MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium

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.

Keywords

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

1. Introduction

It is well established fact that Navier–Stokes equations are not adequate choice in describing the flows of non-Newtonian 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 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 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 stagnation-point 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], [16], [17] and [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 two-dimensional flow over stretching sheet and in the case of shrinking sheet triple solutions can be detected [23], [24] and [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 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\overline{x}\mbox{,}\overline{y}\right)} , the sheet is assumed to coincide with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{y}=0}

and flow takes place in the semi-infinite porous space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\overline{y}>0\right)}
of constant permeability (see Fig. 1). A constant magnetic field of strength Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_0}
is applied perpendicular to the stretching surface. It is assumed that the sheet is stretched with velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{\omega }=b\overline{x}sin\omega t}
in which b   is the stretching rate and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }
denotes the angular velocity. Moreover, sheet is maintained at uniform temperature Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T_w}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(>T_{\infty }\right)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T_{\infty }}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\partial u}{\partial \overline{x}}+\frac{\partial v}{\partial \overline{y}}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}

(1)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial \overline{x}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): v\frac{\partial u}{\partial \overline{y}}=\nu \frac{{\partial }^2u}{\partial {\overline{y}}^2}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\eta }_0}{\rho }\left[\frac{{\partial }^4u}{\partial {\overline{y}}^4}\right]- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\sigma B_0^2}{\rho }u-\frac{\upsilon \phi }{k}u\mbox{,}

(2)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\partial T}{\partial t}+u\frac{\partial T}{\partial \overline{x}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): v\frac{\partial T}{\partial \overline{y}}=\frac{k}{\rho c_p}\frac{{\partial }^2T}{\partial {\overline{y}}^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{Q}{\rho c_p}\left(T-T_{\infty }\right)\mbox{,}

(3)

where u and v   are velocity component along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{x}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{y}}

-directions, respectively, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu }

is the kinematic viscosity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho }
is the density, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\eta }_0}
is the material constant for the couple stress fluid, Q   is the heat source coefficient, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma }
is electric conductivity and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_0}
is the strength of constant applied magnetic field. The flow is subjected to the following boundary conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u=u_{\omega }=b\overline{x}sin\omega t\mbox{,}\quad v= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad \frac{{\partial }^2u}{\partial {\overline{y}}^2}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad T=T_w\quad at\quad \overline{y}=0\mbox{,}\quad t>0\mbox{,}

(4)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u=0\mbox{,}\quad \frac{\partial u}{\partial \overline{y}}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad T\rightarrow T_{\infty }\quad \quad \quad at\quad \quad \quad \overline{y}\rightarrow \infty \mbox{.}

(5)

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y\rightarrow \infty } . To non-dimensionalize the flow problem, we use the dimensionless variables [30] and [31]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): y=\sqrt{\frac{b}{\nu }}\overline{y}\mbox{,}\quad \tau = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t\omega \mbox{,}\quad u=b\overline{x}f_y\left(y\mbox{,}\tau \right)\mbox{,}\quad v= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\sqrt{\nu b}f\left(y\mbox{,}\tau \right)\mbox{,}

(6)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \theta (y\mbox{,}\tau )=\frac{T-T_{\infty }}{T_w-T_{\infty }}\mbox{.}

(7)

In view of above transformations, the continuity equation is identically satisfied and Eqs. (2) and (3) are transformed as follows

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f_{yyy}-{Sf}_{y\tau }-f_y^2+{ff}_{yy}-\beta f_y-{Kf}_{yyyyy}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}

(8)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\theta }_{yy}+Pr\left(f{\theta }_y-S{\theta }_{\tau }+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \lambda \theta \right)=0\mbox{,}

(9)

where the subscript denotes differentiation with respect to y. The boundary conditions (4) and (5) take the following form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f_y\left(0\mbox{,}\tau \right)=sin\tau \mbox{,}\quad f\left(0\mbox{,}\tau \right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad f_{yyy}\left(0\mbox{,}\tau \right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad \theta (0\mbox{,}\tau )=1\mbox{,}

(10)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f_y\left(\infty \mbox{,}\tau \right)=0\mbox{,}\quad f_{yy}\left(\infty \mbox{,}\tau \right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad \theta (\infty \mbox{,}\tau )=0\mbox{.}

(11)

In above equations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K={\eta }_0b/\rho {\upsilon }^2}

is couple stress parameter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=\omega /b}
is unsteady parameter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta =\sigma B_0^2/\rho b+\upsilon \phi /kb}
is the magneto-porous parameter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pr=\mu c_p/k}
is the Prandtl number and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =Q/c\rho c_p}
is the heat source Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\lambda >0\right)}
or sink Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\lambda <0\right)}
parameter. The parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K={\eta }_0\rho b/{\mu }^2=l^2b\rho /\mu }
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=l^2b\rho /\mu }

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b\rho /\mu }

is a typical length scale associated with the flow and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle l=\sqrt{\eta /\mu }}
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.

Figure 1. Geometry of the problem.

The quantities of interest are skin friction coefficient and local Nusselt number which are defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): C_f=\frac{{\tau }_w}{\rho u_w^2}\mbox{,}\quad {Nu}_x= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\overline{x}q_w}{k(T_w-T_{\infty })}\mbox{,}

(12)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\tau }_w=\mu {\left(\frac{\partial u}{\partial \overline{y}}\right)}_{\overline{y}=0}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\eta }_0}{\rho }{\left(\frac{{\partial }^3u}{\partial {\overline{y}}^3}\right)}_{\overline{y}=0}\mbox{,}\quad \quad \quad \quad \quad q_w= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -k{\left(\frac{\partial T}{\partial \overline{y}}\right)}_{\overline{y}=0}\mbox{,}

(13)

are skin friction and heat transfer at the wall, respectively. In view of transformations (6) and (7), we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {Re}_x^{1/2}C_f=f_{yy}(0\mbox{,}\tau )-{Kf}_{yyyy}(0\mbox{,}\tau )\mbox{,}

(14)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {Re}_x^{-1/2}{Nu}_x=-{\theta }_y(0\mbox{,}\tau )\mbox{,}

(15)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {Re}_x=u_w\overline{x}/\nu }

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], [45], [46], [47], [48] and [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], [51] and [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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f(y\mbox{,}\tau )=a_{0\mbox{,}0}^0+\sum_{n=0}^{\infty }\sum_{k=0}^{\infty }\sum_{j=0}^{\infty }a_{n\mbox{,}k}^jy^ksin(j\tau )exp(- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ny)\mbox{,}

(16)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \theta (y\mbox{,}\tau )=b_{0\mbox{,}0}^0+\sum_{n=0}^{\infty }\sum_{k=0}^{\infty }\sum_{j=0}^{\infty }b_{n\mbox{,}k}^jy^ksin(j\tau )exp(- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ny)\mbox{,}

(17)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a_{n\mbox{,}k}^j}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b_{n\mbox{,}k}^j}
are the coefficients to be determined. The appropriate initial guesses for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(y\mbox{,}\tau )}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta (y)}
are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f_0(y\mbox{,}\tau )=\frac{1}{2}sin\tau (3-3exp(-y)- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): yexp(-y))\mbox{,}

(18)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\theta }_0(y)=exp(-y)\mbox{.}

(19)

Further we choose

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\pounds }_f(f)=\frac{{\partial }^3f}{\partial y^3}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\partial f}{\partial y}\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\pounds }_{\theta }(f)=\frac{{\partial }^2f}{\partial y^2}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f\mbox{,}

as auxiliary linear operators having following properties

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\pounds }_f[C_1+C_2exp(-y)+C_3exp(y)]=0\mbox{,}

(20)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\pounds }_{\theta }[C_4exp(y)+C_5exp(-y)]=0\mbox{,}

(21)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_i} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(i=1-5\right)}

are arbitrary constants. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_f}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{\theta }}
denotes the nonzero auxiliary parameters then zeroth order deformation problems are constructed as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (1-p){\pounds }_f[\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f_0(y\mbox{,}\tau )={ph}_fN_f[\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)]\mbox{,}

(22)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (1-p){\pounds }_f[\overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\theta }_0(y\mbox{,}\tau )={ph}_{\theta }N_{\theta }[\overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)\mbox{,}\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)]\mbox{,}

(23)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overset{\mbox{ˆ}}{f}(0\mbox{,}\tau \mbox{;}p)=0\mbox{,}\frac{\partial \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): sin\tau \mbox{,}\quad \frac{\partial \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad \frac{{\partial }^2\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y^2}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}

(24)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overset{\mbox{ˆ}}{\theta }(0\mbox{,}\tau \mbox{;}p)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1\mbox{,}\quad \overset{\mbox{ˆ}}{\theta }(\infty \mbox{,}\tau \mbox{;}p)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p\in [0\mbox{,}1]}

is an embedding parameter and nonlinear operators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_f}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_{\theta }}
are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): N_f[\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -K\frac{{\partial }^5\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y^5}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^3\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y^3}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): S\frac{{\partial }^2\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y\partial \tau }+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)\frac{{\partial }^2\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y^2}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left(\frac{\partial \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y}\right)}^2- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \beta \frac{\partial \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial y}\mbox{,}

(25)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): N_{\theta }[\overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)\mbox{,}\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^2\overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)}{\partial y^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Pr\left(\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)\frac{\partial \overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)}{\partial y}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. S\frac{\partial \overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)}{\partial \tau }+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \lambda \overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)\right)\mbox{.}

(26)

The zeroth-order deformation problems defined above have the following solutions corresponding to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p=0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p=1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}0)=f_0(y\mbox{,}\tau )\mbox{,}\quad \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}1)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f(y\mbox{,}\tau )\mbox{,}

(27)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\theta }_0(y\mbox{,}\tau )\mbox{,}\quad \overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}1)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \theta (y\mbox{,}\tau )\mbox{.}

(28)

Expanding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)}
in a Taylor’s series with respect to p, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)={\overset{\mbox{ˆ}}{f}}_0(y\mbox{,}\tau )+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \sum_{m=1}^{\infty }f_m(y\mbox{,}\tau )p^m\mbox{,}

(29)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\mbox{ˆ}}{\theta }}_0(y\mbox{,}\tau )+\sum_{m=1}^{\infty }{\theta }_m(y\mbox{,}\tau )p^m\mbox{,}

(30)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f_m(y\mbox{,}\tau )=\frac{1}{m!}\frac{{\partial }^m\overset{\mbox{ˆ}}{f}(y\mbox{,}\tau \mbox{;}p)}{\partial p^m}\mbox{,}\quad {\theta }_m(y\mbox{,}\tau )= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{m!}\frac{{\partial }^m\overset{\mbox{ˆ}}{\theta }(y\mbox{,}\tau \mbox{;}p)}{\partial p^m}\mbox{,}

(31)

Differentiating zeroth order deformations equations (22) and (23)m-times with respect to p  , then setting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p=0}

and finally dividing by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m!}

, we get the following mth-order deformation problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\pounds }_f[{\overset{\mbox{ˆ}}{f}}_m(y\mbox{,}\tau )- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\varkappa }_mf_{m-1}(y\mbox{,}\tau )=h_fR_m^f(y\mbox{,}\tau )\mbox{,}

(32)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\pounds }_f[{\overset{\mbox{ˆ}}{\theta }}_m(y\mbox{,}\tau )- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\varkappa }_m{\theta }_{m-1}(y\mbox{,}\tau )=h_{\theta }R_m^{\theta }(y\mbox{,}\tau )\mbox{,}

(33)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f_m(0\mbox{,}\tau \mbox{;}p)=0\mbox{,}{\frac{\partial f_m(y\mbox{,}\tau \mbox{;}0)}{\partial y}}_{y=0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\frac{\partial f_m(y\mbox{,}\tau \mbox{;}0)}{\partial y}}_{y=\infty }= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\frac{\partial f_m^2(y\mbox{,}\tau \mbox{;}0)}{\partial y^2}}_{y=\infty }= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}

(34)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\theta }_m(0)=0\mbox{,}{\theta }_m(\infty )=0\mbox{,}

(35)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): R_m^f(y\mbox{,}\tau )=-K\frac{{\partial }^5f_{m-1}}{\partial y^5}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^3f_{m-1}}{\partial y^3}-S\frac{{\partial }^2f_{m-1}}{\partial y\partial \tau }- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \beta \frac{\partial f_{m-1}}{\partial y}+\sum_{k=0}^{m-1}\left(f_{m-1-k}\frac{{\partial }^2f_k}{\partial y^2}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \frac{f_{m-k-1}}{\partial y}\frac{\partial f_k}{\partial y}\right)\mbox{,}

(36)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): R_m^{\theta }(y\mbox{,}\tau )=\frac{{\partial }^2{\theta }_{m-1}}{\partial y^2}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Pr\left[S\frac{\partial {\theta }_{m-1}}{\partial \tau }\right]+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Pr\left(\sum_{k=0}^{m-1}\left[f_k\frac{\partial {\theta }_{m-1-k}}{\partial y}\right]+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \lambda {\theta }_{m-1}\right)\mbox{,}

(37)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\varkappa }_m=\begin{array}{ll} 0\mbox{,} & m\leqslant 1\mbox{,}\\ 1\mbox{,} & m>1\mbox{,} \end{array}

(38)

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_f}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{\theta }}

. For a particular set of parameters the convergence region can be obtained by plotting h-curves. Figure 2 and Figure 3 present two such curves showing the plausible values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_f}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{\theta }}
for a given set of parameters. We note from these figures that for convergent solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.2\leqslant h_f<-0.2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.5\leqslant h_{\theta }<-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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{''}}(0\mbox{,}\tau )}

obtained by HAM in the present paper are also compared with the corresponding values reported in Refs.  [20], [21], [30] and [31] in  Table 1 and Table 2. Both tables show that our results are in excellent agreement with the existing results in the literature.

Figure 2. The h  -curve for velocity at 7th order of approximation: with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=0.1\mbox{,}\beta =0.2\mbox{,}S=0.1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =0.5\pi } .

Figure 3. The h  -curve for temperature at 7th order of approximation with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=0.1\mbox{,}S=0.1\mbox{,}\beta =0.2\mbox{,}\lambda =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.5\mbox{,}Pr=0.1 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =0.5\pi } .

Table 1. Comparison of values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{''}}(0\mbox{,}\tau )} with Refs. [30] with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=0} and [31].

S	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau }	Zheng et al. [31]	Abbas et al. [30]	Present results
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.0}	12	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.5\pi }	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678565}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678656}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678565}
		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5.5\pi }	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678706}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678707}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678706}
		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 9.5\pi }	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678656}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678656}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 11.678656}

Table 2. Comparison of values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{''}}(0\mbox{,}\tau )} for different values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=0\mbox{,}S=0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =\pi /2} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }	Hayat et al. [20]	Turkyilmazoglu [21]	Present results
0	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.000000}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.00000000}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.000000}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.5}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.224747}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.22474487}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.224747}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.414217}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.41421356}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.414217}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.5}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.581147}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.58113883}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.581147}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2.0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.732057}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.73205081}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -1.732057}

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

on the time series of velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}}
at a fixed location Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y=0.25}
from the sheet, respectively. In Fig. 4(a) the effects of couple stress parameter K   are illustrated by keeping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta =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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

by keeping other parameters constant. This figure shows that the amplitude of velocity decreases by increasing magneto-porous parameter β.

Figure 4. Time series of the velocity profile Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}} at a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y=0.25} from the sheet in the time period Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau \in \left[0\mbox{,}10\pi \right]} (a) The effects of couple stress parameter K   with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta =0.1} , (b) The effects of magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=0.5} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=0.1} .

Fig. 5(a)–(d) demonstrate the effects of couple stress parameter K   on velocity profile Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}}

at four different time instants Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi \mbox{,}\tau =9\pi \mbox{,}\tau =9.5\pi }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =10\pi }

. Fig. 5(a) characterizes the effects of couple stress parameter K   on the velocity profile Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}}

at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi }

. This figure depicts that the velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}}

decrease as we increase K. This decrease in velocity results in decrease in boundary layer thickness. In  Fig. 5(b), the behavior of couple stress parameter K   on the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}}
is shown at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =9\pi }

. This figure shows an increase in magnitude of velocity at this instant. The effects of K   at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =9.5\pi }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =10\pi }
are illustrated in Fig. 5(c) and (d), respectively. It is observed from these figures that at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =9.5\pi }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =10\pi }

, the velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}}

decreases with K  . It is interesting to note that the effects of couple stress are to the increase the amplitude of velocity when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}=1}
at the surface. For other instants when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}=-1}
or 0, the magnitude of velocity decreases due to the presence of couple stresses.

Figure 5. Transverse profiles of the velocity field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}} for four different values of couple stress parameter K   in the fifth period Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau \in \left[8\pi \mbox{,}10\pi \right]} for which a periodic velocity field has been reached: (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi } , (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =9\pi } , (c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =9.5\pi } and (d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =10\pi } , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta =0.1} .

The effects of magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }
are plotted at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi }

. From this figure it can be examined that velocity decreases with increase in magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } . 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } .

Figure 6. Transverse profiles of the velocity field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{{'}}} for four different values of magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } in the fifth period Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau \in \left[8\pi \mbox{,}10\pi \right]} for which a periodic velocity field has been reached: (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi } , (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =9\pi } , (c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =9.5\pi } , and (d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =10\pi } , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=0.5} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=0.1} .

The variation of skin friction coefficient with couple stress parameter K   and magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

in the first five periods Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau \in [0\mbox{,}10\pi ]}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=0.1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta =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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

. Such effects of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

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.

Figure 7. Time series of the skin-friction coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {Re}_x^{1/2}C_f} in the first five periods Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau \in [0\mbox{,}10\pi ]} (a) effects of couple stress parameter K   with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=0.1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta =0.1} and (b) effects of magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=0.5} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=0.1} .

Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 present the effects of Prandtl number Pr  , magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } , couple stress parameter K   and heat source/sink parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda }

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }
on temperature field is explained graphically by keeping other parameters fixed. This figure shows that temperature increases with magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

. The influence of couple stress 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 Figure 11 and Figure 12 by keeping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pr=0.4\mbox{,}\beta =0.5\mbox{,}K=0.1\mbox{,}S=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 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.

Figure 8. The effects of Prandtl number Pr   on temperature field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta } at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi } with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=0.1\mbox{,}S=0.1\mbox{,}\beta =0.2} , (a) in heat source case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =0.3} , (b) in heat sink case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =-0.3} .

Figure 9. The effects of magneto-porous parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } on temperature field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta } at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi } with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pr=1.5\mbox{,}K=0.1\mbox{,}S=0.1\mbox{,}\lambda =0.3} .

Figure 10. The effects of couple stress number K   on temperature field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta } at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi } with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pr=1.5\mbox{,}\beta =5\mbox{,}\lambda =0.4\mbox{,}S=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0.6 .

Figure 11. The effects of heat source parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda } on temperature field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta } at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi } with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pr=0.4\mbox{,}\beta =0.5\mbox{,}K=0.1\mbox{,}S=0.1} .

Figure 12. The effects of heat sink parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda } on temperature field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta } at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =8.5\pi } with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pr=0.4\mbox{,}\beta =0.5\mbox{,}K=0.1\mbox{,}S=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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } and heat source parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda } at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =\pi /2} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=1} .

Pr	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda }	K	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {Re}_x^{-1/2}{Nu}_x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.5}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.2}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5.0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5.0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.596791}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.0}				Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.634885}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.5}				Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.671806}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2.0}				Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.705075}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.5}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.2}			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.596791}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.4}			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.547204}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.6}			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.496478}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.8}			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.44459}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.2}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.2}		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.573683}
		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.4}		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.573021}
		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.8}		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.572122}
		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.0}		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.571885}
		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5.0}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.5}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.643221}
			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1.5}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.630873}
			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2.5}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.619684}
			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 3.5}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0.609657}

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 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. [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), pp. 26–28
2. [2] L.J. Crane; Flow past a stretching plate; Z. Angew. Math. Phy. (ZAMP), 21 (1970), pp. 645–647
3. [3] K.R. Rajagopal, T.Y. Na, A.S. Gupta; Flow of a visco-elastic fluid over a stretching sheet; Rheol. Acta, 23 (1984), pp. 213–215
4. [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), pp. 744–746
5. [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), pp. 215–219
6. [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), p. 385
7. [7] D. Rollins, K. Vajravelu; Heat transfer in a second-order fluid over a continuous stretching surface; Acta Mech., 89 (1991), pp. 167–178
8. [8] K.R. Rajagopal, R.K. Bhatnagar; Exact solutions for some simple flows of an Oldroyd-B fluid; Acta Mech., 113 (1995), pp. 233–239
9. [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), pp. 78–85
10. [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), pp. 1851–1856
11. [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), pp. 1227–1235
12. [12] A. Ishak, R. Nazar, I. Pop; Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet; Mecccanica, 43 (2008), pp. 411–418
13. [13] P.D. Ariel; Axisymmetric flow of a second grade fluid past a stretching sheet; Int. J. Eng. Sci., 39 (2001), pp. 529–553
14. [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), pp. 75–84
15. [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), pp. 2400–2408
16. [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), pp. 4528–4534
17. [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), pp. 96–108
18. [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), pp. 219–229
19. [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), pp. 1576–1585
20. [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), pp. 1183–1196
21. [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), pp. 263–268
22. [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), pp. 150–155
23. [23] M. Turkyilmazoglu; Multiple solutions of hydromagnetic permeable flow and heat for viscoelastic fluid; J. Thermophys. Heat Transfer, 25 (2011), pp. 595–605
24. [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), pp. 886–896
25. [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), pp. 2264–2276
26. [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), pp. 781–789
27. [27] C.Y. Wang; Nonlinear streaming due to the oscillatory stretching of a sheet in a viscous; Acta Mech., 72 (1988), pp. 261–268
28. [28] B. Siddappa, S. Abel, V. Hongunti; Oscillatory motion of a viscoelastic fluid past a stretching sheet; ll Nuovo Cimento D, 17 (1995), pp. 53–60
29. [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), pp. 443–458
30. [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), pp. 783–797
31. [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), pp. 667–675
32. [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), pp. 567–576
33. [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. [34] V.K. Stokes; Couple Stress Fluids, 9 (1966), pp. 1709–1715
35. [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), pp. 353–358
36. [36] L.M. Srivastava; Peristaltic transport of a couple stress fluid; Rheol. Acta, 25 (1986), pp. 638–641
37. [37] E.F. El-Shehawey, K.S. Mekheimer; Couple-stresses in peristaltic transport of fluids; J. Phys. D: Appl. Phys., 27 (1994), p. 1163
38. [38] D. Pal, N. Rudraiah, R. Devanathan; A couple stress model of blood flow in the microcirculation; Bull. Math. Biol., 50 (1988), p. 329
39. [39] H.L. Chiang, J.R. Lin, C.H. Hsu, Y.P. Chang; Linear stability analysis of a rough short journal bearing lubricated with non-newtonian fluids; Tribol. Int., 17 (2004), pp. 867–877
40. [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), pp. 451–458
41. [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), pp. 358–375
42. [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), pp. 125–131
43. [43] S.J. Liao; Advance in the Homotopy Analysis Method, vol. 1, World Scientific Publishing, 5 Toh Tuck Link, Singapore (2014)
44. [44] S. Abbasbandy; The application of homotopy analysis method to nonlinear equations arising in heat transfer; Phys. Lett. A, 360 (2006), pp. 109–113
45. [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), pp. 1735–1744
46. [46] S. Abbasbandy; Homotopy analysis method for heat radiation equations; Int. Commun. Heat Mass Transfer, 34 (2007), pp. 380–387
47. [47] I.A. Abdallah; Homotopy analytical solution of MHD fluid flow and heat transfer problem; Appl. Math. Inform. Sci., 3 (2009), pp. 223–233
48. [48] T. Hayat, M. Sajid; Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid; Int. J. Eng. Sci., 45 (2007), pp. 393–401
49. [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), p. 4149
50. [50] S.J. Liao; A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate; J. Fluid Mech., 385 (1999), pp. 101–128
51. [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), pp. 759–778
52. [52] S.J. Liao; On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet; J. Fluid Mech., 488 (2003), pp. 189–212
53. [53] M. Turkyilmazoglu; Solution of the Thomas-Fermi equation with a convergent approach; Commun. Nonlinear Sci. Numer. Simul., 17 (2012), pp. 4097–4103