119872 = 001 120573 = 03 Λ = 01 and 119861119903 = 4 and120573 = 03119872 = 001 and 119861119903 = 100

The terms V2(119909) and Θ2(119909) are too large to be written abovetherefore their expression is mentioned graphically

5 Formulation of Drainage Problem

The geometry and assumptions of the problem are the sameas in previous problem Consider a film of non-Newtonianliquid draining at volume flow rate 119876 down the vertical beltThe belt is stationary and the fluid drains down the belt dueto gravity The gravity in this case is opposite to the previouscase Therefore the stock number is positively mentioned in(14)The coordinate system is selected the same as in previouscase Assume the flow is steady and laminar while externalpressure is neglected Consider fluid shear forces keep thegravity balanced and the film thickness remains constant

Modeled slip boundary conditions for drainage problemare

V = minus120574119879119909119910 at 119909 = 0

119889V119889119909

= 0 at 119909 = 1(44)

Making use of nondimensional variables the slip boundaryconditions in (42) are

V119899 = minusΛ(119889V119899119889119909

minus119872Θ

119889V119899119889119909

+ 2120573(

119889V119899119889119909

)

3

)

at 119909 = 0

119889V119899119889119909

= 0 at 119909 = 1 when 119899 = 0 1 2 3

(45)

8 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 1004

06

08

10

12

14

16

Λ = 00

Λ = 02

Λ = 04

Λ = 06

Λ = 08

Figure 6 Effect of slip parameter on the lift velocity profile when119878119905 = 1119872 = 01 120573 = 03 and 119861119903 = 50

For heat the boundary conditions are identical as given in(36) (38) and (40) regarding both the problems

6 Solution of Drainage Problem

61 The ADM Solution By making use of ADM in (14)and (15) the Adomian polynomials in (24)ndash(27) for bothproblems are the sameThe different velocity components areobtained as follows

Components of ProblemThe boundary conditions of first andsecond components for drainage velocity profile are the sameas given in (37) The boundary conditions for temperaturedistribution are also the same as given in (38) but the solutionof these components is different dependent on the differentvelocity profile of drainage and lift problemsThe terms V2(119909)and Θ2(119909) are too large to be written above therefore theirexpression is mentioned graphically

Inserting slip boundary conditions from (45) and from(18) and (19) in (29)ndash(34) the different components solutionis obtained as

V0 (119909) =1

2

(2119878119905119909 minus 1198781199051199092)

Θ0 (119909 119905) = 119909

V1 (119909) = (minus1

3

119872

2119878119905 minus 120573119878

3

119905)119909

+ (

Λ119878119905

2

+

3120573119878

3

119905

2

)119909

2

+ (

119872

2119878119905

6

minus

Λ119878119905

3

minus 120573119878

3

119905)119909

3

+ (minus

1

24

119872

2119878119905 +

120573119878

3

119905

4

)119909

4

Table 1 Comparison of present work and existing work [23] for liftvelocity profile when119898 = 119878119905 = 05119872 = 0 120573 = 06 Λ = 0 120572 = 1

119909 Present work Existing work Absolute error00 1 1 001 095485367 095485367 002 091553824 091553824 003 088114335 088114335 004 085118976 085118976 0148 times 10

minus14

05 082550781 082550781 006 080413216 080413216 111 times 10

minus29

07 078721265 0787212652 008 077494144 077494144 minus111 times 10

minus29

09 076749627 0767496272 010 0765 0765 minus111 times 10

minus29

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

Λ1198611199031198782

119905+

1

6

1205731198611199031198784

119905)119909

+ (minus

1

2

1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

Λ1198611199031198782

119905+

2

3

1205731198611199031198784

119905)119909

3

+ (minus

1

12

1198611199031198782

119905minus

1

6

Λ1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

4

+ (

1

20

Λ1198611199031198782

119905+

1

5

1205731198611199031198784

119905)119909

5

+ (minus

1

30

1205731198611199031198784

119905)119909

6

(46)

In the existence work [23]119898 is used as gravitational parame-ter instead of stock number 119878119905 in the present work Magneticparameter is defined as 119872 in the existing work and in thepresent work119872 is embedding parameter If we put magneticparameter = 0 slip parameter Λ = 0 in the existing workand embedding parameter119872 = 0 slip parameter Λ = 0 inpresent work then comparison of existing and present workis as shown in Table 1

OHAM solution of the present work has been obtainedand used only for comparison through Tables 1 2 3 4 and 5

7 Results and Discussion

In this study the heat transfer of a third grade fluid onthin film flow has been examined The geometry of lift anddrainage problems has been shown in Figures 1(a) and 1(b)respectively The velocity and heat transfer analysis for bothlift and drainage problems have been investigated under theinfluence of stock number 119878119905 slip parameterΛ the Brinkmannumber 119861119903 non-Newtonian parameter 120573 and viscosityparameter 119872 The effects of these physical parameters havebeen discussed in Figures 2ndash13 The OHAM solution of thepresent work has been shown graphically and numerically

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

x

u(x)

00 02 04 06 08 1004

06

08

10

12

14

16

Λ = 00

Λ = 02

Λ = 04

Λ = 06

Λ = 08

Figure 6 Effect of slip parameter on the lift velocity profile when119878119905 = 1119872 = 01 120573 = 03 and 119861119903 = 50

For heat the boundary conditions are identical as given in(36) (38) and (40) regarding both the problems

6 Solution of Drainage Problem

61 The ADM Solution By making use of ADM in (14)and (15) the Adomian polynomials in (24)ndash(27) for bothproblems are the sameThe different velocity components areobtained as follows

Components of ProblemThe boundary conditions of first andsecond components for drainage velocity profile are the sameas given in (37) The boundary conditions for temperaturedistribution are also the same as given in (38) but the solutionof these components is different dependent on the differentvelocity profile of drainage and lift problemsThe terms V2(119909)and Θ2(119909) are too large to be written above therefore theirexpression is mentioned graphically

Inserting slip boundary conditions from (45) and from(18) and (19) in (29)ndash(34) the different components solutionis obtained as

V0 (119909) =1

2

(2119878119905119909 minus 1198781199051199092)

Θ0 (119909 119905) = 119909

V1 (119909) = (minus1

3

119872

2119878119905 minus 120573119878

3

119905)119909

+ (

Λ119878119905

2

+

3120573119878

3

119905

2

)119909

2

+ (

119872

2119878119905

6

minus

Λ119878119905

3

minus 120573119878

3

119905)119909

3

+ (minus

1

24

119872

2119878119905 +

120573119878

3

119905

4

)119909

4

Table 1 Comparison of present work and existing work [23] for liftvelocity profile when119898 = 119878119905 = 05119872 = 0 120573 = 06 Λ = 0 120572 = 1

119909 Present work Existing work Absolute error00 1 1 001 095485367 095485367 002 091553824 091553824 003 088114335 088114335 004 085118976 085118976 0148 times 10

minus14

05 082550781 082550781 006 080413216 080413216 111 times 10

minus29

07 078721265 0787212652 008 077494144 077494144 minus111 times 10

minus29

09 076749627 0767496272 010 0765 0765 minus111 times 10

minus29

Θ1 (119909) = (1

4

1198611199031198782

119905minus

1

20

Λ1198611199031198782

119905+

1

6

1205731198611199031198784

119905)119909

+ (minus

1

2

1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

2

+ (

1

3

1198611199031198782

119905+

1

6

Λ1198611199031198782

119905+

2

3

1205731198611199031198784

119905)119909

3

+ (minus

1

12

1198611199031198782

119905minus

1

6

Λ1198611199031198782

119905minus

1

2

1205731198611199031198784

119905)119909

4

+ (

1

20

Λ1198611199031198782

119905+

1

5

1205731198611199031198784

119905)119909

5

+ (minus

1

30

1205731198611199031198784

119905)119909

6

(46)

In the existence work [23]119898 is used as gravitational parame-ter instead of stock number 119878119905 in the present work Magneticparameter is defined as 119872 in the existing work and in thepresent work119872 is embedding parameter If we put magneticparameter = 0 slip parameter Λ = 0 in the existing workand embedding parameter119872 = 0 slip parameter Λ = 0 inpresent work then comparison of existing and present workis as shown in Table 1

OHAM solution of the present work has been obtainedand used only for comparison through Tables 1 2 3 4 and 5

7 Results and Discussion

In this study the heat transfer of a third grade fluid onthin film flow has been examined The geometry of lift anddrainage problems has been shown in Figures 1(a) and 1(b)respectively The velocity and heat transfer analysis for bothlift and drainage problems have been investigated under theinfluence of stock number 119878119905 slip parameterΛ the Brinkmannumber 119861119903 non-Newtonian parameter 120573 and viscosityparameter 119872 The effects of these physical parameters havebeen discussed in Figures 2ndash13 The OHAM solution of thepresent work has been shown graphically and numerically

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

00 02 04 06 08 10

00

02

04

06

08

10

x

u(x)

M = 00

M = 02

M = 04

M = 06

M = 08

(a)

00 02 04 06 08 10

00

05

10

minus05

x

M = 00

M = 02

M = 05

M = 08M = 1

120579(x)

(b)

Figure 7 The effect of viscosity parameter on lift velocity (a) and temperature distribution (b) when 119861119903 = 40 Λ = 01 120573 = 03 and 119878119905 = 05and 119861119903 = 10 120573 = 06 and 119878119905 = 1

00 02 04 06 08 10100

105

110

115

120

125

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 8 Effect of 120573 on lift velocity (a) and temperature distribution (b) when119872 = 01 119878119905 = 04 Λ = 05 and 119861119903 = 50

only for comparison The comparison of analytical methodsADMandOHAMof the presentwork for lift velocity andheattransfer analysis is shown in Figures 2(a) and 2(b) Figures3(a) and 3(b) show the comparison of ADM and OHAM fordrainage velocity and temperature distribution Effect of theBrinkman number 119861119903 for lift velocity profile has been shownin Figure 4(a) Behavior of the lift velocity field V can be seenwhen the velocity of fluid is maximum at the surface of thebelt while it is minimum at the surface of the fluid It hasbeen found out that the velocity of the fluid decreases for largevalue of119861119903 as compared to small values of119861119903The observationof Brinkman number 119861119903 associated with heat in lift problemhas been countered in Figure 4(b) Since 119861119903 is related withthe viscous dissipation term in energy equation higher valuesof the 119861119903 should essentially lead to an increase in amount of

heat being generated by the shear forces in the fluid leadingto increase in the fluid temperature

In Figure 5(a) we observed that the velocity decreaseswith an increase in the stock number 119878119905 due to friction forcethe effect of stock number seems to be smaller near the beltIt can be seen that there is a point in the domain wherethe velocity of the fluid becomes approximately identicalfor different values of stock number 119878119905 The reason is thatthe friction of the belt becomes negligible at this pointOn increasing stock number 119878119905 after this point in liftingflow the velocity decreases due to negligible friction Theeffect of stock number 119878119905 on temperature distribution hasbeen illustrated in Figure 5(b) It has been observed thatthe temperature Θ increases steadily by increasing 119878119905 InFigure 6 it has been noticed that the speed of the fluid near

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

00 02 04 06 08 10

00

02

04

06

x

u(x)

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

Br = 4

Br = 20

Br = 40

Br = 60

Br = 80

120579(x)

(b)

Figure 9 Drainage velocity (a) and heat distribution (b) for Brickman number when 119878119905 = 1 Λ = 01119872 = 02 and 120573 = 03 and119872 = 03119878119905 = 05 120573 = 05 and Λ = 01

00 02 04 06 08 10

000

005

010

015

020

minus005

x

u(x)

St = 01

St = 02

St = 03

St = 04

St = 05

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

St = 01

St = 02

St = 03

St = 04

St = 05

120579(x)

(b)

Figure 10 Effect of stock number on drainage velocity profile (a) and temperature distribution (b) when119872 = 02 120573 = 03 Λ = 01 and119861119903 = 40

the belt is greater than the speed at the surface When weincrease the slip parameter then due to decrease in frictionforce the velocity of the fluid decreases gradually towardsthe surface of the belt The effect of viscosity parameter 119872on lift velocity V has been shown in Figure 7(a) The speedof flow decreases with increasing119872 It can also be seen thatthe thickness of thin film increases in case of dilatant fluidsand decreases in pseudoplastic fluids In Figure 7(b) increasein viscosity parameter 119872 results in increasing temperaturedistribution steadily Figure 8(a) indicates the influence ofnon-Newtonian effect 120573 on lift velocity profile

For small values of 120573 the velocity profile differs littlefrom theNewtonian oneHowever when120573 is increased theseprofiles become more flatten showing the shear thinning

effect Increase in 120573 results in decreasing temperature distri-bution as shown in Figure 8(b) The drain velocity increaseswith increase in Brinkman number as shown in Figure 9(a)Figure 9(b) specifies the effect of 119861119903 for drain temperaturedistribution The temperature distribution increases as the119861119903 increases and becomes more trampled for higher valueof 119861119903 In Figure 10(a) the effect of stock number 119878119905 hasbeen shown for drainage flow It can be seen that there is apoint near the belt where the velocity of the fluid becomesapproximately the same for different values of stock number119878119905 because friction of the belt becomes negligible at thispoint On increasing the stock number 119878119905 after this pointthe curvature of velocity increases due to negligible frictionThe effect of stock number 119878119905 on temperature distribution

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

00 02 04 06 08 10

00

01

02

minus01

minus02

x

u(x)

Λ = 00

Λ = 01

Λ = 02

Λ = 03

Λ = 04

Figure 11 Drain velocity profile for different values of slip parameterwhen119872 = 01 119878119905 = 05 120573 = 03 and 119861119903 = 40

Table 2 Comparison of OHAM and ADM for lift velocity profilewhen 119878119905 = 05119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = 02126191198622 = minus0243971 1198623 = 0016110 1198624 = minus00168232

119909 OHAM ADM Absolute error00 100586 10064 0543 times 10

minus3

01 0962821 0962544 0277 times 10

minus3

02 0923093 0922658 0435 times 10

minus3

03 0887013 0886739 0274 times 10

minus3

04 0854908 0854893 0148 times 10

minus4

05 0827086 0827304 0217 times 10

minus3

06 0803834 0804202 0367 times 10

minus3

07 0785409 0785838 0428 times 10

minus3

08 077204 0772463 0423 times 10

minus3

09 0763914 0764303 0389 times 10

minus3

10 076118 0761549 0369 times 10

minus3

Table 3 Comparison of OHAM and ADM for lift heat distributionwhen 119878119905 = 01119872 = 01 120573 = 03 Λ = 001 119861119903 = 4 1198621 = minus0702421198622 = minus003461 1198623 = minus449647 1198624 = minus1642906

119909 OHAM ADM Absolute error00 0 0 0840 times 10

minus3

01 009998 010083 0135 times 10

minus2

02 019998 020133 0159 times 10

minus2

03 029998 030157 0163 times 10

minus2

04 039998 040162 0152 times 10

minus2

05 049998 050151 0130 times 10

minus2

06 059998 060129 0101 times 10

minus2

07 069998 070101 0693 times 10

minus3

08 079999 080068 0423 times 10

minus3

09 0863914 090034 0349 times 10

minus3

10 1000000 1000000 0294 times 10

minus19

Table 4 Comparison of OHAM and ADM for drainage velocityprofile when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus000002 1198622 = 000007 1198623 = 172844761 1198624 = 15216776

119909 OHAM ADM Absolute error00 minus001012 minus001012 0001 minus000062 minus000067 0461 times 10

minus4

02 000789 000781 0702 times 10

minus4

03 001538 001531 0768 times 10

minus4

04 002188 002181 0708 times 10

minus4

05 002738 002733 0561 times 10

minus4

06 003188 003185 0371 times 10

minus4

07 003538 003536 0168 times 10

minus4

08 003788 0037881 0920 times 10

minus8

09 003938 003939 0134 times 10

minus3

10 003988 003989 0182 times 10

minus3

Table 5 Comparison of OHAM and ADM for drain heat distri-bution when 119878119905 = 01 119872 = 001 120573 = 03 Λ = 01 119861119903 = 041198621 = minus007339 1198622 = 0049829 1198623 = minus112367 1198624 = minus002304

119909 OHAM ADM Absolute error00 0 0 001 0100082 0100081 0818 times 10

minus6

02 0200131 020013 0115 times 10

minus5

03 0300154 0300153 0118 times 10

minus5

04 0400158 0400157 0107 times 10

minus5

05 0500147 0500146 0887 times 10

minus6

06 0600126 0600125 0689 times 10

minus6

07 0700098 0700097 0501 times 10

minus6

08 0800067 0800066 0326 times 10

minus6

09 0900034 0900033 0161 times 10

minus6

10 1000000 1000000 0261 times 10

minus18

is illustrated in Figure 10(b) In Figure 11 it is noticed thatthe speed of the fluid near the belt is smaller than thespeed at the surface When we increase the slip parameterthe velocity of the fluid increases because the friction goeson decreasing It has been observed that the temperatureΘ increases monotonically on increasing 119878119905 Figure 12(a)illustrates the effect of variable viscosity parameter119872 on thedrain velocity profile For higher values of viscosity parameter119872 the velocity of fluid increases gradually towards the surfaceof the fluid Increase in viscosity parameter119872 causes increasein temperature field gradually as shown in Figure 12(b) Theeffect of non-Newtonian parameter 120573 totally depends on theshear thining effect as shown in Figure 13(a) Increase in 120573results in increasing speed of the drain velocity profile butincrease in 120573 decreases temperature profile dependent onthe shear thinning and thickening properties of fluid film asshown in Figure 13(b)

8 Conclusion

We have investigated the temperature dependent viscosityof a third grade thin film fluid flow subject to lifting and

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

00 02 04 06 08 100

2

4

6

8

10

x

u(x)

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

(a)

00 02 04 06 08 1000

02

04

06

08

10

x

120573 = 01

120573 = 03

120573 = 05

120573 = 07

120573 = 09

120579(x)

(b)

Figure 12The effect of non-Newtonian 120573 on velocity profile (a) and temperature distribution (b) when119872 = 02 119878119905 = 1Λ = 03 and 119861119903 = 50and119872 = 03 119878119905 = 05 Λ = 01 and 119861119903 = 50

00 02 04 06 08 10

000

005

010

015

minus005

minus010

x

u(x)

M = 01

M = 02

M = 03

M = 04M = 05

(a)

00 02 04 06 08 10

00

02

04

06

08

10

x

M = 00

M = 01

M = 03

M = 05

M = 07

120579(x)

(b)

Figure 13 Effect of the viscosity parameter119872 on the drainage velocity (a) and temprature (b) when 119878119905 = 05 120573 = 03 Λ = 02 and 119861119903 = 40and 119878119905 = 1 120573 = 05 and 119861119903 = 10

drainage of fluid Problems have been solved analyticallyby using Adomian Decomposition Method (ADM) Thecomparison of ADM and Optimal Homotopy AsymptoticMethod (OHAM) has been derived graphically and numer-ically Expression for velocity fields and temperature dis-tribution has been presented and sketched These solutionsare valid not only for small but also for large values ofemerging parameters The observation of Brinkman number119861119903 associated with heat is countered in both lift and drainageproblems Since119861119903 is relatedwith the viscous dissipation termin heat equation higher values of the 119861119903 should essentiallylead to an increase in the amount of heat being generated bythe shear forces in the fluid leading to increase in the fluidtemperature It has been observed that the temperature Θ

increases monotonically on increasing 119878119905 The speed of flowdecreases with increasing viscosity parameter119872 Increase in119872 increases the temperature distribution It can also be seenthat the thickness of thin film increases in case of dilatantfluids and decreases in pseudoplastic fluids According to thebest of our knowledge no research has been carried out yetregarding our study This is our first attempt to handle thisproblemwith variable viscosity and slip boundary conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 13

Acknowledgment

The authors would like to thank Universiti TeknologiPETRONAS for the financial support

References

[1] D Goussis and R E Kelly ldquoEffects of viscosity variation on thestability of film flow down heated or cooled inclined surfaceslong-wavelength analysisrdquo The Physics of Fluids vol 28 no 11pp 3207ndash3214 1985

[2] D A Goussis and R E Kelly ldquoEffects of viscosity variation onthe stability of a liquid film flow down heated or cooled inclinedsurfaces finite wavelength analysisrdquo Physics of Fluids vol 30pp 974ndash982 1987

[3] C-C Hwang and C-I Weng ldquoNon-linear stability analysis offilm flow down a heated or cooled inclined plane with viscosityvariationrdquo International Journal of Heat and Mass Transfer vol31 pp 1775ndash1784 1988

[4] B Reisfeld and S G Bankoff ldquoNonlinear stability of a heatedthin liquid film with variable viscosityrdquo Physics of Fluids A vol2 pp 2066ndash2067 1990

[5] M C Wu and C C Hwang ldquoNonlinear theory of film rupturewith viscosity variationrdquo International Communications in Heatand Mass Transfer vol 18 pp 705ndash713 1991

[6] A Sansom J R King and D S Riley ldquoDegenerate-diffusionmodels for the spreading of thin non-isothermal gravity cur-rentsrdquo Journal of EngineeringMathematics vol 48 no 1 pp 43ndash68 2004

[7] A R A Khaled and K Vafai ldquoHydrodynomic Squeezed flowand heat transfer over a sensor surfacerdquo International Journal ofEngineering Science vol 42 pp 509ndash519 2004

[8] S Nadeem and M Awais ldquoThin film flow of an unsteadyshrinking sheet through porous medium with variable viscos-ityrdquo Physics Letters A vol 372 pp 4965ndash4972 2008

[9] A J Chamkha T Grosan and I Pop ldquoFully developed free con-vection of amicro polar fluid in a vertical channelrdquo InternationalCommunications in Heat and Mass Transfer vol 29 pp 1119ndash1127 2002

[10] H Saleh and I Hashim ldquoSimulation of micropolar fluid flowin a vertical channel using homotopy-perturbation methodrdquo inProceedings of the Applied Computing Conference pp 1ndash4 Istan-bul Turkey 2008

[11] Y M Aiyesimi and G T Okedao ldquoMHD flow of a third gradefluid with heat transfer down an inclined planerdquoMathematicalTheory and Modeling vol 2 no 9 pp 108ndash109 2012

[12] K Nrgis and T Mahmood ldquoThe influence of slip condition onthe thin film flow of a third order fluidrdquo International Journal ofNonlinear Science vol 1 no 13 pp 105ndash116 2012

[13] M I Anwar I Khan A HussnanM Z Salleh and S SharidanldquoStagnation-point flow of a nanofluid over a nonlinear stretch-ing sheetrdquoWorldApplied Sciences Journal vol 23 no 8 pp 998ndash1006 2013

[14] M Qasim I Khan and S Sharidan ldquoHeat transfer and massdiffusion in nanofluids with convective boundary conditionsrdquoMathematical Problems in Engineering vol 2013 Article ID254973 7 pages 2013

[15] A M Siddiqui R Mahmood and Q K Ghori ldquoHomotopyperturbation method for thin film flow of a third grade fluiddown an inclined planerdquo Chaos Solitons amp Fractals vol 35 no1 pp 140ndash147 2008

[16] S J Liao Beyond Perturbation Introduction to Homotopy Anal-ysis Method Chapman amp Hall Boca Raton Fla USA 2003

[17] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[18] VMarinca andNHerisanu ldquoApplication of optimalHomotopyasymptotic method for solving non-linear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 pp 710ndash715 2008

[19] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[20] A A Juneidi D D Gangi and M Babaelahi ldquoMicropolar flowin a porous channel with high mass transferrdquo InternationalCommunications in Heat and Mass Transfer vol 36 pp 1082ndash1088 2009

[21] A M Siddiqui M Ahmed and Q K Ghori ldquoThin film flowof non-Newtonian fluids on a moving beltrdquo Chaos Solitons ampFractals vol 33 no 3 pp 1006ndash1016 2007

[22] AM Siddiqui RMahmood andQ K Ghori ldquoHomotopy per-turbationmethod for thin filmflow of a fourth grade fluid downa vertical cylinderrdquoPhysics Letters A vol 352 pp 404ndash410 2006

[23] T Gul R A Shah S Islam and M Arif ldquoMHD thin film flowsof a third grade fluid on a vertical belt with slip boundary con-ditionsrdquo Journal of Applied Mathematics vol 2013 Article ID707286 14 pages 2013

[24] T Gul S Islam R A Shah I Khan and S Shafie ldquoThin filmflow in MHD third grade fluid on a vertical belt with temper-ature dependent viscosityrdquo PLoS ONE vol 9 no 6 Article IDe97552 2014

[25] A Costa and G Macedonio ldquoViscous heating in fluids withtemperature-dependent viscosity implications for magmaflowsrdquoNonlinear Processes in Geophysics vol 10 no 6 pp 545ndash555 2003

[26] R Ellahi and A Riaz ldquoAnalytical solutions for MHD flow ina third-grade fluid with variable viscosityrdquo Mathematical andComputer Modelling vol 52 no 9-10 pp 1783ndash1793 2010

[27] Y Aksoy and M Pakdemirly ldquoApproximate analytical solutionfor flow of a third grade fluid through a parallel-plate channelfilled with a porous mediumrdquo Transport in Porous Media vol83 no 2 pp 375ndash395 2010

[28] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht The Netherlands 1994

[29] G Adomian ldquoA review of the decompositionmethod and somerecent results for nonlinear equationsrdquoMathematical and Com-puter Modelling vol 13 no 7 pp 17ndash43 1990

[30] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 pp 652ndash663 2005

[31] A M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Emden-Fowler equationrdquoAppliedMathematicsand Computation vol 161 pp 543ndash560 2005

[32] M K Alam M T Rahim T Haroon S Islam and A M Sid-diqui ldquoSolution of steady thin film flow of Johnson-Segalmanfluid on a vertical moving belt for lifting and drainage problemsusing Adomian decomposition methodrdquo Applied Mathematicsand Computation vol 218 no 21 pp 10413ndash10428 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of