Alexandria Engineering Journal (2016) xxx, xxx–xxx

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium Nasir Ali a, Sami Ullah Khan a,*, Muhammad Sajid a, Zaheer Abbas b a b

Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

Received 25 December 2015; revised 9 February 2016; accepted 20 February 2016

KEYWORDS Boundary layer flow; Couple stress fluid; Oscillatory stretching sheet; Homotopy analysis method

Abstract This article presents the MHD flow and heat transfer of a couple stress fluid over oscillatory stretching sheet embedded in a porous medium in the presence of heat source/sink. The unsteady flow problem is reduced to two coupled partial differential equations using dimensionless variables. Homotopy analysis method is employed to obtain the solution of these equations. Based on the solution of these equations an extensive analysis is performed to investigate the effects of various flow parameters on velocity and temperature fields, skin friction coefficient and Nusselt number. It is found that the presence of couple stress in viscous fluid increases the amplitude of oscillations in velocity and skin friction coefficient. It is also noticed that temperature increases by increasing heat source parameter. Moreover, the numerical values of local Nusselt number are calculated and shown in tabular form. It is found that the Nusselt number increases by increasing Prandtl number while it decreases by increasing couple stress parameter. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction It is well established fact that Navier–Stokes equations are not adequate choice in describing the flows of nonNewtonian fluids. Non-Newtonian fluids frequently occur in many industrial and technological processes. If the flow of non-Newtonian fluid is considered over a stretching sheet then it become relevant to many industrial applications such a extrusion of plastic sheets, paper production, crystal * Corresponding author. Tel.: +92 51 9019756. E-mail address: [email protected] (S. Ullah Khan). Peer review under responsibility of Faculty of Engineering, Alexandria University.

growing and glass blowing, drawing of copper wires, continuous stretching of plastic films. The pioneering work on viscous flow due to continuous moving sheet was performed by Sakiadis [1]. His analysis was extended for stretching sheet by Crane [2]. Crane’s problem was extended by Rajagopal et al. [3] for viscoelastic fluid. Gupta and Gupta [4] extended the analysis of Crane by considering suction/blowing at the sheet surface. Dandapat and Gupta [5] discussed the heat transfer characteristics of viscoelastic fluid over a stretching sheet. A study of uniqueness of boundary value problem arising due to a stretching boundary in a viscous fluid was performed by McLeod and Rajagopal [6]. Rollins and Vajravelu [7] discussed heat transfer characteristics of second-order fluid over a continuos stretching surface with internal heat generation

http://dx.doi.org/10.1016/j.aej.2016.02.018 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

2 or absorption. The stretching sheet problem for viscoelastic Oldroyd-B model was extended by Rajagopal and Bhatnagar [8]. Cortell [9] presented a study of flow and heat transfer of hydromagnetic second-order fluid past a stretching sheet. In another paper [10], Cortell included magnetic and suction effects while discussing the flow and heat transfer of second grade fluid due to stretching sheet. The study of stagnationpoint flow of micropolar fluid toward a stretching sheet was initiated by Nazar et al. [11]. Combined forced and natural convection flow of micropolar fluid over a stretching sheet in the vicinity of stagnation point was discussed by Ishak et al. [12]. Ariel [13] put forward a study of flow of second grade fluid past a radially stretching sheet. Ariel work was extended by Hayat and Sajid [14] to include the heat transfer characteristics. Some other contribution by Hayat and his co-workers regarding heat transfer analysis over a stretching surface can be found in references [15–18]. The investigation of non-similar flow of third order fluid over a stretching sheet was due to Sajid and Hayat [19]. Hayat et al. [20] analyzed combined heat and mass transfer characteristics in boundary layer flow of viscoelastic fluid over linearly stretching vertical surface by using homotopy analysis method. Later on, Turkyilmazoglu [21] computed exact solution of hydromagnetic flow of viscoelastic fluid in the presence of Soret and Dufour effects. Turkyilmazoglu [22] also developed closed-form solutions for three dimensional flow and heat transfer of a electrically conducting viscoelastic fluid over a porous stretching/shrinking sheet. The multiplicity of the solutions in boundary layer flows over stretching/shrinking sheet is also addressed by Turkyilmazoglu. In fact, it was shown by him that multiple solutions exist for twodimensional flow over stretching sheet and in the case of shrinking sheet triple solutions can be detected [23–25]. In another paper, Turkyilmazoglu [26] computed exponential series type solution in the hydromagnetic and heat transfer in two–three dimensional flow over stretching or shrinking surface. According to him exponential series type solution exists with finite and infinite terms in the expansion depending upon prevailing parameters in the governing equations. To best of our knowledge, Wang [27] initiated the study of viscous fluid due to oscillatory stretching sheet. His work was extended by Saddipa et al. [28] by using constitutive equation of Walter’s B model. Later on, Abbas et al. [29] complemented the Wang ’s analysis by including the effects of slip and heat transfer. They also utilized the constitutive relation of second order fluid [30] to study Wang’s problem. Zheng et al. [31] provided an analysis of unsteady heat and mass transfer in MHD flow over an oscillatory stretching sheet with Soret and Dufour effects. They have employed homotopy analysis method and delineated the effects of various emerging parameters on flow and heat transfer characteristics. Their study indicates a decrease in velocity of fluid by increasing unsteady parameter, magnetic parameter and suction/injection parameter. They also found an increase in thermal boundary layer thickness with magnetic parameter and unsteady parameter. Ali et al. [32] analyzed the unsteady flow of Jeffrey fluid over an oscillatory stretching surface. Khan et al. [33] used an implicit finite difference scheme to analyze the heat transfer effects in the hydromagnetic flow of viscoelastic fluid (second grade fluid) over an oscillatory stretching surface. However, so far no one has considered the study of unsteady boundary layer flow of a couple stress fluid through a porous medium over an

N. Ali et al. oscillatory stretching surface in the presence of heat generation or absorption. Hence we feel appropriate to consider the study of such flows, as such type of flows may find a number of industrial applications. The theory of couple stress fluid was developed by Stokes [34] as a simple generalization of classical viscous theory that sustains couple stresses and body couples. The main effects of couple stress fluid are to introduce a size dependent effect which is not present in classical viscous theories. Blood, lubricants containing small amount of additives, electro-rheological and synthetic fluids are examples of couple stress fluid [35]. Couple stress theory has been successfully applied to many problems in biomechanics and lubrications area by Srivastava [36], El-Shehawey and Mekheimer [37], Pal et al. [38], Chiang et al. [39], Naduvinamani et al. [40], Jian and Chen et al. [41] and Lu and Lin [42]. The presentation of the paper is as follows. Flow analysis is presented in Section 2. Homotopy analysis method is employed for solution of governing equations in Section 3. Convergence of solution is discussed in Section 4. Quantitative analysis of results is presented in Section 5. Final remarks are presented at the end of the paper in Section 6. 2. Flow analysis Let us consider the unsteady two-dimensional magnetohydrodynamics flow of an incompressible fluid over an oscillatory stretching sheet. In a rectangular coordinate system ð x; yÞ, the sheet is assumed to coincide with y ¼ 0 and flow takes place in the semi-infinite porous space ðy > 0Þ of constant permeability (see Fig. 1). A constant magnetic field of strength B0 is applied perpendicular to the stretching surface. It is assumed that the sheet is stretched with velocity ux ¼ b x sin xt in which b is the stretching rate and x denotes the angular velocity. Moreover, sheet is maintained at uniform temperature Tw ð> T1 Þ, where T1 is the temperature of ambient fluid. Now problem is to determine the velocity and temperature fields inside the fluid satisfying the boundary conditions at the wall and far away from the wall. To this end, we apply boundary layer approximation on continuity and momentum equations and write @u @v þ ¼ 0; @ x @ y

ð1Þ

Figure 1

Geometry of the problem.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

MHD flow and heat transfer of couple stress fluid

3

@u @u @u @2u g @4u rB2 t/ þu þv ¼m 2 0 0 u u; 4 @t @ x @ y @ y k q @ y q

ð2Þ

@T @T @T k @2T Q þu þv ¼ þ ðT T1 Þ; @t @ x @ y qcp @ y2 qcp

ð3Þ

where u and v are velocity component along x and y-directions, respectively, m is the kinematic viscosity, q is the density, g0 is the material constant for the couple stress fluid, Q is the heat source coefficient, r is electric conductivity and B0 is the strength of constant applied magnetic field. The flow is subjected to the following boundary conditions u ¼ ux ¼ b x sin xt; y ¼ 0;

t > 0;

u ¼ 0;

@u ¼ 0; @ y

v ¼ 0;

@2u ¼ 0; @ y2

T ¼ Tw

at ð4Þ

T ! T1

at

y ! 1:

ð5Þ

The first two boundary conditions in (4) result from the assumption of no-slip at the surface while the third condition is consequence of the assumption of vanishing of couple stresses at the wall. Since flow is in unbounded domain that is why augmented boundary condition is imposed at y ! 1. To nondimensionalize the flow problem, we use the dimensionless variables [30,31] rﬃﬃﬃ pﬃﬃﬃﬃﬃ b y¼ y; s ¼ tx; u ¼ b xfy ðy; sÞ; v ¼ mbfðy; sÞ; ð6Þ m hðy; sÞ ¼

T T1 : Tw T1

ð7Þ

In view of above transformations, the continuity equation is identically satisfied and Eqs. (2) and (3) are transformed as follows fyyy Sfys f2y þ ffyy bfy Kfyyyyy ¼ 0;

ð8Þ

hyy þ Pr fhy Shs þ kh ¼ 0;

ð9Þ

where the subscript denotes differentiation with respect to y. The boundary conditions (4) and (5) take the following form fy ð0; sÞ ¼ sin s;

fð0; sÞ ¼ 0;

fyyy ð0; sÞ ¼ 0;

hð0; sÞ ¼ 1; ð10Þ

fy ð1; sÞ ¼ 0;

fyy ð1; sÞ ¼ 0;

hð1; sÞ ¼ 0:

ð11Þ

In above equations K ¼ g0 b=qt2 is couple stress parameter, S ¼ x=b is unsteady parameter, b ¼ rB20 =qb þ t/=kb is the magneto-porous parameter, Pr ¼ lcp =k is the Prandtl number and k ¼ Q=cqcp is the heat source ðk > 0Þ or sink ðk < 0Þ parameter. The parameter K ¼ g0 qb=l2 ¼ l2 bq=l is called couple stress parameter. It has been pointed out by Stokes [34] that the effects of couple stress are quite large for large values of K ¼ l2 bq=l, where bq=l is a typical length scale associated pﬃﬃﬃﬃﬃﬃﬃﬃ with the flow and l ¼ g=l is a material coefficient which is a function of molecular dimensions of the liquid. The parameter l will be greatly varied for different liquids. For instant, the length of polymer chain may be a million time the diameter of a water molecule. This is the size dependent effect associated with the couple stress fluid theory. For small value of K the

effects of couple stress diminish and fluid behaves like a Newtonian fluid. The quantities of interest are skin friction coefficient and local Nusselt number which are defined as Cf ¼

sw ; qu2w

Nux ¼

xqw ; kðTw T1 Þ

where

@u g0 @ 3 u ; sw ¼ l @ y y¼0 q @ y3 y¼0

ð12Þ @T qw ¼ k ; @ y y¼0

ð13Þ

are skin friction and heat transfer at the wall, respectively. In view of transformations (6) and (7), we get Re1=2 x Cf ¼ fyy ð0; sÞ Kfyyyy ð0; sÞ;

ð14Þ

Nux ¼ hy ð0; sÞ; Re1=2 x

ð15Þ

where Rex ¼ uw x=m is the local Reynolds number. 3. Solution by homotopy analysis method In many physical problems, the resulted differential equations are highly nonlinear nature. The computations of analytical or numerical solutions of such equations have set a challenge for the researchers. Homotopy analysis method (HAM) is one the best analytical methods to compute the series solution of such nonlinear partial as well as ordinary differential equations. This method tackles for strongly nonlinear problems even if a given nonlinear problem does not contain any small/large parameter. This method gives great freedom to choose and adjust the convergence region and rate of approximation. The homotopy analysis method is advantageous over routine numerical methods in the sense that it is free of rounding off errors which arise due to discretization procedure. Moreover, it does not require large computer memory and time [43]. This method was successfully applied by researchers to solve various nonlinear problems in various disciplines of science and engineering [44–49]. It has been emphasized by Liao that the series solution obtained by HAM is purely analytic and he demonstrated this fact in number of articles for instance [50–52]. Moreover, it has been proved by Turkyilmazoglu [53] that HAM can handle any type of nonlinearity by a suitable HAM formulation. For the series solution of Eqs. (8) and (9) subject to the boundary conditions (10) and (11), we employ homotopy analysis method (HAM). According to HAM the velocity and temperature fields can be represented as fðy; sÞ ¼ a00;0 þ

1 X 1 X 1 X j an;k yk sinðjsÞ expðnyÞ;

ð16Þ

n¼0 k¼0 j¼0

hðy; sÞ ¼ b00;0 þ

1 X 1 X 1 X j bn;k yk sinðjsÞ expðnyÞ;

ð17Þ

n¼0 k¼0 j¼0 j j and bn;k are the coefficients to be determined. The where an;k appropriate initial guesses for fðy; sÞ and hðyÞ are

f0 ðy; sÞ ¼

1 sin sð3 3 expðyÞ y expðyÞÞ; 2

h0 ðyÞ ¼ expðyÞ:

ð18Þ ð19Þ

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

4

N. Ali et al.

Further we choose £f ðfÞ ¼

@ f @f ; @y3 @y

£h ðfÞ ¼

@ f f; @y2

3

£f ½C1 þ C2 expðyÞ þ C3 expðyÞ ¼ 0;

ð20Þ

£h ½C4 expðyÞ þ C5 expðyÞ ¼ 0;

ð21Þ

where Ci ði ¼ 1 5Þ are arbitrary constants. If hf and hh denotes the nonzero auxiliary parameters then zeroth order deformation problems are constructed as follows:

2

as auxiliary linear operators having following properties

b s; pÞ f ðy; sÞ ¼ ph Nf ½ fðy; b s; pÞ; ð1 pÞ£f ½ fðy; 0 f

ð22Þ

ð1 pÞ£f ½b hðy; s; pÞ h0 ðy; sÞ b s; pÞ; ¼ phh Nh ½b hðy; s; pÞ; fðy; b b s; pÞ ¼ 0; @ fðy; s; pÞ ¼ sin s; fð0; @y b s; pÞ @ 2 fðy; ¼ 0; ¼ 0; @y2 b hð0; s; pÞ ¼ 1;

ð23Þ b s; pÞ @ fðy; @y ð24Þ

b hð1; s; pÞ ¼ 0;

where p 2 ½0; 1 is an embedding parameter and nonlinear operators Nf and Nh are Figure 2 The h-curve for velocity at 7th order of approximation: with K ¼ 0:1; b ¼ 0:2; S ¼ 0:1 and s ¼ 0:5p.

5b 3b b s; pÞ ¼ K @ fðy; s; pÞ þ @ fðy; s; pÞ S Nf ½ fðy; @y5 @y3 2b b s; pÞ @ fðy; s; pÞ b @ 2 fðy; þ fðy; s; pÞ @[email protected] @y2 !2 b s; pÞ b s; pÞ @ fðy; @ fðy; ; b @y @y

ð25Þ

Figure 3 The h-curve for temperature at 7th order of approximation with K ¼ 0:1; S ¼ 0:1; b ¼ 0:2; k ¼ 0:5; Pr ¼ 0:1 and s ¼ 0:5p. Table 1 Comparison of values of f00 ð0; sÞ with Refs. [30] with K ¼ 0 and [31]. S

b

s

Zheng et al. [31]

Abbas et al. [30]

Present results

1:0

12

1:5p 5:5p 9:5p

11:678565 11:678706 11:678656

11:678656 11:678707 11:678656

11:678565 11:678706 11:678656

Table 2 Comparison of values of f00 ð0; sÞ for different values of b when K ¼ 0; S ¼ 0 and s ¼ p=2. b

Hayat et al. [20]

Turkyilmazoglu [21]

Present results

0 0:5 1:0 1:5 2:0

1:000000 1:224747 1:414217 1:581147 1:732057

1:00000000 1:22474487 1:41421356 1:58113883 1:73205081

1:000000 1:224747 1:414217 1:581147 1:732057

Figure 4 Time series of the velocity profile f0 at a distance y ¼ 0:25 from the sheet in the time period s 2 ½0; 10p: (a) The effects of couple stress parameter K with S ¼ 1 and b ¼ 0:1, (b) The effects of magneto-porous parameter b with S ¼ 0:5 and K ¼ 0:1.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

MHD flow and heat transfer of couple stress fluid

5

b s; pÞ Nh ½ b hðy; s; pÞ; fðy; ¼

Differentiating zeroth order deformations equations (22) and (23) m-times with respect to p, then setting p ¼ 0 and finally dividing by m!, we get the following mth-order deformation problem

2b

b @ hðy; s; pÞ b s; pÞ @ hðy; s; pÞ þ Pr fðy; @y2 @y ! @b hðy; s; pÞ þ kb hðy; s; pÞ : S @s

ð26Þ

The zeroth-order deformation problems defined above have the following solutions corresponding to p ¼ 0 and p ¼ 1 b s; 0Þ ¼ f ðy; sÞ; fðy; 0 b hðy; s; 0Þ ¼ h0 ðy; sÞ;

b s; 1Þ ¼ fðy; sÞ; fðy; b hðy; s; 1Þ ¼ hðy; sÞ:

ð27Þ ð28Þ

b s; pÞ and b Expanding fðy; hðy; s; pÞ in a Taylor’s series with respect to p, we get b s; pÞ ¼ fb0 ðy; sÞ þ fðy;

1 X

fm ðy; sÞpm ;

ð29Þ

£f ½ fbm ðy; sÞ ,m fm1 ðy; sÞ ¼ hf Rmf ðy; sÞ;

ð32Þ

£f ½b h m ðy; sÞ ,m hm1 ðy; sÞ ¼ hh Rhm ðy; sÞ;

ð33Þ

@fm ðy; s; 0Þ @fm ðy; s; 0Þ ¼ @y @y y¼0 y¼1 2 @f ðy; s; 0Þ ¼ m 2 ¼ 0; @y y¼1

fm ð0; s; pÞ ¼ 0;

hm ð0Þ ¼ 0; hm ð1Þ ¼ 0; @ 5 fm1 @ 3 fm1 @ 2 fm1 @f b m1 þ S @y5 @y3 @[email protected] @y m1 2 X @ f f @f fm1k 2k mk1 k ; þ @y @y @y k¼0

ð30Þ

m¼1

fm ðy; sÞ ¼

b s; pÞ 1 @ m fðy; ; @pm m!

hm ðy; sÞ ¼

hðy; s; pÞ 1 @m b ; @pm m!

Rhm ðy; sÞ ¼ ð31Þ

ð35Þ

Rmf ðy; sÞ ¼ K

m¼1 1 X b hm ðy; sÞpm ; hðy; s; pÞ ¼ b h 0 ðy; sÞ þ

ð34Þ

@ 2 hm1 @hm1 Pr S @y2 @s ! m 1 X @hm1k þ Pr fk þ khm1 ; @y k¼0

ð36Þ

ð37Þ

Figure 5 Transverse profiles of the velocity field f0 for four different values of couple stress parameter K in the fifth period s 2 ½8p; 10p for which a periodic velocity field has been reached: (a) s ¼ 8:5p, (b) s ¼ 9p, (c) s ¼ 9:5p and (d) s ¼ 10p, with S ¼ 1 and b ¼ 0:1.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

6

N. Ali et al.

,m ¼

0; 1;

m 6 1; m > 1;

ð38Þ

4. Convergence of HAM solution It is well established fact that the convergence of the HAM solution is largely dependent on the proper choice of the auxiliary parameters hf and hh . For a particular set of parameters the convergence region can be obtained by plotting h-curves. Figs. 2 and 3 present two such curves showing the plausible values of hf and hh for a given set of parameters. We note from these figures that for convergent solution 1:2 6 hf < 0:2 and 1:5 6 hh < 0:2. The purpose of showing such curves is just to emphasize that in principle for any physical choice of parameters of the problem, a convergent solution can be obtained. The values of f00 ð0; sÞ obtained by HAM in the present paper are also compared with the corresponding values reported in Refs. [20,21,30,31] in Tables 1 and 2. Both tables show that our results are in excellent agreement with the existing results in the literature. 5. Results and discussion The homotopy analysis method is a powerfull technique to solve nonlinear partial and ordinary differential equations with

initial and boundary conditions. The set of nonlinear partial differential equations (8) and (9) with boundary conditions (10) and (11) is solved analytically by means of homotopy analysis method and the effects of involved parameters are illustrated through graphs. In this section, we are going to present the detail analysis of flow and heat transfer of couple stress fluid over oscillating stretching surface in the presence of heat source and sink. Fig. 4(a) and (b) are plotted to show the behavior of couple stress parameter K and magneto-porous parameter b on the time series of velocity f0 at a fixed location y ¼ 0:25 from the sheet, respectively. In Fig. 4(a) the effects of couple stress parameter K are illustrated by keeping S ¼ 1 and b ¼ 0:1. From this figure we observe that an increase in couple stress parameter K results in an increase in amplitude of velocity. This is perhaps due to the presence of couple stresses. Fig. 4 (b) points out the effects of magneto-porous parameter b by keeping other parameters constant. This figure shows that the amplitude of velocity decreases by increasing magnetoporous parameter b. Fig. 5(a)–(d) demonstrate the effects of couple stress parameter K on velocity profile f0 at four different time instants s ¼ 8:5p; s ¼ 9p; s ¼ 9:5p and s ¼ 10p. Fig. 5(a) characterizes the effects of couple stress parameter K on the velocity profile f0 at s ¼ 8:5p. This figure depicts that the velocity f0 decrease as we increase K. This decrease in velocity results in decrease in

Figure 6 Transverse profiles of the velocity field f0 for four different values of magneto-porous parameter b in the fifth period s 2 ½8p; 10p for which a periodic velocity field has been reached: (a) s ¼ 8:5p, (b) s ¼ 9p, (c) s ¼ 9:5p, and (d) s ¼ 10p, with S ¼ 0:5 and K ¼ 0:1.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

MHD flow and heat transfer of couple stress fluid boundary layer thickness. In Fig. 5(b), the behavior of couple stress parameter K on the f0 is shown at s ¼ 9p. This figure shows an increase in magnitude of velocity at this instant. The effects of K at s ¼ 9:5p and s ¼ 10p are illustrated in Fig. 5(c) and (d), respectively. It is observed from these figures that at s ¼ 9:5p and s ¼ 10p, the velocity f0 decreases with K. It is interesting to note that the effects of couple stress are to the increase the amplitude of velocity when f0 ¼ 1 at the surface. For other instants when f0 ¼ 1 or 0, the magnitude of velocity decreases due to the presence of couple stresses.

Figure 7 Time series of the skin-friction coefficient Re1=2 x Cf in the first five periods s 2 ½0; 10p: (a) effects of couple stress parameter K with S ¼ 0:1 and b ¼ 0:1 and (b) effects of magneto-porous parameter b with S ¼ 0:5 and K ¼ 0:1.

7 The effects of magneto-porous parameter b on velocity are illustrated in Fig. 6(a)–(d) at four different times keeping other parameters fixed. In Fig. 6(a) the effects of magneto-porous parameter b are plotted at s ¼ 8:5p. From this figure it can be examined that velocity decreases with increase in magneto-porous parameter b. The effects of magneto-porous parameter on velocity at other three time instants are illustrated in Fig. 6(c) and (d). Here, again it is observed that the magnitude of velocity decreases by increasing b. The variation of skin friction coefficient with couple stress parameter K and magneto-porous parameter b in the first five periods s 2 ½0; 10p is shown in Fig. 7(a) and (b), respectively. Fig. 7(a) shows the time series of skin friction coefficient for different values of couple stress parameter K by taking S ¼ 0:1 and b ¼ 0:1. From this figure we observe that skin friction coefficient varies periodically due to periodic motion of the surface and its amplitude increases with couple stress parameter K. This observation reflects that skin friction for a Newtonian fluid flowing over an oscillatory stretching sheet is less in comparison with its value for a couple stress fluid performing the same motion. The effects of magneto-porous parameter b on skin friction coefficient are illustrated in Fig. 7(b). This figure depicts that amplitude of skin friction coefficient increases by increasing magneto-porous parameter b. Such effects of b and M are expected because the fact that magnetic force and drag force offered by the porous medium acts as a resistance to the flow. Figs. 8–12 present the effects of Prandtl number Pr, magneto-porous parameter b, couple stress parameter K and heat source/sink parameter k on the temperature field. Fig. 8 reveals the effects of Prandtl number Pr on temperature profile for both heat source and sink cases. It is seen that temperature is a decreasing function of Pr in both cases. The fact is that the increase in Prandtl number results in low thermal conductivity, as a result conduction as well as thermal boundary layer thickness decreases and hence we observe a decrease in temperature. It is observed that the presence of heat sink enhances the decrease in temperature by increasing Prandtl number. In Fig. 9, the influence of magneto-porous parameter b on temperature field is explained graphically by keeping other parameters fixed. This figure shows that temperature increases with magneto-porous parameter b. The influence of couple stress

Figure 8 The effects of Prandtl number Pr on temperature field h at s ¼ 8:5p with K ¼ 0:1; S ¼ 0:1; b ¼ 0:2, (a) in heat source case k ¼ 0:3, (b) in heat sink case k ¼ 0:3.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

8

Figure 9 The effects of magneto-porous parameter b on temperature field h at s ¼ 8:5p with Pr ¼ 1:5; K ¼ 0:1; S ¼ 0:1; k ¼ 0:3.

N. Ali et al.

Figure 12 The effects of heat sink parameter k on temperature field h at s ¼ 8:5p with Pr ¼ 0:4; b ¼ 0:5; K ¼ 0:1; S ¼ 0:1.

Table 3 The numerical values of the Local Nusselt number for different values of Prandtl number Pr, couple stress parameter K, magneto-porous parameter b and heat source parameter k at time s ¼ p=2 with S ¼ 1.

Figure 10 The effects of couple stress number K on temperature field h at s ¼ 8:5p with Pr ¼ 1:5; b ¼ 5; k ¼ 0:4; S ¼ 0:6.

Figure 11 The effects of heat source parameter k on temperature field h at s ¼ 8:5p with Pr ¼ 0:4; b ¼ 0:5; K ¼ 0:1; S ¼ 0:1.

parameter K on temperature is shown in Fig. 10, which depicts that increase in couple stress parameter results in increase in temperature. Since we are discussing the temperature field in the presence of heat source/sink so effects of this parameter are very important for both the cases. The effects of heat source/sink parameter are illustrated in Figs. 11 and 12 by keeping Pr ¼ 0:4; b ¼ 0:5; K ¼ 0:1; S ¼ 0:1. It is noted from Fig. 11 that as we increase the strength of the heat source, the temperature increases. This is due to the fact that heat source can add more heat to the stretching sheet which increases its temperature and thus the temperature of fluid rises. Moreover, the thermal boundary layer is also found to increase by increasing the strength of heat source. Fig. 12 is

Pr

k

K

b

Rex1=2 Nux

0:5 1:0 1:5 2:0 0:5

0:2

5:0

5:0

0:596791 0:634885 0:671806 0:705075 0:596791 0:547204 0:496478 0:44459 0:573683 0:573021 0:572122 0:571885 0:643221 0:630873 0:619684 0:609657

0:2 0:4 0:6 0:8 0:2

0:2 0:4 0:8 1:0 5:0

0:5 1:5 2:5 3:5

plotted to observe the effects of heat sink by keeping other parameters constant. This figure shows opposite results i.e., the temperature decreases by increasing the strength of heat sink. This is because, when strength of heat sink increases, more heat is removed from the sheet and thus a decrease in thermal boundary-layer thickness and temperature is observed. This result is of key importance for the flows where heat transfer is of prime importance. The numerical values of Nusselt number are shown in Table 3. It is found that the Nusselt number increases with Prandtl number while it decreases with increase in heat source, couple stress and magneto-porous parameter. 6. Concluding remarks Flow and heat transfer of a couple stress fluid over an oscillatory stretching sheet in the presence of heat source/sink is investigated using HAM. It is observed that flow and heat transfer characteristics are greatly influenced by the presence of couple stress and heat source/sink parameter. In fact, the skin friction coefficient increases by increasing couple stress and magneto-porous parameter. It is also interesting to note that rate of heat transfer is enhanced by increasing

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

MHD flow and heat transfer of couple stress fluid magneto-porous parameter, heat source and couple stress parameter, while it decreases for large values of Prandtl number and heat sink parameter. Such observations may have interesting implications where it is desired to reduce skin friction and enhance heat transfer. Acknowledgments We are grateful to the reviewers for their useful comments. The second author is grateful to the Higher Education Commission of Pakistan for financial assistance. References [1] B.C. Sakiadis, Boundary layer behavior on continuous solid surfaces: I boundary layer equations for two dimensional and axisymmetric flow, AIChE J. 7 (1961) 26–28. [2] L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phy. (ZAMP) 21 (1970) 645–647. [3] K.R. Rajagopal, T.Y. Na, A.S. Gupta, Flow of a visco-elastic fluid over a stretching sheet, Rheol. Acta 23 (1984) 213–215. [4] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (1977) 744–746. [5] B.S. Dandapat, A.S. Gupta, Flow and heat transfer in a viscoelastic fluid over a stretching sheet, Int. J. Nonlinear Mech. 24 (1989) 215–219. [6] J.B. McLeod, K.R. Rajagopal, On the uniqueness of flow of a Navier-Stokes fluid due to stretching boundries, Arch. Rat. Mech. Anal. 98 (1987) 385. [7] D. Rollins, K. Vajravelu, Heat transfer in a second-order fluid over a continuous stretching surface, Acta Mech. 89 (1991) 167– 178. [8] K.R. Rajagopal, R.K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mech. 113 (1995) 233– 239. [9] R. Cortell, A note on flow and heat transfer of a viscoelastic liquid over a stretching sheet, Int. J. Non-Linear Mech. 41 (2006) 78–85. [10] R. Cortell, Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to suction and to a transverse magnetic field, Int. J. Heat Mass Transfer 49 (2006) 1851–1856. [11] R. Nazar, N. Amin, D. Filip, I. Pop, Stagnation point flow of a micropolar fluid towards a stretching sheet, Int. J. Non-Linear Mech. 39 (2004) 1227–1235. [12] A. Ishak, R. Nazar, I. Pop, Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet, Mecccanica 43 (2008) 411–418. [13] P.D. Ariel, Axisymmetric flow of a second grade fluid past a stretching sheet, Int. J. Eng. Sci. 39 (2001) 529–553. [14] T. Hayat, M. Sajid, Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet, Int. J. Heat Mass Transfer 50 (2007) 75–84. [15] T. Hayat, Z. Abbas, M. Sajid, Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction, Phys. Lett. A 372 (2008) 2400–2408. [16] T. Hayat, T. Javed, Z. Abbas, Slip flow and heat transfer of a second grade fluid past a stretching sheet through a porous space, Int J. Heat Mass Transfer 51 (2008) 4528–4534. [17] M. Sajid, I. Ahmad, T. Hayat, M. Ayub, Unsteady flow and heat transfer of a second grade fluid over a stretching sheet, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 96–108. [18] T. Hayat, T. Javed, M. Sajid, Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid, Acta Mech. 191 (2007) 219–229.

9 [19] M. Sajid, T. Hayat, Non-similar series solution for boundary layer flow of a third-order fluid over a stretching sheet, Appl. Math. Comput. 189 (2007) 1576–1585. [20] T. Hayat, M. Mustafa, I. Pop, Heat and mass transfer for Soret and Dufour’s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1183–1196. [21] M. Turkyilmazoglu, The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface, Int. J. Mech. Sci. 77 (2013) 263– 268. [22] M. Turkyilmazoglu, Three dimensional MHD flow and heat transfer over a stretching/shrinking surface in a viscoelastic fluid with various physical effects, Int. J. Heat Mass Transfer 78 (2014) 150–155. [23] M. Turkyilmazoglu, Multiple solutions of hydromagnetic permeable flow and heat for viscoelastic fluid, J. Thermophys. Heat Transfer 25 (2011) 595–605. [24] M. Turkyilmazoglu, Analytic heat and mass transfer of the mixed hydrodynamic/thermal slip MHD viscous flow over a stretching sheet, Int. J. Mech. Sci. 53 (2011) 886–896. [25] M. Turkyilmazoglu, Multiple solutions of heat and mass transfer of MHD slip flow for the viscoelastic fluid over a stretching sheet, Int. J. Therm. Sci. 50 (2011) 2264–2276. [26] M. Turkyilmazoglu, An analytical treatment for the exact solutions of MHD flow and heat over two–three dimensional deforming bodies, Int. J. Heat Mass Transfer 90 (2015) 781– 789. [27] C.Y. Wang, Nonlinear streaming due to the oscillatory stretching of a sheet in a viscous, Acta Mech. 72 (1988) 261– 268. [28] B. Siddappa, S. Abel, V. Hongunti, Oscillatory motion of a viscoelastic fluid past a stretching sheet, ll Nuovo Cimento D 17 (1995) 53–60. [29] Z. Abbas, Y. Wang, T. Hayat, M. Oberlack, Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface, Int. J. Numer. Methods Fluids 59 (2009) 443– 458. [30] Z. Abbas, Y. Wang, T. Hayat, M. Oberlack, Hydromagnetic flow in a viscoelastic fluid due to the oscillatory stretching surface, Int. J. Nonlinear Mech. 43 (2008) 783–797. [31] L.C. Zheng, X. Jin, X.X. Zhang, J.H. Zhang, Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects, Acta Mech. Sinica 29 (5) (2013) 667–675. [32] N. Ali, S.U. Khan, Z. Abbas, Hydromagnetic flow and heat transfer of a jeffrey fluid over an oscillatory stretching surface, Z. Naturforsch. A 70 (7) (2015) 567–576. [33] S.U. Khan, N. Ali, Z. Abbas, Hydromagnetic flow and heat transfer over a porous oscillating stretching surface in a viscoelastic fluid with porous medium, PLOS ONE (2015), http://dx.doi.org/10.1371/journal.pone.0144299, December 14. [34] V.K. Stokes, Couple Stress Fluids 9 (1966) 1709–1715. [35] D. Srinivasacharya, K. Kaladhar, Soret and Dufour effects in a mixed convection couple stress fluid with heat and mass fluxes, Lat. Am. Appl. Res. 41 (2011) 353–358. [36] L.M. Srivastava, Peristaltic transport of a couple stress fluid, Rheol. Acta 25 (1986) 638–641. [37] E.F. El-Shehawey, K.S. Mekheimer, Couple-stresses in peristaltic transport of fluids, J. Phys. D: Appl. Phys. 27 (1994) 1163. [38] D. Pal, N. Rudraiah, R. Devanathan, A couple stress model of blood flow in the microcirculation, Bull. Math. Biol. 50 (1988) 329. [39] H.L. Chiang, J.R. Lin, C.H. Hsu, Y.P. Chang, Linear stability analysis of a rough short journal bearing lubricated with nonnewtonian fluids, Tribol. Int. 17 (2004) 867–877.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

10 [40] N.B. Naduvinamani, P.S. Hiremath, G. Gurubasavaraj, Effect of surface roughness on the couple-stress squeeze film between a sphere and a flat plate, Tribol. Int. 38 (2005) 451–458. [41] C.W.C. Jian, C.K. Chen, Chaos and bifurcation of a flexible rotor supported by porous squeeze couple stress fluid film journal bearings with non-linear suspension, Chaos Solit. Fract. 35 (2008) 358–375. [42] R.F. Lu, J.R. Lin, A theoretical study of combined effects of non-Newtonian rheology and viscosity-pressure dependence in the sphere-plate squeeze-film system, Tribo. In. 40 (1) (2007) 125–131. [43] S.J. Liao, Advance in the Homotopy Analysis Method, vol. 1, World Scientific Publishing, 5 Toh Tuck Link, Singapore, 2014. [44] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A 360 (2006) 109–113. [45] M. Turkyilmazoglu, Analytic approximate solutions of rotating disk boundary layer flow subject to a uniform suction or injection, Int. J. Mech. Sci. 52 (2010) 1735–1744. [46] S. Abbasbandy, Homotopy analysis method for heat radiation equations, Int. Commun. Heat Mass Transfer 34 (2007) 380– 387.

N. Ali et al. [47] I.A. Abdallah, Homotopy analytical solution of MHD fluid flow and heat transfer problem, Appl. Math. Inform. Sci. 3 (2009) 223–233. [48] T. Hayat, M. Sajid, Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid, Int. J. Eng. Sci. 45 (2007) 393–401. [49] B. Raftari, K. Vajravelu, Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4149. [50] S.J. Liao, A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. Fluid Mech. 385 (1999) 101–128. [51] S.J. Liao, An explicit, totally analytic approximation of Blasius’ viscous flow problems totally analytic approximation of Blasius’ viscous flow problems, Int. J. Non-Linear Mech. 34 (4) (1999) 759–778. [52] S.J. Liao, On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. Fluid Mech. 488 (2003) 189–212. [53] M. Turkyilmazoglu, Solution of the Thomas-Fermi equation with a convergent approach, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4097–4103.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium Nasir Ali a, Sami Ullah Khan a,*, Muhammad Sajid a, Zaheer Abbas b a b

Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

Received 25 December 2015; revised 9 February 2016; accepted 20 February 2016

KEYWORDS Boundary layer flow; Couple stress fluid; Oscillatory stretching sheet; Homotopy analysis method

Abstract This article presents the MHD flow and heat transfer of a couple stress fluid over oscillatory stretching sheet embedded in a porous medium in the presence of heat source/sink. The unsteady flow problem is reduced to two coupled partial differential equations using dimensionless variables. Homotopy analysis method is employed to obtain the solution of these equations. Based on the solution of these equations an extensive analysis is performed to investigate the effects of various flow parameters on velocity and temperature fields, skin friction coefficient and Nusselt number. It is found that the presence of couple stress in viscous fluid increases the amplitude of oscillations in velocity and skin friction coefficient. It is also noticed that temperature increases by increasing heat source parameter. Moreover, the numerical values of local Nusselt number are calculated and shown in tabular form. It is found that the Nusselt number increases by increasing Prandtl number while it decreases by increasing couple stress parameter. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction It is well established fact that Navier–Stokes equations are not adequate choice in describing the flows of nonNewtonian fluids. Non-Newtonian fluids frequently occur in many industrial and technological processes. If the flow of non-Newtonian fluid is considered over a stretching sheet then it become relevant to many industrial applications such a extrusion of plastic sheets, paper production, crystal * Corresponding author. Tel.: +92 51 9019756. E-mail address: [email protected] (S. Ullah Khan). Peer review under responsibility of Faculty of Engineering, Alexandria University.

growing and glass blowing, drawing of copper wires, continuous stretching of plastic films. The pioneering work on viscous flow due to continuous moving sheet was performed by Sakiadis [1]. His analysis was extended for stretching sheet by Crane [2]. Crane’s problem was extended by Rajagopal et al. [3] for viscoelastic fluid. Gupta and Gupta [4] extended the analysis of Crane by considering suction/blowing at the sheet surface. Dandapat and Gupta [5] discussed the heat transfer characteristics of viscoelastic fluid over a stretching sheet. A study of uniqueness of boundary value problem arising due to a stretching boundary in a viscous fluid was performed by McLeod and Rajagopal [6]. Rollins and Vajravelu [7] discussed heat transfer characteristics of second-order fluid over a continuos stretching surface with internal heat generation

http://dx.doi.org/10.1016/j.aej.2016.02.018 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

2 or absorption. The stretching sheet problem for viscoelastic Oldroyd-B model was extended by Rajagopal and Bhatnagar [8]. Cortell [9] presented a study of flow and heat transfer of hydromagnetic second-order fluid past a stretching sheet. In another paper [10], Cortell included magnetic and suction effects while discussing the flow and heat transfer of second grade fluid due to stretching sheet. The study of stagnationpoint flow of micropolar fluid toward a stretching sheet was initiated by Nazar et al. [11]. Combined forced and natural convection flow of micropolar fluid over a stretching sheet in the vicinity of stagnation point was discussed by Ishak et al. [12]. Ariel [13] put forward a study of flow of second grade fluid past a radially stretching sheet. Ariel work was extended by Hayat and Sajid [14] to include the heat transfer characteristics. Some other contribution by Hayat and his co-workers regarding heat transfer analysis over a stretching surface can be found in references [15–18]. The investigation of non-similar flow of third order fluid over a stretching sheet was due to Sajid and Hayat [19]. Hayat et al. [20] analyzed combined heat and mass transfer characteristics in boundary layer flow of viscoelastic fluid over linearly stretching vertical surface by using homotopy analysis method. Later on, Turkyilmazoglu [21] computed exact solution of hydromagnetic flow of viscoelastic fluid in the presence of Soret and Dufour effects. Turkyilmazoglu [22] also developed closed-form solutions for three dimensional flow and heat transfer of a electrically conducting viscoelastic fluid over a porous stretching/shrinking sheet. The multiplicity of the solutions in boundary layer flows over stretching/shrinking sheet is also addressed by Turkyilmazoglu. In fact, it was shown by him that multiple solutions exist for twodimensional flow over stretching sheet and in the case of shrinking sheet triple solutions can be detected [23–25]. In another paper, Turkyilmazoglu [26] computed exponential series type solution in the hydromagnetic and heat transfer in two–three dimensional flow over stretching or shrinking surface. According to him exponential series type solution exists with finite and infinite terms in the expansion depending upon prevailing parameters in the governing equations. To best of our knowledge, Wang [27] initiated the study of viscous fluid due to oscillatory stretching sheet. His work was extended by Saddipa et al. [28] by using constitutive equation of Walter’s B model. Later on, Abbas et al. [29] complemented the Wang ’s analysis by including the effects of slip and heat transfer. They also utilized the constitutive relation of second order fluid [30] to study Wang’s problem. Zheng et al. [31] provided an analysis of unsteady heat and mass transfer in MHD flow over an oscillatory stretching sheet with Soret and Dufour effects. They have employed homotopy analysis method and delineated the effects of various emerging parameters on flow and heat transfer characteristics. Their study indicates a decrease in velocity of fluid by increasing unsteady parameter, magnetic parameter and suction/injection parameter. They also found an increase in thermal boundary layer thickness with magnetic parameter and unsteady parameter. Ali et al. [32] analyzed the unsteady flow of Jeffrey fluid over an oscillatory stretching surface. Khan et al. [33] used an implicit finite difference scheme to analyze the heat transfer effects in the hydromagnetic flow of viscoelastic fluid (second grade fluid) over an oscillatory stretching surface. However, so far no one has considered the study of unsteady boundary layer flow of a couple stress fluid through a porous medium over an

N. Ali et al. oscillatory stretching surface in the presence of heat generation or absorption. Hence we feel appropriate to consider the study of such flows, as such type of flows may find a number of industrial applications. The theory of couple stress fluid was developed by Stokes [34] as a simple generalization of classical viscous theory that sustains couple stresses and body couples. The main effects of couple stress fluid are to introduce a size dependent effect which is not present in classical viscous theories. Blood, lubricants containing small amount of additives, electro-rheological and synthetic fluids are examples of couple stress fluid [35]. Couple stress theory has been successfully applied to many problems in biomechanics and lubrications area by Srivastava [36], El-Shehawey and Mekheimer [37], Pal et al. [38], Chiang et al. [39], Naduvinamani et al. [40], Jian and Chen et al. [41] and Lu and Lin [42]. The presentation of the paper is as follows. Flow analysis is presented in Section 2. Homotopy analysis method is employed for solution of governing equations in Section 3. Convergence of solution is discussed in Section 4. Quantitative analysis of results is presented in Section 5. Final remarks are presented at the end of the paper in Section 6. 2. Flow analysis Let us consider the unsteady two-dimensional magnetohydrodynamics flow of an incompressible fluid over an oscillatory stretching sheet. In a rectangular coordinate system ð x; yÞ, the sheet is assumed to coincide with y ¼ 0 and flow takes place in the semi-infinite porous space ðy > 0Þ of constant permeability (see Fig. 1). A constant magnetic field of strength B0 is applied perpendicular to the stretching surface. It is assumed that the sheet is stretched with velocity ux ¼ b x sin xt in which b is the stretching rate and x denotes the angular velocity. Moreover, sheet is maintained at uniform temperature Tw ð> T1 Þ, where T1 is the temperature of ambient fluid. Now problem is to determine the velocity and temperature fields inside the fluid satisfying the boundary conditions at the wall and far away from the wall. To this end, we apply boundary layer approximation on continuity and momentum equations and write @u @v þ ¼ 0; @ x @ y

ð1Þ

Figure 1

Geometry of the problem.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

MHD flow and heat transfer of couple stress fluid

3

@u @u @u @2u g @4u rB2 t/ þu þv ¼m 2 0 0 u u; 4 @t @ x @ y @ y k q @ y q

ð2Þ

@T @T @T k @2T Q þu þv ¼ þ ðT T1 Þ; @t @ x @ y qcp @ y2 qcp

ð3Þ

where u and v are velocity component along x and y-directions, respectively, m is the kinematic viscosity, q is the density, g0 is the material constant for the couple stress fluid, Q is the heat source coefficient, r is electric conductivity and B0 is the strength of constant applied magnetic field. The flow is subjected to the following boundary conditions u ¼ ux ¼ b x sin xt; y ¼ 0;

t > 0;

u ¼ 0;

@u ¼ 0; @ y

v ¼ 0;

@2u ¼ 0; @ y2

T ¼ Tw

at ð4Þ

T ! T1

at

y ! 1:

ð5Þ

The first two boundary conditions in (4) result from the assumption of no-slip at the surface while the third condition is consequence of the assumption of vanishing of couple stresses at the wall. Since flow is in unbounded domain that is why augmented boundary condition is imposed at y ! 1. To nondimensionalize the flow problem, we use the dimensionless variables [30,31] rﬃﬃﬃ pﬃﬃﬃﬃﬃ b y¼ y; s ¼ tx; u ¼ b xfy ðy; sÞ; v ¼ mbfðy; sÞ; ð6Þ m hðy; sÞ ¼

T T1 : Tw T1

ð7Þ

In view of above transformations, the continuity equation is identically satisfied and Eqs. (2) and (3) are transformed as follows fyyy Sfys f2y þ ffyy bfy Kfyyyyy ¼ 0;

ð8Þ

hyy þ Pr fhy Shs þ kh ¼ 0;

ð9Þ

where the subscript denotes differentiation with respect to y. The boundary conditions (4) and (5) take the following form fy ð0; sÞ ¼ sin s;

fð0; sÞ ¼ 0;

fyyy ð0; sÞ ¼ 0;

hð0; sÞ ¼ 1; ð10Þ

fy ð1; sÞ ¼ 0;

fyy ð1; sÞ ¼ 0;

hð1; sÞ ¼ 0:

ð11Þ

In above equations K ¼ g0 b=qt2 is couple stress parameter, S ¼ x=b is unsteady parameter, b ¼ rB20 =qb þ t/=kb is the magneto-porous parameter, Pr ¼ lcp =k is the Prandtl number and k ¼ Q=cqcp is the heat source ðk > 0Þ or sink ðk < 0Þ parameter. The parameter K ¼ g0 qb=l2 ¼ l2 bq=l is called couple stress parameter. It has been pointed out by Stokes [34] that the effects of couple stress are quite large for large values of K ¼ l2 bq=l, where bq=l is a typical length scale associated pﬃﬃﬃﬃﬃﬃﬃﬃ with the flow and l ¼ g=l is a material coefficient which is a function of molecular dimensions of the liquid. The parameter l will be greatly varied for different liquids. For instant, the length of polymer chain may be a million time the diameter of a water molecule. This is the size dependent effect associated with the couple stress fluid theory. For small value of K the

effects of couple stress diminish and fluid behaves like a Newtonian fluid. The quantities of interest are skin friction coefficient and local Nusselt number which are defined as Cf ¼

sw ; qu2w

Nux ¼

xqw ; kðTw T1 Þ

where

@u g0 @ 3 u ; sw ¼ l @ y y¼0 q @ y3 y¼0

ð12Þ @T qw ¼ k ; @ y y¼0

ð13Þ

are skin friction and heat transfer at the wall, respectively. In view of transformations (6) and (7), we get Re1=2 x Cf ¼ fyy ð0; sÞ Kfyyyy ð0; sÞ;

ð14Þ

Nux ¼ hy ð0; sÞ; Re1=2 x

ð15Þ

where Rex ¼ uw x=m is the local Reynolds number. 3. Solution by homotopy analysis method In many physical problems, the resulted differential equations are highly nonlinear nature. The computations of analytical or numerical solutions of such equations have set a challenge for the researchers. Homotopy analysis method (HAM) is one the best analytical methods to compute the series solution of such nonlinear partial as well as ordinary differential equations. This method tackles for strongly nonlinear problems even if a given nonlinear problem does not contain any small/large parameter. This method gives great freedom to choose and adjust the convergence region and rate of approximation. The homotopy analysis method is advantageous over routine numerical methods in the sense that it is free of rounding off errors which arise due to discretization procedure. Moreover, it does not require large computer memory and time [43]. This method was successfully applied by researchers to solve various nonlinear problems in various disciplines of science and engineering [44–49]. It has been emphasized by Liao that the series solution obtained by HAM is purely analytic and he demonstrated this fact in number of articles for instance [50–52]. Moreover, it has been proved by Turkyilmazoglu [53] that HAM can handle any type of nonlinearity by a suitable HAM formulation. For the series solution of Eqs. (8) and (9) subject to the boundary conditions (10) and (11), we employ homotopy analysis method (HAM). According to HAM the velocity and temperature fields can be represented as fðy; sÞ ¼ a00;0 þ

1 X 1 X 1 X j an;k yk sinðjsÞ expðnyÞ;

ð16Þ

n¼0 k¼0 j¼0

hðy; sÞ ¼ b00;0 þ

1 X 1 X 1 X j bn;k yk sinðjsÞ expðnyÞ;

ð17Þ

n¼0 k¼0 j¼0 j j and bn;k are the coefficients to be determined. The where an;k appropriate initial guesses for fðy; sÞ and hðyÞ are

f0 ðy; sÞ ¼

1 sin sð3 3 expðyÞ y expðyÞÞ; 2

h0 ðyÞ ¼ expðyÞ:

ð18Þ ð19Þ

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

4

N. Ali et al.

Further we choose £f ðfÞ ¼

@ f @f ; @y3 @y

£h ðfÞ ¼

@ f f; @y2

3

£f ½C1 þ C2 expðyÞ þ C3 expðyÞ ¼ 0;

ð20Þ

£h ½C4 expðyÞ þ C5 expðyÞ ¼ 0;

ð21Þ

where Ci ði ¼ 1 5Þ are arbitrary constants. If hf and hh denotes the nonzero auxiliary parameters then zeroth order deformation problems are constructed as follows:

2

as auxiliary linear operators having following properties

b s; pÞ f ðy; sÞ ¼ ph Nf ½ fðy; b s; pÞ; ð1 pÞ£f ½ fðy; 0 f

ð22Þ

ð1 pÞ£f ½b hðy; s; pÞ h0 ðy; sÞ b s; pÞ; ¼ phh Nh ½b hðy; s; pÞ; fðy; b b s; pÞ ¼ 0; @ fðy; s; pÞ ¼ sin s; fð0; @y b s; pÞ @ 2 fðy; ¼ 0; ¼ 0; @y2 b hð0; s; pÞ ¼ 1;

ð23Þ b s; pÞ @ fðy; @y ð24Þ

b hð1; s; pÞ ¼ 0;

where p 2 ½0; 1 is an embedding parameter and nonlinear operators Nf and Nh are Figure 2 The h-curve for velocity at 7th order of approximation: with K ¼ 0:1; b ¼ 0:2; S ¼ 0:1 and s ¼ 0:5p.

5b 3b b s; pÞ ¼ K @ fðy; s; pÞ þ @ fðy; s; pÞ S Nf ½ fðy; @y5 @y3 2b b s; pÞ @ fðy; s; pÞ b @ 2 fðy; þ fðy; s; pÞ @[email protected] @y2 !2 b s; pÞ b s; pÞ @ fðy; @ fðy; ; b @y @y

ð25Þ

Figure 3 The h-curve for temperature at 7th order of approximation with K ¼ 0:1; S ¼ 0:1; b ¼ 0:2; k ¼ 0:5; Pr ¼ 0:1 and s ¼ 0:5p. Table 1 Comparison of values of f00 ð0; sÞ with Refs. [30] with K ¼ 0 and [31]. S

b

s

Zheng et al. [31]

Abbas et al. [30]

Present results

1:0

12

1:5p 5:5p 9:5p

11:678565 11:678706 11:678656

11:678656 11:678707 11:678656

11:678565 11:678706 11:678656

Table 2 Comparison of values of f00 ð0; sÞ for different values of b when K ¼ 0; S ¼ 0 and s ¼ p=2. b

Hayat et al. [20]

Turkyilmazoglu [21]

Present results

0 0:5 1:0 1:5 2:0

1:000000 1:224747 1:414217 1:581147 1:732057

1:00000000 1:22474487 1:41421356 1:58113883 1:73205081

1:000000 1:224747 1:414217 1:581147 1:732057

Figure 4 Time series of the velocity profile f0 at a distance y ¼ 0:25 from the sheet in the time period s 2 ½0; 10p: (a) The effects of couple stress parameter K with S ¼ 1 and b ¼ 0:1, (b) The effects of magneto-porous parameter b with S ¼ 0:5 and K ¼ 0:1.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

MHD flow and heat transfer of couple stress fluid

5

b s; pÞ Nh ½ b hðy; s; pÞ; fðy; ¼

Differentiating zeroth order deformations equations (22) and (23) m-times with respect to p, then setting p ¼ 0 and finally dividing by m!, we get the following mth-order deformation problem

2b

b @ hðy; s; pÞ b s; pÞ @ hðy; s; pÞ þ Pr fðy; @y2 @y ! @b hðy; s; pÞ þ kb hðy; s; pÞ : S @s

ð26Þ

The zeroth-order deformation problems defined above have the following solutions corresponding to p ¼ 0 and p ¼ 1 b s; 0Þ ¼ f ðy; sÞ; fðy; 0 b hðy; s; 0Þ ¼ h0 ðy; sÞ;

b s; 1Þ ¼ fðy; sÞ; fðy; b hðy; s; 1Þ ¼ hðy; sÞ:

ð27Þ ð28Þ

b s; pÞ and b Expanding fðy; hðy; s; pÞ in a Taylor’s series with respect to p, we get b s; pÞ ¼ fb0 ðy; sÞ þ fðy;

1 X

fm ðy; sÞpm ;

ð29Þ

£f ½ fbm ðy; sÞ ,m fm1 ðy; sÞ ¼ hf Rmf ðy; sÞ;

ð32Þ

£f ½b h m ðy; sÞ ,m hm1 ðy; sÞ ¼ hh Rhm ðy; sÞ;

ð33Þ

@fm ðy; s; 0Þ @fm ðy; s; 0Þ ¼ @y @y y¼0 y¼1 2 @f ðy; s; 0Þ ¼ m 2 ¼ 0; @y y¼1

fm ð0; s; pÞ ¼ 0;

hm ð0Þ ¼ 0; hm ð1Þ ¼ 0; @ 5 fm1 @ 3 fm1 @ 2 fm1 @f b m1 þ S @y5 @y3 @[email protected] @y m1 2 X @ f f @f fm1k 2k mk1 k ; þ @y @y @y k¼0

ð30Þ

m¼1

fm ðy; sÞ ¼

b s; pÞ 1 @ m fðy; ; @pm m!

hm ðy; sÞ ¼

hðy; s; pÞ 1 @m b ; @pm m!

Rhm ðy; sÞ ¼ ð31Þ

ð35Þ

Rmf ðy; sÞ ¼ K

m¼1 1 X b hm ðy; sÞpm ; hðy; s; pÞ ¼ b h 0 ðy; sÞ þ

ð34Þ

@ 2 hm1 @hm1 Pr S @y2 @s ! m 1 X @hm1k þ Pr fk þ khm1 ; @y k¼0

ð36Þ

ð37Þ

Figure 5 Transverse profiles of the velocity field f0 for four different values of couple stress parameter K in the fifth period s 2 ½8p; 10p for which a periodic velocity field has been reached: (a) s ¼ 8:5p, (b) s ¼ 9p, (c) s ¼ 9:5p and (d) s ¼ 10p, with S ¼ 1 and b ¼ 0:1.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

6

N. Ali et al.

,m ¼

0; 1;

m 6 1; m > 1;

ð38Þ

4. Convergence of HAM solution It is well established fact that the convergence of the HAM solution is largely dependent on the proper choice of the auxiliary parameters hf and hh . For a particular set of parameters the convergence region can be obtained by plotting h-curves. Figs. 2 and 3 present two such curves showing the plausible values of hf and hh for a given set of parameters. We note from these figures that for convergent solution 1:2 6 hf < 0:2 and 1:5 6 hh < 0:2. The purpose of showing such curves is just to emphasize that in principle for any physical choice of parameters of the problem, a convergent solution can be obtained. The values of f00 ð0; sÞ obtained by HAM in the present paper are also compared with the corresponding values reported in Refs. [20,21,30,31] in Tables 1 and 2. Both tables show that our results are in excellent agreement with the existing results in the literature. 5. Results and discussion The homotopy analysis method is a powerfull technique to solve nonlinear partial and ordinary differential equations with

initial and boundary conditions. The set of nonlinear partial differential equations (8) and (9) with boundary conditions (10) and (11) is solved analytically by means of homotopy analysis method and the effects of involved parameters are illustrated through graphs. In this section, we are going to present the detail analysis of flow and heat transfer of couple stress fluid over oscillating stretching surface in the presence of heat source and sink. Fig. 4(a) and (b) are plotted to show the behavior of couple stress parameter K and magneto-porous parameter b on the time series of velocity f0 at a fixed location y ¼ 0:25 from the sheet, respectively. In Fig. 4(a) the effects of couple stress parameter K are illustrated by keeping S ¼ 1 and b ¼ 0:1. From this figure we observe that an increase in couple stress parameter K results in an increase in amplitude of velocity. This is perhaps due to the presence of couple stresses. Fig. 4 (b) points out the effects of magneto-porous parameter b by keeping other parameters constant. This figure shows that the amplitude of velocity decreases by increasing magnetoporous parameter b. Fig. 5(a)–(d) demonstrate the effects of couple stress parameter K on velocity profile f0 at four different time instants s ¼ 8:5p; s ¼ 9p; s ¼ 9:5p and s ¼ 10p. Fig. 5(a) characterizes the effects of couple stress parameter K on the velocity profile f0 at s ¼ 8:5p. This figure depicts that the velocity f0 decrease as we increase K. This decrease in velocity results in decrease in

Figure 6 Transverse profiles of the velocity field f0 for four different values of magneto-porous parameter b in the fifth period s 2 ½8p; 10p for which a periodic velocity field has been reached: (a) s ¼ 8:5p, (b) s ¼ 9p, (c) s ¼ 9:5p, and (d) s ¼ 10p, with S ¼ 0:5 and K ¼ 0:1.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

MHD flow and heat transfer of couple stress fluid boundary layer thickness. In Fig. 5(b), the behavior of couple stress parameter K on the f0 is shown at s ¼ 9p. This figure shows an increase in magnitude of velocity at this instant. The effects of K at s ¼ 9:5p and s ¼ 10p are illustrated in Fig. 5(c) and (d), respectively. It is observed from these figures that at s ¼ 9:5p and s ¼ 10p, the velocity f0 decreases with K. It is interesting to note that the effects of couple stress are to the increase the amplitude of velocity when f0 ¼ 1 at the surface. For other instants when f0 ¼ 1 or 0, the magnitude of velocity decreases due to the presence of couple stresses.

Figure 7 Time series of the skin-friction coefficient Re1=2 x Cf in the first five periods s 2 ½0; 10p: (a) effects of couple stress parameter K with S ¼ 0:1 and b ¼ 0:1 and (b) effects of magneto-porous parameter b with S ¼ 0:5 and K ¼ 0:1.

7 The effects of magneto-porous parameter b on velocity are illustrated in Fig. 6(a)–(d) at four different times keeping other parameters fixed. In Fig. 6(a) the effects of magneto-porous parameter b are plotted at s ¼ 8:5p. From this figure it can be examined that velocity decreases with increase in magneto-porous parameter b. The effects of magneto-porous parameter on velocity at other three time instants are illustrated in Fig. 6(c) and (d). Here, again it is observed that the magnitude of velocity decreases by increasing b. The variation of skin friction coefficient with couple stress parameter K and magneto-porous parameter b in the first five periods s 2 ½0; 10p is shown in Fig. 7(a) and (b), respectively. Fig. 7(a) shows the time series of skin friction coefficient for different values of couple stress parameter K by taking S ¼ 0:1 and b ¼ 0:1. From this figure we observe that skin friction coefficient varies periodically due to periodic motion of the surface and its amplitude increases with couple stress parameter K. This observation reflects that skin friction for a Newtonian fluid flowing over an oscillatory stretching sheet is less in comparison with its value for a couple stress fluid performing the same motion. The effects of magneto-porous parameter b on skin friction coefficient are illustrated in Fig. 7(b). This figure depicts that amplitude of skin friction coefficient increases by increasing magneto-porous parameter b. Such effects of b and M are expected because the fact that magnetic force and drag force offered by the porous medium acts as a resistance to the flow. Figs. 8–12 present the effects of Prandtl number Pr, magneto-porous parameter b, couple stress parameter K and heat source/sink parameter k on the temperature field. Fig. 8 reveals the effects of Prandtl number Pr on temperature profile for both heat source and sink cases. It is seen that temperature is a decreasing function of Pr in both cases. The fact is that the increase in Prandtl number results in low thermal conductivity, as a result conduction as well as thermal boundary layer thickness decreases and hence we observe a decrease in temperature. It is observed that the presence of heat sink enhances the decrease in temperature by increasing Prandtl number. In Fig. 9, the influence of magneto-porous parameter b on temperature field is explained graphically by keeping other parameters fixed. This figure shows that temperature increases with magneto-porous parameter b. The influence of couple stress

Figure 8 The effects of Prandtl number Pr on temperature field h at s ¼ 8:5p with K ¼ 0:1; S ¼ 0:1; b ¼ 0:2, (a) in heat source case k ¼ 0:3, (b) in heat sink case k ¼ 0:3.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

8

Figure 9 The effects of magneto-porous parameter b on temperature field h at s ¼ 8:5p with Pr ¼ 1:5; K ¼ 0:1; S ¼ 0:1; k ¼ 0:3.

N. Ali et al.

Figure 12 The effects of heat sink parameter k on temperature field h at s ¼ 8:5p with Pr ¼ 0:4; b ¼ 0:5; K ¼ 0:1; S ¼ 0:1.

Table 3 The numerical values of the Local Nusselt number for different values of Prandtl number Pr, couple stress parameter K, magneto-porous parameter b and heat source parameter k at time s ¼ p=2 with S ¼ 1.

Figure 10 The effects of couple stress number K on temperature field h at s ¼ 8:5p with Pr ¼ 1:5; b ¼ 5; k ¼ 0:4; S ¼ 0:6.

Figure 11 The effects of heat source parameter k on temperature field h at s ¼ 8:5p with Pr ¼ 0:4; b ¼ 0:5; K ¼ 0:1; S ¼ 0:1.

parameter K on temperature is shown in Fig. 10, which depicts that increase in couple stress parameter results in increase in temperature. Since we are discussing the temperature field in the presence of heat source/sink so effects of this parameter are very important for both the cases. The effects of heat source/sink parameter are illustrated in Figs. 11 and 12 by keeping Pr ¼ 0:4; b ¼ 0:5; K ¼ 0:1; S ¼ 0:1. It is noted from Fig. 11 that as we increase the strength of the heat source, the temperature increases. This is due to the fact that heat source can add more heat to the stretching sheet which increases its temperature and thus the temperature of fluid rises. Moreover, the thermal boundary layer is also found to increase by increasing the strength of heat source. Fig. 12 is

Pr

k

K

b

Rex1=2 Nux

0:5 1:0 1:5 2:0 0:5

0:2

5:0

5:0

0:596791 0:634885 0:671806 0:705075 0:596791 0:547204 0:496478 0:44459 0:573683 0:573021 0:572122 0:571885 0:643221 0:630873 0:619684 0:609657

0:2 0:4 0:6 0:8 0:2

0:2 0:4 0:8 1:0 5:0

0:5 1:5 2:5 3:5

plotted to observe the effects of heat sink by keeping other parameters constant. This figure shows opposite results i.e., the temperature decreases by increasing the strength of heat sink. This is because, when strength of heat sink increases, more heat is removed from the sheet and thus a decrease in thermal boundary-layer thickness and temperature is observed. This result is of key importance for the flows where heat transfer is of prime importance. The numerical values of Nusselt number are shown in Table 3. It is found that the Nusselt number increases with Prandtl number while it decreases with increase in heat source, couple stress and magneto-porous parameter. 6. Concluding remarks Flow and heat transfer of a couple stress fluid over an oscillatory stretching sheet in the presence of heat source/sink is investigated using HAM. It is observed that flow and heat transfer characteristics are greatly influenced by the presence of couple stress and heat source/sink parameter. In fact, the skin friction coefficient increases by increasing couple stress and magneto-porous parameter. It is also interesting to note that rate of heat transfer is enhanced by increasing

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

MHD flow and heat transfer of couple stress fluid magneto-porous parameter, heat source and couple stress parameter, while it decreases for large values of Prandtl number and heat sink parameter. Such observations may have interesting implications where it is desired to reduce skin friction and enhance heat transfer. Acknowledgments We are grateful to the reviewers for their useful comments. The second author is grateful to the Higher Education Commission of Pakistan for financial assistance. References [1] B.C. Sakiadis, Boundary layer behavior on continuous solid surfaces: I boundary layer equations for two dimensional and axisymmetric flow, AIChE J. 7 (1961) 26–28. [2] L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phy. (ZAMP) 21 (1970) 645–647. [3] K.R. Rajagopal, T.Y. Na, A.S. Gupta, Flow of a visco-elastic fluid over a stretching sheet, Rheol. Acta 23 (1984) 213–215. [4] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (1977) 744–746. [5] B.S. Dandapat, A.S. Gupta, Flow and heat transfer in a viscoelastic fluid over a stretching sheet, Int. J. Nonlinear Mech. 24 (1989) 215–219. [6] J.B. McLeod, K.R. Rajagopal, On the uniqueness of flow of a Navier-Stokes fluid due to stretching boundries, Arch. Rat. Mech. Anal. 98 (1987) 385. [7] D. Rollins, K. Vajravelu, Heat transfer in a second-order fluid over a continuous stretching surface, Acta Mech. 89 (1991) 167– 178. [8] K.R. Rajagopal, R.K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mech. 113 (1995) 233– 239. [9] R. Cortell, A note on flow and heat transfer of a viscoelastic liquid over a stretching sheet, Int. J. Non-Linear Mech. 41 (2006) 78–85. [10] R. Cortell, Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to suction and to a transverse magnetic field, Int. J. Heat Mass Transfer 49 (2006) 1851–1856. [11] R. Nazar, N. Amin, D. Filip, I. Pop, Stagnation point flow of a micropolar fluid towards a stretching sheet, Int. J. Non-Linear Mech. 39 (2004) 1227–1235. [12] A. Ishak, R. Nazar, I. Pop, Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet, Mecccanica 43 (2008) 411–418. [13] P.D. Ariel, Axisymmetric flow of a second grade fluid past a stretching sheet, Int. J. Eng. Sci. 39 (2001) 529–553. [14] T. Hayat, M. Sajid, Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet, Int. J. Heat Mass Transfer 50 (2007) 75–84. [15] T. Hayat, Z. Abbas, M. Sajid, Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction, Phys. Lett. A 372 (2008) 2400–2408. [16] T. Hayat, T. Javed, Z. Abbas, Slip flow and heat transfer of a second grade fluid past a stretching sheet through a porous space, Int J. Heat Mass Transfer 51 (2008) 4528–4534. [17] M. Sajid, I. Ahmad, T. Hayat, M. Ayub, Unsteady flow and heat transfer of a second grade fluid over a stretching sheet, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 96–108. [18] T. Hayat, T. Javed, M. Sajid, Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid, Acta Mech. 191 (2007) 219–229.

9 [19] M. Sajid, T. Hayat, Non-similar series solution for boundary layer flow of a third-order fluid over a stretching sheet, Appl. Math. Comput. 189 (2007) 1576–1585. [20] T. Hayat, M. Mustafa, I. Pop, Heat and mass transfer for Soret and Dufour’s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1183–1196. [21] M. Turkyilmazoglu, The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface, Int. J. Mech. Sci. 77 (2013) 263– 268. [22] M. Turkyilmazoglu, Three dimensional MHD flow and heat transfer over a stretching/shrinking surface in a viscoelastic fluid with various physical effects, Int. J. Heat Mass Transfer 78 (2014) 150–155. [23] M. Turkyilmazoglu, Multiple solutions of hydromagnetic permeable flow and heat for viscoelastic fluid, J. Thermophys. Heat Transfer 25 (2011) 595–605. [24] M. Turkyilmazoglu, Analytic heat and mass transfer of the mixed hydrodynamic/thermal slip MHD viscous flow over a stretching sheet, Int. J. Mech. Sci. 53 (2011) 886–896. [25] M. Turkyilmazoglu, Multiple solutions of heat and mass transfer of MHD slip flow for the viscoelastic fluid over a stretching sheet, Int. J. Therm. Sci. 50 (2011) 2264–2276. [26] M. Turkyilmazoglu, An analytical treatment for the exact solutions of MHD flow and heat over two–three dimensional deforming bodies, Int. J. Heat Mass Transfer 90 (2015) 781– 789. [27] C.Y. Wang, Nonlinear streaming due to the oscillatory stretching of a sheet in a viscous, Acta Mech. 72 (1988) 261– 268. [28] B. Siddappa, S. Abel, V. Hongunti, Oscillatory motion of a viscoelastic fluid past a stretching sheet, ll Nuovo Cimento D 17 (1995) 53–60. [29] Z. Abbas, Y. Wang, T. Hayat, M. Oberlack, Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface, Int. J. Numer. Methods Fluids 59 (2009) 443– 458. [30] Z. Abbas, Y. Wang, T. Hayat, M. Oberlack, Hydromagnetic flow in a viscoelastic fluid due to the oscillatory stretching surface, Int. J. Nonlinear Mech. 43 (2008) 783–797. [31] L.C. Zheng, X. Jin, X.X. Zhang, J.H. Zhang, Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects, Acta Mech. Sinica 29 (5) (2013) 667–675. [32] N. Ali, S.U. Khan, Z. Abbas, Hydromagnetic flow and heat transfer of a jeffrey fluid over an oscillatory stretching surface, Z. Naturforsch. A 70 (7) (2015) 567–576. [33] S.U. Khan, N. Ali, Z. Abbas, Hydromagnetic flow and heat transfer over a porous oscillating stretching surface in a viscoelastic fluid with porous medium, PLOS ONE (2015), http://dx.doi.org/10.1371/journal.pone.0144299, December 14. [34] V.K. Stokes, Couple Stress Fluids 9 (1966) 1709–1715. [35] D. Srinivasacharya, K. Kaladhar, Soret and Dufour effects in a mixed convection couple stress fluid with heat and mass fluxes, Lat. Am. Appl. Res. 41 (2011) 353–358. [36] L.M. Srivastava, Peristaltic transport of a couple stress fluid, Rheol. Acta 25 (1986) 638–641. [37] E.F. El-Shehawey, K.S. Mekheimer, Couple-stresses in peristaltic transport of fluids, J. Phys. D: Appl. Phys. 27 (1994) 1163. [38] D. Pal, N. Rudraiah, R. Devanathan, A couple stress model of blood flow in the microcirculation, Bull. Math. Biol. 50 (1988) 329. [39] H.L. Chiang, J.R. Lin, C.H. Hsu, Y.P. Chang, Linear stability analysis of a rough short journal bearing lubricated with nonnewtonian fluids, Tribol. Int. 17 (2004) 867–877.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018

10 [40] N.B. Naduvinamani, P.S. Hiremath, G. Gurubasavaraj, Effect of surface roughness on the couple-stress squeeze film between a sphere and a flat plate, Tribol. Int. 38 (2005) 451–458. [41] C.W.C. Jian, C.K. Chen, Chaos and bifurcation of a flexible rotor supported by porous squeeze couple stress fluid film journal bearings with non-linear suspension, Chaos Solit. Fract. 35 (2008) 358–375. [42] R.F. Lu, J.R. Lin, A theoretical study of combined effects of non-Newtonian rheology and viscosity-pressure dependence in the sphere-plate squeeze-film system, Tribo. In. 40 (1) (2007) 125–131. [43] S.J. Liao, Advance in the Homotopy Analysis Method, vol. 1, World Scientific Publishing, 5 Toh Tuck Link, Singapore, 2014. [44] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A 360 (2006) 109–113. [45] M. Turkyilmazoglu, Analytic approximate solutions of rotating disk boundary layer flow subject to a uniform suction or injection, Int. J. Mech. Sci. 52 (2010) 1735–1744. [46] S. Abbasbandy, Homotopy analysis method for heat radiation equations, Int. Commun. Heat Mass Transfer 34 (2007) 380– 387.

N. Ali et al. [47] I.A. Abdallah, Homotopy analytical solution of MHD fluid flow and heat transfer problem, Appl. Math. Inform. Sci. 3 (2009) 223–233. [48] T. Hayat, M. Sajid, Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid, Int. J. Eng. Sci. 45 (2007) 393–401. [49] B. Raftari, K. Vajravelu, Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4149. [50] S.J. Liao, A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. Fluid Mech. 385 (1999) 101–128. [51] S.J. Liao, An explicit, totally analytic approximation of Blasius’ viscous flow problems totally analytic approximation of Blasius’ viscous flow problems, Int. J. Non-Linear Mech. 34 (4) (1999) 759–778. [52] S.J. Liao, On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. Fluid Mech. 488 (2003) 189–212. [53] M. Turkyilmazoglu, Solution of the Thomas-Fermi equation with a convergent approach, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4097–4103.

Please cite this article in press as: N. Ali et al., MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.02.018