nasir ali, sami ullah khan* and zaheer abbas hydromagnetic

10
Z. Naturforsch. 2015; 70(7)a: 567–576 *Corresponding author: Sami Ullah Khan, Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan, Tel.: +92 51 9019756, E-mail: [email protected] Nasir Ali: Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan Zaheer Abbas: Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan Nasir Ali, Sami Ullah Khan* and Zaheer Abbas Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid over an Oscillatory Stretching Surface DOI 10.1515/zna-2014-0273 Received September 28, 2014; accepted April 24, 2015; previously published online June 13, 2015 Abstract: The flow and heat transfer of a Jeffrey fluid over an oscillatory stretching sheet is investigated using the boundary-layer approximations. The flow is induced due to infinite elastic sheet that is stretched periodically. The number of independent variables in the governing equa- tions was reduced by using appropriate dimensionless variables. This dimensionless system has been solved by using the homotopy analysis method (HAM) and a finite difference scheme, in which a coordinate transformation was used to transform the semi-infinite physical space to a bounded computational domain. A comparison of both solutions is provided. The effects of involved parameters are illustrated through graphs and discussed in detail. Keywords: Boundary Layer Flow; Finite Difference Scheme; Homotopy Analysis Method; Jeffrey Fluid; Oscillatory Stretching Sheet. 1 Introduction The phenomenon of flow and heat transfer of non-New- tonian fluids over continuous stretching sheet plays a vital role in many industrial processes. The most valuable applications of flow of non-Newtonian fluids over stretch- ing sheet include paper production, glass blowing, wire drawing, and aerodynamic extrusion of plastic sheet. The fundamental work on flow over continuous moving sheet was provided by Sakiadis [1]. Crane [2] extended the work of Sakiadis [1] for a stretching sheet. The study of flows of non-Newtonian fluids has attracted the attention of many researchers because of its wide range of applications in technology and industry. The dynamics of non-Newtonian fluids is entirely different from that of Newtonian fluids. This is because non-Newtonian fluids exhibit many typical characteristics such as shear-thinning or thicken- ing, yield stress, the Weissenberg effects, fluid memory, and die swelling, which are absent in Newtonian fluids. Therefore, these fluids do not obey the well-known New- ton’s law of viscosity, which is valid for Newtonian fluids. Due to the diversity of non-Newtonian fluids, many con- stitutive relations are proposed in the literature. Fluid models in which viscosity depends on the instantaneous rate of deformation are known as generalised Newtonian fluid (GNF) models. These models are capable of describ- ing shear thinning, shear thickening, and yield shear stress effects. The stretching sheet problems using GNF models were analysed by Anderson et al. [3], Hassanien [4], Chen [5], Nadeem et al. [6, 7], Mukhopadhyay [8], etc. The Weissenberg or rod-climbing effect is associated with nonlinear effects and normal stresses. Such an effect is predicted by those models for which normal stress dif- ferences are non-zero. Second-grade fluid is one of them. Studies pertaining to the flow of second-grade fluid over a stretching sheet can be found in [9–20]. To account for vis- coelastic effects like fluid memory or die swelling, models like linear/convected Maxwell and Jeffrey models have been proposed in the literature. Some studies regarding flow over a stretching sheet using these models were also carried out in the literature [21–28]. Fluid models that can simultaneously predict shear thinning/shear thickening and normal stress effects have also been used to study flow over stretching sheets. For instance, Sajid and Hayat [29] discussed the boundary layer of a third-grade fluid over a stretching sheet. In another study, they carried out flow and heat transfer analysis of a third-grade fluid over a stretching surface [30]. The literature reveals that very little attention is paid to the flow and heat transfer analysis due to the oscil- latory stretching sheet. The viscous flow analysis over the oscillatory stretching sheet was initially performed by Wang [31]. Later, Abbas et al. [32] extended Wang’s problem by including wall slip and performing heat transfer analysis. Furthermore they [33], complemented

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Page 1: Nasir Ali, Sami Ullah Khan* and Zaheer Abbas Hydromagnetic

Z. Naturforsch. 2015; 70(7)a: 567–576

*Corresponding author: Sami Ullah Khan, Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan, Tel.: +92 51 9019756, E-mail: [email protected] Ali: Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, PakistanZaheer Abbas: Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

Nasir Ali, Sami Ullah Khan* and Zaheer Abbas

Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid over an Oscillatory Stretching Surface

DOI 10.1515/zna-2014-0273Received September 28, 2014; accepted April 24, 2015; previously published online June 13, 2015

Abstract: The flow and heat transfer of a Jeffrey fluid over an oscillatory stretching sheet is investigated using the boundary-layer approximations. The flow is induced due to infinite elastic sheet that is stretched periodically. The number of independent variables in the governing equa-tions was reduced by using appropriate dimensionless variables. This dimensionless system has been solved by using the homotopy analysis method (HAM) and a finite difference scheme, in which a coordinate transformation was used to transform the semi-infinite physical space to a bounded computational domain. A comparison of both solutions is provided. The effects of involved parameters are illustrated through graphs and discussed in detail.

Keywords: Boundary Layer Flow; Finite Difference Scheme; Homotopy Analysis Method; Jeffrey Fluid; Oscillatory Stretching Sheet.

1 IntroductionThe phenomenon of flow and heat transfer of non-New-tonian fluids over continuous stretching sheet plays a vital role in many industrial processes. The most valuable applications of flow of non-Newtonian fluids over stretch-ing sheet include paper production, glass blowing, wire drawing, and aerodynamic extrusion of plastic sheet. The fundamental work on flow over continuous moving sheet was provided by Sakiadis [1]. Crane [2] extended the work of Sakiadis [1] for a stretching sheet. The study of flows of

non-Newtonian fluids has attracted the attention of many researchers because of its wide range of applications in technology and industry. The dynamics of non- Newtonian fluids is entirely different from that of Newtonian fluids. This is because non-Newtonian fluids exhibit many typical characteristics such as shear-thinning or thicken-ing, yield stress, the Weissenberg effects, fluid memory, and die swelling, which are absent in Newtonian fluids. Therefore, these fluids do not obey the well-known New-ton’s law of viscosity, which is valid for Newtonian fluids. Due to the diversity of non-Newtonian fluids, many con-stitutive relations are proposed in the literature. Fluid models in which viscosity depends on the instantaneous rate of deformation are known as generalised Newtonian fluid (GNF) models. These models are capable of describ-ing shear thinning, shear thickening, and yield shear stress effects. The stretching sheet problems using GNF models were analysed by Anderson et al. [3], Hassanien [4], Chen [5], Nadeem et al. [6, 7], Mukhopadhyay [8], etc. The Weissenberg or rod-climbing effect is associated with nonlinear effects and normal stresses. Such an effect is predicted by those models for which normal stress dif-ferences are non-zero. Second-grade fluid is one of them. Studies pertaining to the flow of second-grade fluid over a stretching sheet can be found in [9–20]. To account for vis-coelastic effects like fluid memory or die swelling, models like linear/convected Maxwell and Jeffrey models have been proposed in the literature. Some studies regarding flow over a stretching sheet using these models were also carried out in the literature [21–28]. Fluid models that can simultaneously predict shear thinning/shear thickening and normal stress effects have also been used to study flow over stretching sheets. For instance, Sajid and Hayat [29] discussed the boundary layer of a third-grade fluid over a stretching sheet. In another study, they carried out flow and heat transfer analysis of a third-grade fluid over a stretching surface [30].

The literature reveals that very little attention is paid to the flow and heat transfer analysis due to the oscil-latory stretching sheet. The viscous flow analysis over the oscillatory stretching sheet was initially performed by Wang [31]. Later, Abbas et  al. [32] extended Wang’s problem by including wall slip and performing heat transfer analysis. Furthermore they [33], complemented

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568      N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid

Wang’s problem by carrying out a study of the unsteady magneto hydrodynamic (MHD) boundary flow of a sec-ond-grade fluid over an oscillatory stretching surface. Some recent attempts regarding flow due to oscillatory stretching sheet can be found in [34–36]. Despite recent developments in non-Newtonian fluids, it is still of inter-est to analyse stretching flows involving non-Newtonian fluids. For instance, no attempts are available in the lit-erature that deal with the flow and heat transfer of Jeffrey fluid over an oscillatory stretching surface. The purpose of this article is to provide such an analysis. The Jeffrey fluid model is a fairly simple linear model using a time derivative instead of a convected derivative as used in the Oldroyd-B model. This model has three material param-eters: the viscosity, the ratio of relaxation to retardation time, and retardation time. The Jeffrey model is capable of predicting elastic and memory effects exhibited by dilute polymer solutions and biological liquids. Relaxation time is a time constant associated with viscoelastic fluids. For viscoelastic fluids, the internal molecular configuration of the fluid can sustain stress for some time. This time, called relaxation time, varies widely among materials. In fact, the Jeffrey model is a simple extension of the Newtonian model to include elastic and memory effects. By choosing the Jeffrey fluid model, it becomes possible to study the oscillatory stretching sheet problem for various Newto-nian and non-Newtonian fluids. However, this model is not capable of predicting shear-thinning and shear-thick-ening effects and thus cannot be used to discuss the fluids that are driven to flow at high shear rates.

The problem presented here has many useful engineer-ing and industrial applications. In fact, the production of sheeting material is involved in many manufacturing pro-cesses and includes both metal and polymer sheets. The rate of heat transfer over a surface has to play a useful role in the quality of the final product. Industrial applications of stretching sheet include hot rolling, fibers spinning, man-ufacturing of plastic and rubber sheet, continuous casting, and glass blowing. The present study also describes MHD effects that are very prominent in MHD power-generating systems, plasma studies, cooling of nuclear reactors, geo-thermal energy extraction, and many others.

The structure of the article is as follows. The problem is formulated with underlying assumptions in section 2. A solution using the homotopy analysis method (HAM) is provided in Section 3. A scheme of the numerical solu-tion is illustrated in Section 4. The convergence of the HAM solution and its comparison with a numerical solu-tion are presented in Section 5. A discussion of the results obtained is given in Section 6. The main conclusions of the study are summarised in Section 7.

2 Flow AnalysisWe consider the unsteady, two-dimensional, and MHDs flow of an incompressible Jeffrey fluid over an oscillatory stretching sheet coinciding with the plane y̅ = 0. The elastic sheet is stretched back and forth periodically with velo city u

ω = bx̅ sin ωt (b is the maximum stretching rate, x̅ is the

coordinate along the sheet, and ω represents the frequency). A constant magnetic field of strength B0 is applied perpen-dicular to the stretching surface. The magnetic Reynolds number is assumed to be small so that the induced mag-netic field can be neglected. Let Tw be the temperature of the surface of the sheet and T∞ is the temperature of the fluid far away from the sheet, where Tw > T∞. A schematic diagram of the flow geometry is illustrated in Figure 1. In addition to these assumptions, the boundary-layer approximations are used and viscous dissipation effects are neglected. Thus, the governing equations for the flow under consideration in the presence of magnetic field are [28]

0,u v

x y∂ ∂+ =∂ ∂

(1)

2 3 3

12 2 2

22 2 30

2 3

1

,

u u u u u uu v ut x y y y t x y

Bu u u u uv ux y x yy y

νλ

λ

σ

ρ

∂ ∂ ∂ ∂ ∂ ∂+ + = + + ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂− + + −∂ ∂ ∂ ∂∂ ∂ (2)

2

2 ,pT T T Tc u v kt x y y

ρ ∂ ∂ ∂ ∂

+ + = ∂ ∂ ∂ ∂ (3)

where u and v are velocity components along x̅ and y̅ direc-tions, respectively. In the above equations, ν represents the kinematic viscosity, which is the ratio of the dynamic viscosity of a fluid to its density; λ is the ratio of relaxation to retardation time (ratio parameter); λ1 is the retardation parameter; T represents the temperature; cp is the specific

Tw >T∞

y

B0

x

T = Twuw = bx sin ωt0

Figure 1: Geometry of the problem.

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N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid      569

heat, which shows the relationship between heat and temperature changes; and k is the thermal conductivity, which primarily appears in Fourier’s law for heat conduc-tion. The thermal conductivity is the property of a mate-rial’s ability to conduct heat.

The boundary conditions to be satisfied by the veloc-ity components and temperature are

sin , 0, at 0, 0,wu u bx t v T T y tω

ω= = = = = > (4)

0, 0, as .uu T T y

y ∞

∂→ → → → ∞

∂ (5)

The second boundary condition in (5) is the aug-mented boundary condition [33] and is imposed due to the consideration of the flow in the unbounded domain. To nondimensionalise the flow problem, we use the follow-ing dimensionless variables [31, 32]:

( , ), ( , ), and ,y

bu bxf y v b f y y y tτ ν τ τ ων

= = − = =

(6)

( , ) .

w

T Ty

T Tθ τ ∞

−=

(7)

With the help of (6) and (7), the continuity equation is identically satisfied, and (2) and (3) take the form

( ) ( )2 2 2( 1 ) ,y y yy y yyy yyy yy yyyySf f ff M f f Sf f ffτ τ

λ β+ + − + = + + − (8)

Pr( ) 0,yy yf Sτ

θ θ θ+ − = (9)

where subscripts denote differentiation. It is pointed out that a reduction in the number of independent variables by one is also achieved as an implication of transforming (2) and (3) in the dimensionless form. The boundary con-ditions (4) and (5) becomes

( , ) sin , ( , ) 0, ( , ) 1 at 0,yf y f y y yτ τ τ θ τ= = = = (10)

( , ) 0, ( , ) 0, ( , ) 0 as .y yyf y f y y yτ τ θ τ→ → → → ∞ (11)

Here, S ≡ ω/b is the ratio of oscillation frequency of the sheet to its stretching rate, β = λ1b represents the Deborah number, Pr = μcp/k is the Prandtl number, and

20= /M B bσ ρ represents the Hartmann number.

We define the skin friction coefficient Cf and the local Nusselt number Nux as

2 , Nu ,

( )w w

f xww

xqC

k T Tuτ

ρ ∞

= =−

(12)

where τw and qw are the shear stress and heat flux at wall, respectively, which are defined as [28]

2 2 2

1 2

0

0

, 1

.

w

y

wy

u u u uu vy t y y x y

Tq ky

µτ λ

λ=

=

∂ ∂ ∂ ∂= + + + + ∂ ∂ ∂ ∂ ∂ ∂

∂= − ∂

(13)

In view of (6) and (7), (12) gives

1/ 20

1/ 2

1Re [ ( )] , 1

Re Nu ( 0, ),

x f yy yy y yy yyy y

x x y

C f Sf f f ffτ

βλ

θ τ

=

= + + −+

= −

(14)

where Re /x wu x ν= denotes the local Reynolds number.

3 Solution of the Problem

We have employed two methods for obtaining the solution of (8) and (9) subject to boundary conditions (10) and (11). Firstly, we have used HAM to solve nonlinear partial dif-ferential equations. The procedural details of this method can be found in the latest book by Liao [37]. Secondly, a numerical method based on an implicit finite difference scheme is used. The details of numerical method are explained in the next section.

4 Direct Numerical Solution of the Problem

The set of nonlinear partial differential equations (8) and (9) with boundary conditions (10) and (11) are solved numerically by using the finite difference scheme. To transform the semi-infinite physical domain y ∈ [0, ∞) to a finite calculation domain η ∈ [0, 1], a coordinate trans-formation η = 1/(y + 1) is employed. With the help of this coordinate transformation, (8) and (9) become

2 4 32 4 3

3 2

22 4 2 3

32 4

3

23 2 4

2

22 2 3 45 6 5 6

2 2 3 4

(( 1 ) 6 ) 6

(( 1 ) 4 ) ( 6 24

( 1 ) 2( 1 ) )

( 6 ( 1 ) 36 )

4 12

f f fS S S

f f

f fM f

ff f

f f f f ff f

λ βη βη βητ η η τ η τ

λ η βη η βηη

λ λ η ηη η

η λ η βηη

βη βη βη βηη η η η η

∂ ∂ ∂+ − − −∂ ∂ ∂ ∂ ∂ ∂

∂= + − + + ∂

∂ ∂− + − + +

∂ ∂∂+ − + +∂

∂ ∂ ∂ ∂ ∂− − + + ∂ ∂ ∂ ∂ ∂ ,

(15)

24 3 2

2 2 Pr 0,f Sθ θ θ θη η η

η η τη

∂ ∂ ∂ ∂+ − + = ∂ ∂ ∂ ∂

(16)

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570      N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid

0, 0, 0 at 0,f fη ηη

θ η= = = = (17)

0, sin , 1 at 1.f fη

τ θ η= = − = = (18)

Equations (15) and (16) are discretised for L uniformly distributed discrete points in η = (η1, η2, ……, η

{L}) ∈

(0, 1) with a space grid size of Δη = 1/(L + 1) and time level t = (t1, t2, …). Hence, the discrete values ( )1 2, , , n n n

Lf f f… and ( )1 2, , , n n n

Lθ θ θ… at these grid points for time levels tn = nΔt (Δt is the time step size) can be numerically obtained by solving a system of linear equations. Such a system of linear equa-tions can be obtained by constructing a semi-implicit time difference for f and θ and assuming that only linear equation for the new time step (n + 1) needs to be solved. The following semi-implicit time difference for f and θ is constructed:

( 1) ( )2

( 1)3 3 ( )4

3 3

( 1)2 2 ( )3

2 2

2( )2 4

( 1)2 2

( 1( ) 23 ( ) 3

1( ( 1 ) 6 )

1

16

( (1 ) 4 )

( 6 (1 ) )

(24 2(1 ) ) 6

n n

n n

n n

n

n

nnn

f fSt

f fSt

f fSt

f

fM

f ff

λ βη∆ η η

βη∆ η η

βη∆ η η

λ η βηη

η λη

βη λ η ηη

+

+

+

+

+

∂ ∂+ − − ∂ ∂ ∂ ∂− − ∂ ∂

∂ ∂− − ∂ ∂

∂= + − ∂

∂+ − +∂

∂ ∂+ − + +∂

)

2

2 ( )4 2 ( )

2

2( ) 2 ( ) 2 ( )5 6

2 2

( 1)3 3 ( ) 4 ( )4 5 ( ) 6 ( )

3 3 4

( 36 (1 ) )

4

12 .

nn

n n n

n n nn n

ff

f f f

f f ff f

η

βη λ ηη

βη βηη η η

η βη βηη η η

+

∂∂

+ − +∂

∂ ∂ ∂− − ∂ ∂ ∂

∂ ∂ ∂+ + +∂ ∂ ∂

(19)

( 1) ( ) ( 1) ( 1)24 3

2

( 1)( ) 2

( )Pr 2

Pr .

n n n n

nn

St

f

θ θ θ θη η

∆ ηη

θη

η

+ + +

+

− ∂ ∂= +∂∂

∂−

∂ (20)

Equations (19) and (20) are converted into a system of linear equations by means of the finite difference method, which can be solved by any method, e.g. Gaussian elimi-nation. The simulations are started from motionless veloc-ity field and a uniform temperature distribution equal to temperature at infinity as

( , 0) 0 and ( , 0) 0.f η τ θ η τ= = = = (21)

5 Convergence of HAM Solution and its Comparison with a Numerical Solution

It is a well-established fact that the convergence of the HAM solution is largely dependent on the proper choice of the auxiliary parameters hf and h

θ. For a particular set

of parameters, the convergence region can be obtained by plotting h-curves. Figure 2a and b presents two such curves showing the plausible values of hf and h

θ for a

given set of parameters. This figure indicates for conver-gent solution –1.1  ≤  hf < 0 and –1.7  ≤  h

θ < –0.2. Table  1

presents values of f ″(0, τ) at different order of approxi-mations for a specific set of parameters. It is observed from this table that the convergent values of f ″(0, τ) are achieved at the 10th-order of approximation. A compari-son of the HAM solution with a numerical solution is also presented in this section in Figures 3 and 4. These figures show an excellent agreement between both the solutions

8

6

4

2

0

–2

–4–1.5 –1.0 –0.5 0.0 –2.0

–0.5

–1.5

–2.0

–2.5

–3.0

–1.0

–1.0 –0.5 0.0–1.5

hfh

θ

f′′(0

, τ)

θ′ (

0, τ

)

λ = 0.1, β = 0.1, M = 1.5, S = 0.2 and τ = π/2 λ = 0.1, β = 0.1, M = 1.5, S = 0.2, Pr = 0.5 and τ = π/2a b

Figure 2: The h-curves at 10th-order of approximation (a) for velocity and (b) temperature profile.

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N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid      571

6 Results and DiscussionIn this section, the effects of pertinent parameters of interest on the velocity component f ′, skin friction coefficient, tem-perature profile θ, and the Nusselt number are illustrated through various plots. The effects of the magnetic field on various flow characteristics are not shown to reduce the length of the article. The interested reader may consult [33, 35] for illustration of such effects in similar situations.

Figure 5 shows the effects of the Deborah number β and ratio of relaxation to retardation parameter λ on dimension-less velocity at a fixed distance y = 0.25 from the sheet as a function of time. In Figure 5a, the effects of the Deborah number β are illustrated. This figure shows an increase in the amplitude of velocity f ′ by increasing the Deborah number β. Moreover, a phase difference occurs for the non-zero value of β. Thus, a viscoelastic fluid with larger retardation time oscillates in time at a fixed location from the sheet with a different amplitude and phase than that of a Newtonian fluid. Figure 5b shows the effects of λ on the

as the order of approximation increases. This is because at higher order of approximation, the HAM solution becomes closer to the exact solution. The values of f ″(0, τ) obtained by HAM and numerical solution for a Newtonian fluid are also compared with the corresponding values in [33, 35]. Table 2 shows the comparison of such values. It can be seen that our results are in excellent agreement with those presented in [33, 35].

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

y

f′ f′

S = 0.2, M = 5, β = 0.5, λ = 0.5 and τ = 0.5 π

(5th-order)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

y

S = 0.2, M = 5, β = 0.5, λ = 0.5 and τ = 0.5 π

(30th-order)

a b

Figure 3: Comparison of f ′(y, τ) obtained from the HAM solution (solid lines) and the numerical solution (open circles).

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

y

θ

(5th-order)

S = 0.2, M = 5, β = 0.5, λ = 0.5, Pr = 0.5 and τ = 0.5 π

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

y

θ

(30th-order)

S = 0.2, M = 5, β = 0.5, λ = 0.5, Pr = 0.5 and τ = 0.5 πa b

Figure 4: Comparison of θ(y, τ) obtained from the HAM solution (solid lines) and the numerical solution (open circles).

Table 1: Convergence of HAM solution of f ″(0, τ) with S = 0.5, M = 3, β = 0.1, and λ = 0.1.

Order of approximations

  τ = 0  τ = π  τ = 1.5π

3   –0.002304  0.002304  0.9444485   –0.002632  0.002574  0.94312410   –0.002670  0.002588  0.94301715   –0.002670  0.002588  0.94301720   –0.002670  0.002588  0.94301730   –0.002670  0.002588  0.943017

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572      N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid

velocity profile f ′. It is observed that the amplitude of flow oscillations decreases by increasing the ratio parameter.

The transverse profiles of velocity f ′ for various values of β and at four different time instants in the fifth period are shown in Figure 6a–d. Figure 6a shows that at τ = 8.5π, the velocity f ′ increases by increasing β. Moreover, at this time instant, there are no oscillations in the velocity profile f ′. At time instant τ = 9π (Fig. 6b), the velocity f ′ oscillates near the wall and amplitude of oscillation increases by increasing β. Figure 6c elucidates that at τ = 9.5π, velocity f ′ increases from –1 at the sheet to zero at infinity. More-over, f ′ decreases by increasing β at this time instant. When τ = 10π, again oscillation in f ′ is observed near the oscillating sheet and its amplitude increases with β.

Figure 7 shows the effects of ratio of relaxation to retardation time, λ on the transverse profiles of velocity f ′ at four different instants τ = 8.5π, τ = 9π, τ = 9.5π, and τ = 10π. It is observed from this figure that, for τ = 8.5π, velocity f ′ decreases from unity at the wall to zero far away from the wall. Moreover, it decreases by increasing λ. At time instant τ = 9π, f ′ has zero value both at the wall and far away from the wall. Furthermore, it oscillates near the wall and amplitude of oscillation decreases by increas-ing λ. Similar results can be observed at τ = 10π. However,

Table 2: Comparison of values of f ″(0, τ) for λ = β = 0 with [33, 35].

S    M   τ   Ref. [33] with K = 0

   Ref. [35]   Present results

Numerical solution

  HAM solution

1.0  12  1.5π  11.678656  11.678565  11.678656  11.6785657    5.5π  11.678707  11.678706  11.678707  11.6787065    9.5π  11.678656  11.678656  11.678656  11.6786561

0 5 10 15 20 25 30−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

τ

f′ f′

β = 0.5 β = 1.0β = 1.5β = 2.0

S = 4, M = 12 and λ = 0.1

0 5 10 15 20 25 30−0.04

−0.02

0

0.02

0.04

τ

λ = 0.0λ = 0.5λ = 1.0λ = 1.5

S = 0.5, M =12 and β = 0.1a b

Figure 5: The velocity profile f ′ as a function of time in the first five periods τ ∈ [0, 10π] at a fixed distance from the sheet, y = 0.25: (a) effects of β and (b) effects of λ.

at τ = 9.5π, f ′ decreases from –1 at the wall to zero far away from the wall. Further, its magnitude decreases by increasing λ.

The variation of skin friction coefficient 1/ 2Rex fC with time for different values of β and λ is depicted in Figure 8. We observe from this figure that, like velocity profile f ′, skin friction coefficient also oscillates with time and amplitude of oscillation increases by increasing β and λ. Further, like velocity profile f ′, the oscillations in the skin friction for β = 0 are not in phase with the oscillations for non-zero of β. In fact, a phase lag is observed in skin friction for the non-zero value of β. The profile of temperature θ for differ-ent values of the Prandtl number Pr, β, M, and λ are shown in Figure 9. It is observed through this figure that the tem-perature θ of the of the fluid decreases by increasing the Prandtl number and β. However, it increases by increasing M and λ. The variation of the local Nusselt number with time for different values of Pr, β, M, and λ is displayed in Figure 10a–d. Figure 10a illustrates the effects of the Prandtl number on the local Nusselt number 1/ 2Re Nu .x x

− It is noted that the local Nusselt number increases with the increase of Pr. In Figure 10b, the effects of the Deborah number β are illustrated. This figure predicts an increase in the local Nusselt number by increasing the Deborah number β. The profiles of the local Nusselt number follow a reverse trend in Figure 10c and d. This figure indicates that the local Nusselt number decreases by increasing ratio parameter λ and the Hartmann number M.

7 Concluding RemarksIn this article, we have studied the unsteady flow and heat transfer of Jeffrey fluid over an oscillatory

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N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid      573

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

f ′

y

β = 0.0 β = 0.5β = 1.0β = 1.5

τ = 8.5 πa

f ′−0.12 −0.1 −0.08 −0.06 −0.04 −0.02 00

0.2

0.4

0.6

0.8

1

y

β = 0.0 β = 0.5β = 1.0β = 1.5

τ = 9 πb

f ′0 0.05 0.1 0.15 0.2 0.25

0

0.2

0.4

0.6

0.8

1

y

β = 0.0 β = 0.5β = 1.0β = 1.5

τ = 10 πd

f ′−0.8 −0.6 −0.4 −0.2 0

0

0.2

0.4

0.6

0.8

1

y

β = 0.0 β = 0.5β = 1.0β = 1.5

τ = 9.5 πc

Figure 6: Transverse profiles of velocity field f ′ for different values of β in the fifth period τ ∈ [8π, 10π] with S = 1, M = 12, and λ = 0.1.

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

y

λ = 0.0 λ = 0.5λ = 1.0λ = 1.5

τ = 8.5 π

−0.007 −0.006 −0.005 −0.004 −0.003 −0.002 −0.001 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f ′

y

λ = 0.0 λ = 0.5λ = 1.0λ = 1.5

τ = 9 π

−0.8 −0.6 −0.4 −0.2 00

0.2

0.4

0.6

0.8

1

y

λ = 0.0 λ = 0.5λ = 1.0λ = 1.5

τ = 9.5 π

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.0080

0.2

0.4

0.6

0.8

1

y

λ = 0.0 λ = 0.5λ = 1.0λ = 1.5

τ = 10 π

f ′

f ′ f ′

a b

c d

Figure 7: Transverse profiles of velocity field f ′ for different values λ in the fifth period τ ∈ [8π, 10π] with S = 0.5, M = 12, and β = 0.1.

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574      N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid

0 5 10 15 20 25 30−30

−20

−10

0

10

20

β = 0.0β = 0.5 β = 1.0β = 1.5

S = 2, M = 12 and λ = 0.1

0 5 10 15 20 25 30

−15

−10

−5

0

5

10

15

20

ττ

λ = 0.0 λ = 0.5λ = 1.5λ = 2.0

S = 2, M = 12 and β = 0.5R

e1/2

Cf

x

Re1/

2 C

fx

a b

Figure 8: The skin friction coefficient 1/2Rex fC in first five periods τ ∈ [0, 10π]: (a) effects of β and (b) effects of λ.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

θ

y

M = 5.0 M =7.0M = 9.0M = 12.0

S = 0.7, β = 2.5, λ = 0.2 and Pr = 5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

θ

y

β = 0.5 β = 1.0β = 2.0β = 3.0

S = 0.8, Pr = 10, λ = 0.1 and M =12

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

θ

y

λ = 0.5 λ = 2.0λ = 4.0λ = 8.0

S = 0.7, β = 2.5, Pr = 12 and M = 15

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

θ

y

Pr = 5.0 Pr = 7.0Pr = 9.0Pr = 12.0

S = 5, β = 0.2, λ = 0.7 and M = 12a b

c d

Figure 9: Transverse profiles of the temperature field θ at the time point τ = 8π (a) effects of Pr, (b) effects of β, (c) effects of M, and (d) effects of λ.

stretching surface, which is maintained at a constant temperature. The resulting nonlinear partial differential equations are solved by using HAM and a finite differ-ence scheme. The main conclusions of the present study are as follows:

– The amplitude of flow velocity increases by increas-ing the Deborah number β, while it decreases with ratio of relaxation to retardation time λ. Moreover, a phase shift is observed for non-zero values of the Deborah number β.

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N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid      575

– The amplitude of skin friction coefficient increases by increasing the Deborah number β and ratio of relaxa-tion to retardation time λ.

– An increase in temperature is found by increasing the Hartmann number M and ratio of relaxation to retar-dation time λ. However, it decreases by increasing the Deborah number and the Prandtl number Pr. More-over, the rate of heat transfer increases by increasing the Prandtl number Pr and the Deborah number β.

– The solution presented in this article is more general and includes the solution corresponding to radiative problem as a special case. This conclusion is also sup-ported by recent papers by Magyari and Pantokratoras [38], Fetecau et al. [39], and Vieru et al. [40], where it is explicitly shown that an evaluation of the effect of thermal radiation in the linearized Rosseland approx-imation does not require any additional research effort. Once the problem has been solved for a com-prehensive set of values of the Prandtl number in the absence of radiation, it has been automatically solved for the radiative case too. In view of above argument, the solution of problem with radiative effects can be

obtained by replacing the Prandtl number in our solu-tion by the effective Prandtl number Preff = Pr/(1 + Nr), where Nr is the radiation parameter.

Acknowledgments: We are thankful to the anonymous reviewers for their useful comments to improve the earlier version of the paper. The second author is grateful to the Higher Education Commission of Pakistan for financial assistance.

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5 10 15 20 25 30

0.5

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−θ y

(0, τ

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576      N. Ali et al.: Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid

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