research article gravity modulation effects of

Research ArticleGravity Modulation Effects of Hydromagnetic Elastico-ViscousFluid Flow past a Porous Plate in Slip Flow Regime

Debasish Dey

Department of Mathematics Dibrugarh University Dibrugarh Assam 786 004 India

Correspondence should be addressed to Debasish Dey debasish41092gmailcom

Received 17 February 2014 Accepted 11 March 2014 Published 21 May 2014

Academic Editors I Hashim and Y Liu

Copyright copy 2014 Debasish DeyThis 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

The two-dimensional hydromagnetic free convective flow of elastico-viscous fluid (Walters liquid Model B1015840) with simultaneousheat and mass transfer past an infinite vertical porous plate under the influence of gravity modulation effects has been analysedGeneralized Navierrsquos boundary condition has been used to study the characteristics of slip flow regime Fluctuating characteristicsof temperature and concentration are considered in the neighbourhood of the surface having periodic suction The governingequations of fluidmotion are solved analytically by using perturbation technique Various fluid flow characteristics (velocity profileviscous drag etc) are analyzed graphically for various values of flowparameters involved in the solution A special emphasis is givenon the gravity modulation effects on both Newtonian and non-Newtonian fluids

1 Introduction

The analysis of viscoelastic fluid flow is one of the importantfields of fluid dynamics The complex stress-strain rela-tionships of viscoelastic fluid flow mechanisms are usedin geophysics chemical engineering (absorption filtration)petroleum engineering hydrology soil-physics biophysicsand paper and pulp technologyThe viscosity of the viscoelas-tic fluid signifies the physics of the energy dissipated duringthe flow and its elasticity represents the energy stored duringthe flow AsWalters liquid (Model B1015840) contains both viscosityand elasticity it is different from Newtonian fluid because theNewtonian fluid does not have the concept of elasticity as itdiscusses only the concept of energy dissipation

The phenomenon of transient free convection flow froma vertical plate has been analysed by Siegel [1] Gebhart[2] has studied the fluid motion in the presence of naturalconvection from vertical elements Chung and Anderson [3]have investigated the nature of unsteady fluid flow includingthe effects of natural convection Schetz and Eichhorn [4]have investigated the above unsteady problem in the vicinityof a doubly infinite vertical plate Goldstein and Briggs [5]have studied the problem of free convection about verticalplates and circular cylinders Two- and three-dimensional

oscillatory convection in gravitationally modulated fluidlayers have been investigated by Clever et al [6 7] A study ofthermal convection in an enclosure induced simultaneouslyby gravity and vibration has been done by Fu and Shieh [8]Convection phenomenon of material processing in space hasbeen described by Ostrach [9] Li [10] has investigated theeffect of magnetic fields on low frequency oscillating naturalconvection Deka and Soundalgekar [11] have analysed theproblem of free convection flow influenced by gravity mod-ulation by using Laplace transform technique Rajvanshi andSaini [12] have studied the free convection MHD flow pasta moving vertical porous surface with gravity modulationat constant heat flux The influence of combined heat andmass transfer and gravity modulation on unsteady flow pasta porous vertical plate in slip flow regime has been examinedby Jain and Rajvanshi [13]

In this study an analysis is carried out to study theeffects of gravity modulation on free convection unsteadyflow of a viscoelastic fluid past a vertical permeable platewith slip flow regime under the action of transverse magneticfield The velocity field and magnitude of shearing stress atthe plate are obtained and illustrated graphically to observethe viscoelastic effects in combination with other flowparameters

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2014 Article ID 492906 7 pageshttpdxdoiorg1011552014492906

2 ISRN Applied Mathematics

The constitutive equation for Walters liquid (Model B1015840) is

120590119894119896= minus119901119892

119894119896+ 1205901015840

119894119896 1205901015840119894119896

= 21205780119890119894119896minus 211989601198901015840119894119896 (1)

where 120590119894119896 is the stress tensor 119901 is isotropic pressure 119892119894119896is the

metric tensor of a fixed coordinate system 119909119894 V119894is the velocity

vector and the contravariant form of 1198901015840119894119896 is given by

1198901015840119894119896

=

120597119890119894119896

120597119905

+ V119898119890119894119896119898minus V119896119898119890119894119898minus V119894119898119890119898119896 (2)

It is the convected derivative of the deformation rate tensor119890119894119896 defined by

2119890119894119896= V119894119896+ V119896119894 (3)

Here 1205780is the limiting viscosity at the small rate of shear

which is given by

1205780= int

infin

0

119873(120591) 119889120591 1198960= int

infin

0

120591119873 (120591) 119889120591 (4)

119873(120591) being the relaxation spectrumThis idealizedmodel is avalid approximation of Walters liquid (Model B1015840) taking veryshort memories into account so that terms involving

int

infin

0

119905119899119873(120591) 119889120591 119899 ge 2 (5)

have been neglectedWalter [14] reported that the mixture of polymethyl

methacrylate and pyridine at 25∘C containing 305 gm ofpolymer per litre and having density 098 gmmL fits verynearly to this model

2 Mathematical Formulation

An unsteady two-dimensional free convective flow of anelectrically conducting elastico-viscous fluid past a verticalporous plate has been analysed in the presence of gravitymodulation and slip flow regime119860magnetic field of uniformstrength 119861

0is applied in the direction normal to the plate

Induced magnetic field is neglected by assuming very smallvalues ofmagnetic Reynolds number (Crammer andPai [15])The electrical conductivity of the fluid is also assumed to beof smaller order of magnitude Let 1199091015840-axis be taken along thevertical plate and 119910

1015840-axis is taken normal to the plate Let119879119908and 119862

119908be respectively the temperature and the molar

species concentration of the fluid at the plate and let 119879infin

and 119862infin

be respectively the equilibrium temperature andequilibrium molar species concentration of the fluid Thegeometry of the problem is shown by Figure 1

g

y998400

u998400

998400

x998400

Tw and Cw

998400 = minusV0(1 + 120598Aei120596998400 t998400 )

Figure 1 Geometry of the problem

The governing equations of the fluid motion are asfollows

120597V1015840

1205971199101015840= 0 997904rArr V1015840 = minus119881

0(1 + 120598119860119890

11989412059610158401199051015840

)

1205971199061015840

1205971199051015840+ V1015840

1205971199061015840

1205971199101015840= 119892120573 (119879

1015840minus 119879infin) + 119892120573 (119862

1015840minus 119862infin) + ]

12059721199061015840

12059711991010158402

minus

1198960

120588

(

12059731199061015840

12059711991010158402

1205971199051015840

+ V101584012059731199061015840

12059711991010158403) minus

1205901198612

01199061015840

120588

1205971198791015840

1205971199051015840+ V1015840

1205971198791015840

1205971199101015840=

119870

120588119862119901

12059721198791015840

12059711991010158402

1205971198621015840

1205971199051015840+ V1015840

1205971198621015840

1205971199101015840= 119863

12059721198621015840

12059711991010158402

(6)

In fact nearly two hundred years ago Navier [16] pro-posed amore general boundary conditionwhich includes thepossibility of fluid slipNavierrsquos proposed boundary conditionassumes that the velocity at a solid surface is proportional tothe shear rate at the surface

The boundary conditions of the problem are

1199061015840= ℎ10158401205901015840

119909119910 119879

1015840= 119879119908+ 120598 (119879

119908minus 119879infin) 11989011989412059610158401199051015840

1198621015840= 119862119908+ 120598 (119862

119908minus 119862infin) 11989011989412059610158401199051015840

at 1199101015840 = 0

1199061015840997888rarr 0 119879

1015840997888rarr 119879infin 119862

1015840997888rarr 119862

infin

as 1199101015840 997888rarr infin

(7)

where 1205901015840119909119910

= 1205780(12059711990610158401205971199101015840) minus 1198960((1205972119906101584012059711991010158401205971199051015840) + V1015840(1205973119906101584012059711991010158403))

3 Method of Solution

The gravitational acceleration is considered as

119892 = 1198920minus 119894119892111989011989412059610158401199051015840

1198921= 120598119886119892

0 (8)

ISRN Applied Mathematics 3

Let us introduce the following nondimensional quantities

119910 =

11991010158401198810

] 119905 =

11990510158401198810

2

4] 120596 =

4]1205961015840

1198810

2

119906 =

1199061015840

1198810

120579 =

1198791015840minus 119879infin

119879119908minus 119879infin

119862 =

1198621015840minus 119862infin

119862119908minus 119862infin

Gr =1198920120573] (119879119908minus 119879infin)

1198810

3 Gr =

11989201205730] (119862119908minus 119862infin)

1198810

3

119872 =

1205901198612

0]

1205881198810

2 119896 =

11989601198810

2

120588]2

Pr =1205780119862119875

119870

Sc = ]119863

ℎ = ℎ10158401205881198812

0

(9)

where 119910 is the displacement variable 119905 is the time 120596 isthe frequency of oscillation 119906 is the dimension velocity 120579is the dimensionless temperature 119862 is the dimensionlessconcentration Gr is Grashof number for heat transfer Gm isGrashof number for mass transfer119872 is magnetic parameter119896 is viscoelastic parameter Pr is the Prandtl number Sc is theSchmidt number 119886 is gravity modulation parameter and ℎ isthe slip parameter

Then the nondimensional forms of the governing equa-tions of motions are as follows

1

4

120597119906

120597119905

minus (1 + 120598119860119890119894120596119905)

120597119906

120597119910

= (Gr120579 + Gm119862) (1 minus 119894120598120572119890119894120596119905) +

1205972119906

1205971199102minus119872119906

minus 119896[

1

4

1205973119906

1205971199102120597119905

minus (1 + 120598119860119890119894120596119905)

1205973119906

1205971199103]

1

4

120597120579

120597119905

minus (1 + 120598119860119890119894120596119905)

120597120579

120597119910

=

1

Pr1205972120579

1205971199102

1

4

120597119862

120597119905

minus (1 + 120598119860119890119894120596119905)

120597119862

120597119910

=

1

Pr1205972119862

1205971199102

(10)

Boundary conditions in the dimensionless form are given asfollows

119906 = ℎ [

120597119906

120597119910

minus 119896

1

4

1205972119906

120597119910120597119905

minus (1 + 120598119860119890119894120596119905)

1205972119906

1205971199102]

120579 = 1 + 120598119890119894120596119905 119862 = 1 + 120598119890

119894120596119905

at 119910 = 0

119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0

as 119910 997888rarr infin

(11)

Assuming small amplitude of oscillations (120598 ≪ 1) in theneighbourhood of the plate the velocity temperature andconcentration are considered as

119906 = 1199060+ 1205981198901198941205961199051199061+ 119900 (120598

2)

120579 = 1205790+ 1205981198901198941205961199051205791+ 119900 (120598

2)

119862 = 1198620+ 1205981198901198941205961199051198621+ 119900 (120598

2)

(12)

Using (12) in (10) and equating the like powers of 120598 andneglecting the higher powers of 120598 we get

119896119906101584010158401015840

0+ 11990610158401015840

0+ 1199061015840

0minus119872119906

0= minus (Gr120579

0+ Gm119862

0)

119896119906101584010158401015840

1+ 11990610158401015840

1(1 minus 119896

119894120596

4

) + 1199061015840

1minus (119872 +

119894120596

4

) 1199061

= Gr (1205721205790minus 1205791) + Gm (120572119862

0minus 1198621)

minus 1198601199061015840

0minus 119896119860119906

101584010158401015840

0

(13)

1

Pr12057910158401015840

0+ 1205791015840

0= 0

1

Pr12057910158401015840

1+ 1205791015840

1minus

119894120596

4

1205791= minus1205791015840

0

1

Sc11986210158401015840

0+ 1198621015840

0= 0

1

Sc11986210158401015840

1+ 1198621015840

1minus

119894120596

4

1198621= minus1198621015840

0

(14)

Relevant boundary conditions for solving the above equa-tions are as follows

1199060= ℎ [119906

1015840

0+ 11989611990610158401015840

0]

1199061= ℎ [119906

1015840

1minus 119896

119894120596

4

1199061015840

1minus 11986011990610158401015840

0minus 11990610158401015840

1]

1205790= 1205791= 1198620= 1198621= 1

at 119910 = 0

1199060= 1199061= 1205790= 1205791= 1198620= 1198621= 0


(15)

Equations (14) are solved by using the boundary conditions(15) and their solutions are given by

1205790= 119890minusPr119910

1205791= 11986231198901205721119910+ 1198624119890minus1205722119910+

1198944Pr120596

119890minusPr119910

1198620= 119890minusSc119910

1198621= 11986251198901205723119910+ 1198626119890minus1205724119910+

1198944Sc120596

119890minusSc119910

(16)

The presence of elasticity in the governing fluid motionconstitutes a third-order differential equation (13) but for


Newtonian fluid 119896 = 0 then the differential equation redu-ces to of order two And also it is seen that as there areinsufficient boundary conditions for solving (13) we usemultiparameter perturbation technique Since 119896 which is ameasure (dimensionless) of relaxation time is very small fora viscoelastic fluid with small memory thus following RayMahapatra and Gupta [17] and Reza and Gupta [18] we usethe multiparameter perturbation technique and we consider

1199060= 11990600+ 11989611990601+ 119900 (119896

2)

1199061= 11990610+ 11989611990611+ 119900 (119896

2)

(17)

Using the perturbation scheme (17) in (13) and equating thelike powers of 119896 and neglecting the higher orders power of 119896we get

11990610158401015840

00+ 1199061015840

00minus119872119906

00= minus (Gr120579

0+ Gm119862

0)

11990610158401015840

00+ 1199061015840

00minus119872119906

00= minus119906101584010158401015840

00

11990610158401015840

10+ 1199061015840

10minus (119872 +

119894120596

4

) 11990610

= 120572 (Gr1205790+ Gm119862

0) minus 119860119906

1015840

0minus (Gr120579

1+ Gm119862

1)

11990610158401015840

11+ 1199061015840

11minus (119872 +

119894120596

4

) 11990611

= minus1198601199061015840

01minus 119860119906101584010158401015840

00minus 119906101584010158401015840

10minus

119894120596

4

11990610158401015840

10

(18)

The modified boundary conditions for solving the aboveequations are

11990600= ℎ1199061015840

00 119906

01= ℎ [119906

1015840

01+ 11990610158401015840

00] 119906

10= ℎ1199061015840

10

11990611= ℎ [119906

1015840

11minus

119894120596

4

1199061015840

10+ 11986011990610158401015840

00+ 11990610158401015840

10]

at 119910 = 0

11990600997888rarr 0 119906

01997888rarr 0 119906

10997888rarr 0 119906

11997888rarr 0


(19)

The solutions of (18) relevant to the above boundary condi-tions are presented as follows

11990600= 1198628119890minus1205726119910+ 1198601119890minus1198601119910+ 1198602119890minus1198602119910

11990601= 11986210119890minus1205726119910+ 1198603119910119890minus1205726119910+ 1198604119890minus1198601119910+ 1198605119890minus1198602119910

11990610= 11986212119890minus1205728119910+ (1198606+ 1198941198607) 119890minusPr119910

+ (1198608+ 1198941198609) 119890minusSc119910

+ 11989411986010119890minus1205726119910

+ (11986011+ 11989411986012) 119890minus1198601119910+ (11986013+ 11989411986014) 119890minus1198602119910

+ (11986015+ 11989411986016) 119890minus1205722119910+ (11986017+ 11989411986018) 119890minus1205724119910

11990610= 11986214119890minus1205728119910+ (11986036+ 11989411986037) 119890minus1205726119910+ 11989411986038119910119890minus1205726119910

+ (11986039+ 11989411986040) 119890minus1198601119910+ (11986041+ 11989411986042) 119890minus1198602119910

+ (11986043+ 11989411986044) 119910119890minus1205728119910+ (11986045+ 11989411986046) 119890minusPr119910

+ (11986047+ 11989411986048) 119890minusSc119910

+ (11986049+ 11989411986050) 119890minus1205722119910

+ (11986051+ 11989411986052) 119890minus1205724119910

(20)

The constants of the solutions are not presented here for thesake of brevity

Knowing the velocity field the shearing stress at the plateis defined as

Sh = [

120597119906

120597119910

minus 119896

1

4

1205972119906

120597119910120597119905

minus (1 + 120598119860119890119894120596119905)

1205972119906

1205971199102]

119910=0

(21)

The effects of gravity modulation on free convectiveflow of an elastico-viscous fluid with simultaneous heat andmass transfer past an infinite vertical porous plate under theinfluence of transverse magnetic field have been analyzedFigures 2 to 5 represent the pattern of velocity profile against119910 for various values of flow parameters involved in thesolution The effect of viscoelasticity is exhibited throughthe nondimensional parameter ldquo119896rdquo The nonzero values of119896 characterize viscoelastic fluid where 119896 = 0 character-izes Newtonian fluid flow mechanisms In this study mainemphasis is given on the effect of viscoelasticity on thegoverning fluid motion and also on the difference in flowpattern of both Newtonian and non-Newtonian fluids in thepresence of various physical properties considered in thisproblem

It is seen from Figure 2 that both Newtonian and non-Newtonian fluid flows accelerate asymptotically in the neigh-bourhood of the plate and then they experience a declinetrend as we move away from the plate The nonzero value ofvelocity profile at 119910 = 0 represents the strength of slip at theplate Also it can be concluded that the growth in viscoelas-ticity slows down the speed of fluid flow Effects of gravitymodulation parameter 119886 on the governing fluid motions areshown in Figure 3 and it is observed that as 119886 increases thespeed of fluid flow increases along with the increasing valuesof 119896 = (0 02 and 04) Application of transverse magneticfield leads to the generalization of Lorentz force and its effectis displayed by the nondimensional parameter 119872 Figure 4shows the impact of magnetic parameter on the fluid flowagainst the displacement variable 119910 As magnetic parameterincreases the strength of Lorentz force rises and as a resultthe flow is retarded It is also noticed that as119872 decreases theeffect of viscoelasticity is seen prominent


0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

k = 0

k = 04k = 02

Figure 2 Pr = 10 Sc = 7 Gm = 5 Gr = 10119872 = 2 119890 = 001 119886 = 1ℎ = 01 120596 = 5 119860 = 1 120596119905 = 120587

0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

a = 1 k = 0a = 1 k = 02a = 1 k = 0

a = 3 k = 04a = 3 k = 02a = 3 k = 0

Figure 3 Pr = 10 Sc = 7 Gm = 5 Gr = 10119872 = 2 119890 = 001 ℎ = 01120596 = 5 119860 = 1 120596119905 = 120587

Grashof number is defined as the ratio of buoyancy forceto viscous force and its positive value identifies the flow pastan externally cooled plate and the negative value characterizesthe flow past an externally heated plate Figure 5 shows thatas Gr increases the resistance of fluid flow diminishes and asa consequence the speed enhances along with the increasingvalues of visco-elastic parameter Again when the flow passesan externally heated plate (Gr = minus10) a back flow is noticedin case of both Newtonian and non-Newtonian fluids

4 Results and Discussions

Knowing the velocity field it is important from a practicalpoint of view to know the effect of viscoelastic parameteron shearing stress or viscous drag Figures 6 to 8 depictthe magnitude of shearing stress at the plate of viscoelasticfluid in comparison with the Newtonian fluid for the variousvalues of flow parameters involved in the solution Thefigures notify that the growth in viscoelasticity subdues themagnitude of viscous drag at the surface Prandtl numberplays a significant role in heat transfer flow problems as it

0 2 4 6 8 10 120

05

1

15

2

25

3

35

u

y

M = 5 M = 3 M = 2 M = 1

Figure 4 Pr = 10 Sc = 7 Gm = 5 Gr = 10 119890 = 001 ℎ = 01 120596 = 5119860 = 1 120596119905 = 120587

minus3 minus2 minus1 0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

Gr = 7 Gr = 10Gr = minus10

Figure 5 Pr = 10 Sc = 7 Gm = 5 119890 = 001 ℎ = 01 120596 = 5 119860 = 1120596119905 = 120587119872 = 2

helps to study the simultaneous effect of momentum andthermal diffusion in fluid flow Effect of Prandtl number onthe magnitude of shearing stress is seen in Figure 6 and it canbe concluded that the magnitude of shearing stress increasesrapidly for Pr lt 8 of both Newtonian and non-Newtonianfluids but for higher values of Pr (gt8) it increases steadily Inmass transfer problems the importance of Schmidt numbercannot be neglected as it studies the combined effect ofmomentum and mass diffusion The same phenomenon asthat in case of growth in Prandtl number is experiencedagainst Schmidt number for various values of viscoelasticparameter (Figure 7) Effect of gravity modulation parameter119886 on the viscous drag is shown in Figure 8 and it is revealedthat as 119886 increases themagnitude of shearing stress increases

The rate of heat transfer and rate of mass transfer are notsignificantly affected by the presence of viscoelasticity of thegoverning fluid flow

5 Conclusions

From the present study we make the following conclusions

(i) Both Newtonian and non-Newtonian fluid flowsaccelerate asymptotically in the neighbourhood of theplate

(ii) Growth in viscoelasticity slows down the speed offluid flow


55 6 65 7 75 8 85 9 9502468

101214161820

k = 0

k = 04k = 02

Magnitude of shearing stress

Pran

dtl n

umbe

r

Figure 6 Sc = 7 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587

6 65 7 75 8 85 9 9502468

10121416

k = 0

k = 04k = 02


Schm

idt n

umbe

r

Figure 7 Pr = 10 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587

(iii) Effect of viscoelasticity is seen prominent during thelower order magnitude of magnetic parameter

(iv) A back flow is experienced in the motion governingfluid flow past an externally heated plate

(v) As gravity modulation parameter 119886 increases themagnitude of shearing stress experienced by Newto-nian as well as non-Newtonian fluid flows increase

Conflict of Interests

The author does not have any conflict of interests regardingthe publication of paper

Acknowledgment

Theauthor acknowledges Professor Rita Choudhury Depart-ment of Mathematics Gauhati University for her encourage-ment throughout this work

75 8 85 9 95 100

1

2

3

4

5

6

k = 0

k = 04k = 02


Gra

vity

mod

ulat

ion

para

met

er

Figure 8 Pr = 10 Gm = 10 Gr = 7119872 = 2 119890 = 001 Sc = 7 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587

References

[1] R Siegel ldquoTransient-free convection from a vertical flat platerdquoTransactions of the American Society of Mechanical Engineersvol 80 pp 347ndash359 1958

[2] B Gebhart ldquoTransient natural convection from vertical ele-mentsrdquo Journal of Heat Transfer vol 83 pp 61ndash70 1961

[3] P M Chung and A D Anderson ldquoUnsteady laminar freeconvectionrdquo Journal of Heat Transfer vol 83 pp 474ndash478 1961

[4] J A Schetz and R Eichhorn ldquoUnsteady natural convection inthe vicinity of a doubly infinite vertical platerdquo Journal of HeatTransfer vol 84 pp 334ndash338 1962

[5] R J Goldstein and D G Briggs ldquoTransient-free convectionabout vertical plates and circular cylindersrdquo Journal of HeatTransfer vol 86 pp 490ndash500 1964

[6] R Clever G Schubert and F H Busse ldquoTwo-dimensionaloscillatory convection in a gravitationally modulated fluidlayerrdquo Journal of Fluid Mechanics vol 253 pp 663ndash680 1993

[7] R Clever G Schubert and F H Busse ldquoThree-dimensionaloscillatory convection in a gravitational modulated fluid layerrdquoPhysics of Fluids A vol 5 no 10 pp 2430ndash2437 1992

[8] W S Fu and W J Shieh ldquoA study of thermal convection inan enclosure induced simultaneously by gravity and vibrationrdquoInternational Journal of Heat and Mass Transfer vol 35 no 7pp 1695ndash1710 1992

[9] S Ostrach ldquoConvection phenomena of importance for materi-als processing in spacerdquo in Proceedings of the COSPAR Sympo-sium on Materials Sciences in Space pp 3ndash33 Philadelphia PaUSA 1976

[10] B Q Li ldquoThe effect of magnetic fields on low frequency oscil-lating natural convectionrdquo International Journal of EngineeringScience vol 34 no 12 pp 1369ndash1383 1996

[11] R K Deka and V M Soundalgekar ldquoGravity modulationeffects on transient-free convection flow past an infinite verticalisothermal platerdquoDefence Science Journal vol 56 no 4 pp 477ndash483 2006

[12] S C Rajvanshi and B S Saini ldquoFree convection MHD flowpast a moving vertical porous surface with gravity modulationat constant heat fluxrdquo International Journal of Theoretical andApplied Sciences vol 2 no 1 pp 29ndash33 2010

[13] S R Jain and S C Rajvanshi ldquoInfluence of gravity modulationviscous heating and concentration on flow past a vertical


plate in slip flow regime with periodic temperature variationsrdquoInternational Journal of Theoretical and Applied Science vol 3no 1 pp 16ndash22 2011

[14] K Walter ldquoNon-Newtonian effects in some elastico-viscousliquids whose behavior at small rates of shear is characterizedby a general linear equation of staterdquo The Quarterly Journal ofMechanics and Applied Mathematics vol 15 pp 63ndash76 1962

[15] K R Crammer and S Pai Magneto-Fluid Dynamics for Engi-neers and Applied Physicist McGraw- Hill New York NY USA1973

[16] C L M H Navier Memoirs de LrsquoAcademie Royale des Sciencesde Lrsquoinstitut de France vol 1 1823

[17] T R Mahapatra and A S Gupta ldquoStagnation-point flow ofa viscoelastic fluid towards a stretching surfacerdquo InternationalJournal of Non-Linear Mechanics vol 39 no 5 pp 811ndash8202004

[18] M Reza and A S Gupta ldquoMomentum and heat transfer in theflow of a viscoelastic fluid past a porous flat plate subject tosuction or blowingrdquoWorld Academy of Science Engineering andTechnology vol 23 pp 960ndash963 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The constitutive equation for Walters liquid (Model B1015840) is

120590119894119896= minus119901119892

119894119896+ 1205901015840

119894119896 1205901015840119894119896

= 21205780119890119894119896minus 211989601198901015840119894119896 (1)

where 120590119894119896 is the stress tensor 119901 is isotropic pressure 119892119894119896is the

metric tensor of a fixed coordinate system 119909119894 V119894is the velocity

vector and the contravariant form of 1198901015840119894119896 is given by

1198901015840119894119896

=

120597119890119894119896

120597119905

+ V119898119890119894119896119898minus V119896119898119890119894119898minus V119894119898119890119898119896 (2)

It is the convected derivative of the deformation rate tensor119890119894119896 defined by

2119890119894119896= V119894119896+ V119896119894 (3)

Here 1205780is the limiting viscosity at the small rate of shear

which is given by

1205780= int

infin

0

119873(120591) 119889120591 1198960= int

infin

0

120591119873 (120591) 119889120591 (4)

119873(120591) being the relaxation spectrumThis idealizedmodel is avalid approximation of Walters liquid (Model B1015840) taking veryshort memories into account so that terms involving

int

infin

0

119905119899119873(120591) 119889120591 119899 ge 2 (5)

have been neglectedWalter [14] reported that the mixture of polymethyl

methacrylate and pyridine at 25∘C containing 305 gm ofpolymer per litre and having density 098 gmmL fits verynearly to this model

2 Mathematical Formulation

An unsteady two-dimensional free convective flow of anelectrically conducting elastico-viscous fluid past a verticalporous plate has been analysed in the presence of gravitymodulation and slip flow regime119860magnetic field of uniformstrength 119861

0is applied in the direction normal to the plate

Induced magnetic field is neglected by assuming very smallvalues ofmagnetic Reynolds number (Crammer andPai [15])The electrical conductivity of the fluid is also assumed to beof smaller order of magnitude Let 1199091015840-axis be taken along thevertical plate and 119910

1015840-axis is taken normal to the plate Let119879119908and 119862

119908be respectively the temperature and the molar

species concentration of the fluid at the plate and let 119879infin

and 119862infin

be respectively the equilibrium temperature andequilibrium molar species concentration of the fluid Thegeometry of the problem is shown by Figure 1

g

y998400

u998400

998400

x998400

Tw and Cw

998400 = minusV0(1 + 120598Aei120596998400 t998400 )

Figure 1 Geometry of the problem

The governing equations of the fluid motion are asfollows

120597V1015840

1205971199101015840= 0 997904rArr V1015840 = minus119881

0(1 + 120598119860119890

11989412059610158401199051015840

)

1205971199061015840

1205971199051015840+ V1015840

1205971199061015840

1205971199101015840= 119892120573 (119879

1015840minus 119879infin) + 119892120573 (119862

1015840minus 119862infin) + ]

12059721199061015840

12059711991010158402

minus

1198960

120588

(

12059731199061015840

12059711991010158402

1205971199051015840

+ V101584012059731199061015840

12059711991010158403) minus

1205901198612

01199061015840

120588

1205971198791015840

1205971199051015840+ V1015840

1205971198791015840

1205971199101015840=

119870

120588119862119901

12059721198791015840

12059711991010158402

1205971198621015840

1205971199051015840+ V1015840

1205971198621015840

1205971199101015840= 119863

12059721198621015840

12059711991010158402

(6)

In fact nearly two hundred years ago Navier [16] pro-posed amore general boundary conditionwhich includes thepossibility of fluid slipNavierrsquos proposed boundary conditionassumes that the velocity at a solid surface is proportional tothe shear rate at the surface

The boundary conditions of the problem are

1199061015840= ℎ10158401205901015840

119909119910 119879

1015840= 119879119908+ 120598 (119879

119908minus 119879infin) 11989011989412059610158401199051015840

1198621015840= 119862119908+ 120598 (119862

119908minus 119862infin) 11989011989412059610158401199051015840

at 1199101015840 = 0

1199061015840997888rarr 0 119879

1015840997888rarr 119879infin 119862

1015840997888rarr 119862

infin

as 1199101015840 997888rarr infin

(7)

where 1205901015840119909119910

= 1205780(12059711990610158401205971199101015840) minus 1198960((1205972119906101584012059711991010158401205971199051015840) + V1015840(1205973119906101584012059711991010158403))

3 Method of Solution

The gravitational acceleration is considered as

119892 = 1198920minus 119894119892111989011989412059610158401199051015840

1198921= 120598119886119892

0 (8)



119910 =

11991010158401198810

] 119905 =

11990510158401198810

2

4] 120596 =

4]1205961015840

1198810

2

119906 =

1199061015840

1198810

120579 =

1198791015840minus 119879infin

119879119908minus 119879infin

119862 =

1198621015840minus 119862infin

119862119908minus 119862infin

Gr =1198920120573] (119879119908minus 119879infin)

1198810

3 Gr =

11989201205730] (119862119908minus 119862infin)

1198810

3

119872 =

1205901198612

0]

1205881198810

2 119896 =

11989601198810

2

120588]2

Pr =1205780119862119875

119870

Sc = ]119863

ℎ = ℎ10158401205881198812

0

(9)



1

4

120597119906

120597119905

minus (1 + 120598119860119890119894120596119905)

120597119906

120597119910

= (Gr120579 + Gm119862) (1 minus 119894120598120572119890119894120596119905) +

1205972119906

1205971199102minus119872119906

minus 119896[

1

4

1205973119906

1205971199102120597119905

minus (1 + 120598119860119890119894120596119905)

1205973119906

1205971199103]

1

4

120597120579

120597119905

minus (1 + 120598119860119890119894120596119905)

120597120579

120597119910

=

1

Pr1205972120579

1205971199102

1

4

120597119862

120597119905

minus (1 + 120598119860119890119894120596119905)

120597119862

120597119910

=

1

Pr1205972119862

1205971199102

(10)


119906 = ℎ [

120597119906

120597119910

minus 119896

1

4

1205972119906

120597119910120597119905

minus (1 + 120598119860119890119894120596119905)

1205972119906

1205971199102]

120579 = 1 + 120598119890119894120596119905 119862 = 1 + 120598119890

119894120596119905

at 119910 = 0

119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0


(11)


119906 = 1199060+ 1205981198901198941205961199051199061+ 119900 (120598

2)

120579 = 1205790+ 1205981198901198941205961199051205791+ 119900 (120598

2)

119862 = 1198620+ 1205981198901198941205961199051198621+ 119900 (120598

2)

(12)


119896119906101584010158401015840

0+ 11990610158401015840

0+ 1199061015840

0minus119872119906

0= minus (Gr120579

0+ Gm119862

0)

119896119906101584010158401015840

1+ 11990610158401015840

1(1 minus 119896

119894120596

4

) + 1199061015840

1minus (119872 +

119894120596

4

) 1199061

= Gr (1205721205790minus 1205791) + Gm (120572119862

0minus 1198621)

minus 1198601199061015840

0minus 119896119860119906

101584010158401015840

0

(13)

1

Pr12057910158401015840

0+ 1205791015840

0= 0

1

Pr12057910158401015840

1+ 1205791015840

1minus

119894120596

4

1205791= minus1205791015840

0

1

Sc11986210158401015840

0+ 1198621015840

0= 0

1

Sc11986210158401015840

1+ 1198621015840

1minus

119894120596

4

1198621= minus1198621015840

0

(14)


1199060= ℎ [119906

1015840

0+ 11989611990610158401015840

0]

1199061= ℎ [119906

1015840

1minus 119896

119894120596

4

1199061015840

1minus 11986011990610158401015840

0minus 11990610158401015840

1]

1205790= 1205791= 1198620= 1198621= 1

at 119910 = 0

1199060= 1199061= 1205790= 1205791= 1198620= 1198621= 0


(15)


1205790= 119890minusPr119910

1205791= 11986231198901205721119910+ 1198624119890minus1205722119910+

1198944Pr120596

119890minusPr119910

1198620= 119890minusSc119910

1198621= 11986251198901205723119910+ 1198626119890minus1205724119910+

1198944Sc120596

119890minusSc119910

(16)




1199060= 11990600+ 11989611990601+ 119900 (119896

2)

1199061= 11990610+ 11989611990611+ 119900 (119896

2)

(17)


11990610158401015840

00+ 1199061015840

00minus119872119906

00= minus (Gr120579

0+ Gm119862

0)

11990610158401015840

00+ 1199061015840

00minus119872119906

00= minus119906101584010158401015840

00

11990610158401015840

10+ 1199061015840

10minus (119872 +

119894120596

4

) 11990610

= 120572 (Gr1205790+ Gm119862

0) minus 119860119906

1015840

0minus (Gr120579

1+ Gm119862

1)

11990610158401015840

11+ 1199061015840

11minus (119872 +

119894120596

4

) 11990611

= minus1198601199061015840

01minus 119860119906101584010158401015840

00minus 119906101584010158401015840

10minus

119894120596

4

11990610158401015840

10

(18)


11990600= ℎ1199061015840

00 119906

01= ℎ [119906

1015840

01+ 11990610158401015840

00] 119906

10= ℎ1199061015840

10

11990611= ℎ [119906

1015840

11minus

119894120596

4

1199061015840

10+ 11986011990610158401015840

00+ 11990610158401015840

10]

at 119910 = 0

11990600997888rarr 0 119906

01997888rarr 0 119906

10997888rarr 0 119906

11997888rarr 0


(19)




11990610= 11986212119890minus1205728119910+ (1198606+ 1198941198607) 119890minusPr119910

+ (1198608+ 1198941198609) 119890minusSc119910

+ 11989411986010119890minus1205726119910

+ (11986011+ 11989411986012) 119890minus1198601119910+ (11986013+ 11989411986014) 119890minus1198602119910

+ (11986015+ 11989411986016) 119890minus1205722119910+ (11986017+ 11989411986018) 119890minus1205724119910


+ (11986039+ 11989411986040) 119890minus1198601119910+ (11986041+ 11989411986042) 119890minus1198602119910

+ (11986043+ 11989411986044) 119910119890minus1205728119910+ (11986045+ 11989411986046) 119890minusPr119910

+ (11986047+ 11989411986048) 119890minusSc119910

+ (11986049+ 11989411986050) 119890minus1205722119910

+ (11986051+ 11989411986052) 119890minus1205724119910

(20)



Sh = [

120597119906

120597119910

minus 119896

1

4

1205972119906

120597119910120597119905

minus (1 + 120598119860119890119894120596119905)

1205972119906

1205971199102]

119910=0

(21)




0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

k = 0

k = 04k = 02


0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

a = 1 k = 0a = 1 k = 02a = 1 k = 0

a = 3 k = 04a = 3 k = 02a = 3 k = 0





0 2 4 6 8 10 120

05

1

15

2

25

3

35

u

y

M = 5 M = 3 M = 2 M = 1



05

1

15

2

25

3

35

u

y


Figure 5 Pr = 10 Sc = 7 Gm = 5 119890 = 001 ℎ = 01 120596 = 5 119860 = 1120596119905 = 120587119872 = 2



5 Conclusions





55 6 65 7 75 8 85 9 9502468

101214161820

k = 0

k = 04k = 02


Pran

dtl n

umbe

r

Figure 6 Sc = 7 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587

6 65 7 75 8 85 9 9502468

10121416

k = 0

k = 04k = 02


Schm

idt n

umbe

r

Figure 7 Pr = 10 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587






Acknowledgment


75 8 85 9 95 100

1

2

3

4

5

6

k = 0

k = 04k = 02


Gra

vity

mod

ulat

ion

para

met

er


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119910 =

11991010158401198810

] 119905 =

11990510158401198810

2

4] 120596 =

4]1205961015840

1198810

2

119906 =

1199061015840

1198810

120579 =

1198791015840minus 119879infin

119879119908minus 119879infin

119862 =

1198621015840minus 119862infin

119862119908minus 119862infin

Gr =1198920120573] (119879119908minus 119879infin)

1198810

3 Gr =

11989201205730] (119862119908minus 119862infin)

1198810

3

119872 =

1205901198612

0]

1205881198810

2 119896 =

11989601198810

2

120588]2

Pr =1205780119862119875

119870

Sc = ]119863

ℎ = ℎ10158401205881198812

0

(9)



1

4

120597119906

120597119905

minus (1 + 120598119860119890119894120596119905)

120597119906

120597119910

= (Gr120579 + Gm119862) (1 minus 119894120598120572119890119894120596119905) +

1205972119906

1205971199102minus119872119906

minus 119896[

1

4

1205973119906

1205971199102120597119905

minus (1 + 120598119860119890119894120596119905)

1205973119906

1205971199103]

1

4

120597120579

120597119905

minus (1 + 120598119860119890119894120596119905)

120597120579

120597119910

=

1

Pr1205972120579

1205971199102

1

4

120597119862

120597119905

minus (1 + 120598119860119890119894120596119905)

120597119862

120597119910

=

1

Pr1205972119862

1205971199102

(10)


119906 = ℎ [

120597119906

120597119910

minus 119896

1

4

1205972119906

120597119910120597119905

minus (1 + 120598119860119890119894120596119905)

1205972119906

1205971199102]

120579 = 1 + 120598119890119894120596119905 119862 = 1 + 120598119890

119894120596119905

at 119910 = 0

119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0


(11)


119906 = 1199060+ 1205981198901198941205961199051199061+ 119900 (120598

2)

120579 = 1205790+ 1205981198901198941205961199051205791+ 119900 (120598

2)

119862 = 1198620+ 1205981198901198941205961199051198621+ 119900 (120598

2)

(12)


119896119906101584010158401015840

0+ 11990610158401015840

0+ 1199061015840

0minus119872119906

0= minus (Gr120579

0+ Gm119862

0)

119896119906101584010158401015840

1+ 11990610158401015840

1(1 minus 119896

119894120596

4

) + 1199061015840

1minus (119872 +

119894120596

4

) 1199061

= Gr (1205721205790minus 1205791) + Gm (120572119862

0minus 1198621)

minus 1198601199061015840

0minus 119896119860119906

101584010158401015840

0

(13)

1

Pr12057910158401015840

0+ 1205791015840

0= 0

1

Pr12057910158401015840

1+ 1205791015840

1minus

119894120596

4

1205791= minus1205791015840

0

1

Sc11986210158401015840

0+ 1198621015840

0= 0

1

Sc11986210158401015840

1+ 1198621015840

1minus

119894120596

4

1198621= minus1198621015840

0

(14)


1199060= ℎ [119906

1015840

0+ 11989611990610158401015840

0]

1199061= ℎ [119906

1015840

1minus 119896

119894120596

4

1199061015840

1minus 11986011990610158401015840

0minus 11990610158401015840

1]

1205790= 1205791= 1198620= 1198621= 1

at 119910 = 0

1199060= 1199061= 1205790= 1205791= 1198620= 1198621= 0


(15)


1205790= 119890minusPr119910

1205791= 11986231198901205721119910+ 1198624119890minus1205722119910+

1198944Pr120596

119890minusPr119910

1198620= 119890minusSc119910

1198621= 11986251198901205723119910+ 1198626119890minus1205724119910+

1198944Sc120596

119890minusSc119910

(16)




1199060= 11990600+ 11989611990601+ 119900 (119896

2)

1199061= 11990610+ 11989611990611+ 119900 (119896

2)

(17)


11990610158401015840

00+ 1199061015840

00minus119872119906

00= minus (Gr120579

0+ Gm119862

0)

11990610158401015840

00+ 1199061015840

00minus119872119906

00= minus119906101584010158401015840

00

11990610158401015840

10+ 1199061015840

10minus (119872 +

119894120596

4

) 11990610

= 120572 (Gr1205790+ Gm119862

0) minus 119860119906

1015840

0minus (Gr120579

1+ Gm119862

1)

11990610158401015840

11+ 1199061015840

11minus (119872 +

119894120596

4

) 11990611

= minus1198601199061015840

01minus 119860119906101584010158401015840

00minus 119906101584010158401015840

10minus

119894120596

4

11990610158401015840

10

(18)


11990600= ℎ1199061015840

00 119906

01= ℎ [119906

1015840

01+ 11990610158401015840

00] 119906

10= ℎ1199061015840

10

11990611= ℎ [119906

1015840

11minus

119894120596

4

1199061015840

10+ 11986011990610158401015840

00+ 11990610158401015840

10]

at 119910 = 0

11990600997888rarr 0 119906

01997888rarr 0 119906

10997888rarr 0 119906

11997888rarr 0


(19)




11990610= 11986212119890minus1205728119910+ (1198606+ 1198941198607) 119890minusPr119910

+ (1198608+ 1198941198609) 119890minusSc119910

+ 11989411986010119890minus1205726119910

+ (11986011+ 11989411986012) 119890minus1198601119910+ (11986013+ 11989411986014) 119890minus1198602119910

+ (11986015+ 11989411986016) 119890minus1205722119910+ (11986017+ 11989411986018) 119890minus1205724119910


+ (11986039+ 11989411986040) 119890minus1198601119910+ (11986041+ 11989411986042) 119890minus1198602119910

+ (11986043+ 11989411986044) 119910119890minus1205728119910+ (11986045+ 11989411986046) 119890minusPr119910

+ (11986047+ 11989411986048) 119890minusSc119910

+ (11986049+ 11989411986050) 119890minus1205722119910

+ (11986051+ 11989411986052) 119890minus1205724119910

(20)



Sh = [

120597119906

120597119910

minus 119896

1

4

1205972119906

120597119910120597119905

minus (1 + 120598119860119890119894120596119905)

1205972119906

1205971199102]

119910=0

(21)




0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

k = 0

k = 04k = 02


0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

a = 1 k = 0a = 1 k = 02a = 1 k = 0

a = 3 k = 04a = 3 k = 02a = 3 k = 0





0 2 4 6 8 10 120

05

1

15

2

25

3

35

u

y

M = 5 M = 3 M = 2 M = 1



05

1

15

2

25

3

35

u

y


Figure 5 Pr = 10 Sc = 7 Gm = 5 119890 = 001 ℎ = 01 120596 = 5 119860 = 1120596119905 = 120587119872 = 2



5 Conclusions





55 6 65 7 75 8 85 9 9502468

101214161820

k = 0

k = 04k = 02


Pran

dtl n

umbe

r

Figure 6 Sc = 7 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587

6 65 7 75 8 85 9 9502468

10121416

k = 0

k = 04k = 02


Schm

idt n

umbe

r

Figure 7 Pr = 10 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587






Acknowledgment


75 8 85 9 95 100

1

2

3

4

5

6

k = 0

k = 04k = 02


Gra

vity

mod

ulat

ion

para

met

er


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199060= 11990600+ 11989611990601+ 119900 (119896

2)

1199061= 11990610+ 11989611990611+ 119900 (119896

2)

(17)


11990610158401015840

00+ 1199061015840

00minus119872119906

00= minus (Gr120579

0+ Gm119862

0)

11990610158401015840

00+ 1199061015840

00minus119872119906

00= minus119906101584010158401015840

00

11990610158401015840

10+ 1199061015840

10minus (119872 +

119894120596

4

) 11990610

= 120572 (Gr1205790+ Gm119862

0) minus 119860119906

1015840

0minus (Gr120579

1+ Gm119862

1)

11990610158401015840

11+ 1199061015840

11minus (119872 +

119894120596

4

) 11990611

= minus1198601199061015840

01minus 119860119906101584010158401015840

00minus 119906101584010158401015840

10minus

119894120596

4

11990610158401015840

10

(18)


11990600= ℎ1199061015840

00 119906

01= ℎ [119906

1015840

01+ 11990610158401015840

00] 119906

10= ℎ1199061015840

10

11990611= ℎ [119906

1015840

11minus

119894120596

4

1199061015840

10+ 11986011990610158401015840

00+ 11990610158401015840

10]

at 119910 = 0

11990600997888rarr 0 119906

01997888rarr 0 119906

10997888rarr 0 119906

11997888rarr 0


(19)




11990610= 11986212119890minus1205728119910+ (1198606+ 1198941198607) 119890minusPr119910

+ (1198608+ 1198941198609) 119890minusSc119910

+ 11989411986010119890minus1205726119910

+ (11986011+ 11989411986012) 119890minus1198601119910+ (11986013+ 11989411986014) 119890minus1198602119910

+ (11986015+ 11989411986016) 119890minus1205722119910+ (11986017+ 11989411986018) 119890minus1205724119910


+ (11986039+ 11989411986040) 119890minus1198601119910+ (11986041+ 11989411986042) 119890minus1198602119910

+ (11986043+ 11989411986044) 119910119890minus1205728119910+ (11986045+ 11989411986046) 119890minusPr119910

+ (11986047+ 11989411986048) 119890minusSc119910

+ (11986049+ 11989411986050) 119890minus1205722119910

+ (11986051+ 11989411986052) 119890minus1205724119910

(20)



Sh = [

120597119906

120597119910

minus 119896

1

4

1205972119906

120597119910120597119905

minus (1 + 120598119860119890119894120596119905)

1205972119906

1205971199102]

119910=0

(21)




0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

k = 0

k = 04k = 02


0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

a = 1 k = 0a = 1 k = 02a = 1 k = 0

a = 3 k = 04a = 3 k = 02a = 3 k = 0





0 2 4 6 8 10 120

05

1

15

2

25

3

35

u

y

M = 5 M = 3 M = 2 M = 1



05

1

15

2

25

3

35

u

y


Figure 5 Pr = 10 Sc = 7 Gm = 5 119890 = 001 ℎ = 01 120596 = 5 119860 = 1120596119905 = 120587119872 = 2



5 Conclusions





55 6 65 7 75 8 85 9 9502468

101214161820

k = 0

k = 04k = 02


Pran

dtl n

umbe

r

Figure 6 Sc = 7 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587

6 65 7 75 8 85 9 9502468

10121416

k = 0

k = 04k = 02


Schm

idt n

umbe

r

Figure 7 Pr = 10 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587






Acknowledgment


75 8 85 9 95 100

1

2

3

4

5

6

k = 0

k = 04k = 02


Gra

vity

mod

ulat

ion

para

met

er


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

k = 0

k = 04k = 02


0 1 2 3 4 5 6 70

05

1

15

2

25

3

35

u

y

a = 1 k = 0a = 1 k = 02a = 1 k = 0

a = 3 k = 04a = 3 k = 02a = 3 k = 0





0 2 4 6 8 10 120

05

1

15

2

25

3

35

u

y

M = 5 M = 3 M = 2 M = 1



05

1

15

2

25

3

35

u

y


Figure 5 Pr = 10 Sc = 7 Gm = 5 119890 = 001 ℎ = 01 120596 = 5 119860 = 1120596119905 = 120587119872 = 2



5 Conclusions





55 6 65 7 75 8 85 9 9502468

101214161820

k = 0

k = 04k = 02


Pran

dtl n

umbe

r

Figure 6 Sc = 7 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587

6 65 7 75 8 85 9 9502468

10121416

k = 0

k = 04k = 02


Schm

idt n

umbe

r

Figure 7 Pr = 10 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587






Acknowledgment


75 8 85 9 95 100

1

2

3

4

5

6

k = 0

k = 04k = 02


Gra

vity

mod

ulat

ion

para

met

er


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










55 6 65 7 75 8 85 9 9502468

101214161820

k = 0

k = 04k = 02


Pran

dtl n

umbe

r

Figure 6 Sc = 7 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587

6 65 7 75 8 85 9 9502468

10121416

k = 0

k = 04k = 02


Schm

idt n

umbe

r

Figure 7 Pr = 10 Gm = 10 Gr = 7119872 = 2 119890 = 001 119886 = 1 ℎ = 03120596 = 5 119860 = 6 120596119905 = 120587






Acknowledgment


75 8 85 9 95 100

1

2

3

4

5

6

k = 0

k = 04k = 02


Gra

vity

mod

ulat

ion

para

met

er


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article gravity modulation effects of

Documents