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MHD Carreau fluid slip flow over a porous stretching sheet with viscous dissipation and variable thermal conductivity
 Rehan Ali Shah^{1}Email author,
 Tariq Abbas^{2},
 Muhammad Idrees^{2} and
 Murad Ullah^{2}
 Received: 4 January 2017
 Accepted: 26 May 2017
 Published: 26 June 2017
Abstract
The aim of this article is to investigate MHD Carreau fluid slip flow with viscous dissipation and heat transfer by taking the effect of thermal radiation over a stretching sheet embedded in a porous medium with variable thickness and variable thermal conductivity. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The constitutive equations of Carreau fluid are modeled in the form of partial differential equations (PDEs). Concerning boundary conditions available, the PDEs are converted to ordinary differential equations (ODEs) by means of similarity transformation. The homotopy analysis method (HAM) is used for solution of the system of nonlinear problems. The effects of various parameters such as Weissenberg number \(\mathit{We}^{2}\), magnetic parameter \(M^{2}\), power law index n, porosity parameter D, wall thickness parameter α, power index parameter m, slip parameter λ, thermal conductivity parameter ε, radiation parameter R and Prandtl number on velocity and temperature profiles are analyzed and studied graphically.
Keywords
 magnetohydrodynamics (MHD)
 Carreau fluid flow
 stretching sheet
 slip flow
 variable thickness
 variable thermal conductivity
 thermal radiation
1 Introduction
The study of heat transfer and boundary layer flow over a stretching sheet has received a great deal of attention from many researchers due to its importance in many engineering and industrial applications, such as paper production, glassfiber production, solidification of liquid crystals, petroleum production, exotic lubricants, suspension solutions, wire drawing, continuous cooling and fibers spinning, manufacturing plastic films and extraction of polymer sheet. Crane [1] was the first person who studied the boundary layer flow past a stretching sheet. He concluded that velocity is proportional to the distance from the slit. Gupta and Gupta [2] discussed the problem of the continuous moving surface with constant temperature. The constant surface velocity case with a power law temperature variation was studied by Soundalgekar et al. [3]. Elbashbeshy [4] examined the heat transfer over a stretching surface with variable heat flux and uniform surface heat flux. Grubka et al. [5] studied the stretching flow problem with a variable surface temperature. Hayat et al. [6] obtained the series solutions for stretching sheet problem with mixed convection by using the homotopy analysis method (HAM). In the presence of a transverse magnetic field, Chaim [7] studied boundary layer flow due to a plate stretching with a power law velocity. Effect of variable thermal conductivity and heat source/sink on flow near a stagnation point on a nonconducting stretching sheet was studied by Sharma et al. [8].
Georgiou [9] investigated timedependent Poiseuille flow of Carreau fluid in the presence of slip effect and concluded that the wavelength and amplitude of oscillations in radial direction decrease with an increase in the slip effect. The peristaltic flow characteristics of Carreau fluid in a uniform tube and the heat transfer characteristics of Carreau fluid were discussed by El Hakeem [10]. Malik et al. [11] studied the pressuredependent viscosity in Carreau fluid through a porous medium. The effect of transpiration on magnetohydrodynamic stagnationpoint flow of Carreau nanofluid toward a stretching/shrinking sheet in the presence of thermophoresis and Brownian motion was numerically investigated by Sulochana et al. [12]. Akbar et al. [13] investigated numerically the flow of peristaltic Carreau nanofluid past an asymmetric channel and found that increasing values of magnetic parameter encourage the velocity profiles. Ali and Hayat [14] presented the analytic solution of mathematical modeling for the flow of incompressible Carreau fluid in an asymmetric channel with sinusoidal wall variations. Suneetha et al. [15] investigated the effect of thermal radiation on a twodimensional stagnation point flow of an incompressible MHD Carreau fluid towards a shrinking surface in the presence of convective boundary conditions. The unsteady peristaltic flow of an incompressible Carreau fluid in eccentric cylinders was investigated by Nadeem et al. [16]. The boundary layer flow and heat transfer to a Carreau model over a nonlinear stretching surface was discussed by Khan et al. [17]. Masood et al. [18] investigated the effect of magnetic field on the stagnation point flow of a generalized Newtonian Carreau fluid.
The problem of free convection about a vertical impermeable flat plate in a Darcy porous medium was studied by Cheng et al. [19]. The heat transfer and flow in a porous medium over a stretching surface with internal heat generation and suction or blowing when the surface is held at a constant temperature was studied by Elbashbeshy et al. [20]. Using the homotopy analysis method, analytic solution was obtained by Hayat et al. [21] for the flow through a porous medium. Fang et al. [22] studied the boundary layer over a continuously stretching sheet with variable thickness. The progress of thermal diffusive flow over a stretching sheet with variable thickness was investigated by Subhashini et al. [23]. Khader et al. [24] obtained the numerical solution for the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium in the presence of thermal radiation by using the finite difference method. Mostafa et al. [25] studied the flow and heat transfer of an electrically conducting nonNewtonian power law fluid within a thin liquid film over an unsteady stretching sheet in the presence of a transverse magnetic field with variable viscosity and variable thermal conductivity. Anjali Devi et al. [26] studied the boundary layer and heat transfer characteristics of hydromagnetic flow over a stretching sheet with variable thickness. Numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet with a power law surface velocity, slip velocity and variable thickness was studied by Megahed et al. [27]. Eid et al. [28] studied numerical solutions for the slip flow and heat transfer of a Newtonian fluid due to an impermeable stretching sheet which is embedded in a porous medium with a power law surface velocity and variable thickness in the presence of thermal radiation, viscous dissipation and slip velocity effects. The heat and mass transfer in Carreau fluid flow over a permeable stretching sheet with convective slip conditions in the presence of applied magnetic field, nonlinear thermal radiation, cross diffusion and suction/injection effects was numerically studied by Gnaneswara et al. [29]. Groza et al. [30] presented a Newton interpolating series for approximate solutions of the entire functions of multipoint boundary value problems for differential equations. Marin [31] extended the concept of domain of influence, proposed by Cowin and Nunziato, in order to cover the elasticity of micro stretch materials. Ghita et al. [32] formulated some problems modeling the local hardening behavior of a plastic material following a PrandtlReuss law. Directional linear hardening, which is similar to Bauschinger’s effect in metals, is characterized by an anisotropic factor. The magnetohydrodynamic (MHD) flow of nonNewtonian nanofluid in a pipe was studied by Ellahi [33]. Ellahi et al. [34] theoretically investigated the problem of the peristaltic flow of Jeffrey fluid in a nonuniform rectangular duct under the effects of Hall and ion slip. Marin et al. [35] considered a right cylinder composed of a physically micropolar thermoelastic material for which one plane end is subjected to an excitation harmonic in time.
The aim of the present work is to model and analyze the steady boundary layer flow of MHD Carreau fluid slip flow with viscous dissipation and heat transfer by taking the effects of thermal radiation over a stretching sheet embedded in a porous medium with variable thickness and variable thermal conductivity. The system of nonlinear partial differential equations is transformed into a system of ordinary differential equations using appropriate similarity transformations. A model system of equations is solved analytically by means of the homotopy analysis method (HAM).
2 Mathematical formulation
2.1 Description of the problem
Consider twodimensional steady boundary layer flow of MHD Carreau fluid slip flow over a stretching sheet embedded in a porous medium. The origin is located at the slit, through the sheet is drawn in the fluid medium. The xaxis is taken in the direction of sheet motion, and the yaxis is normal to it. The sheet is stretched with velocity \(U_{w}= U_{0}(x+b)^{m}\), where \(U_{0}\) is the reference velocity. Assume that the sheet is not flat, which is specified as \(y=A(x+b)^{\frac{1m}{2}}\), where the coefficient A is chosen small for the sheet to be sufficiently thin, and m is the velocity power index. The problem is valid for \(m\neq1\) because for \(m=1\), the problem reduces to a flat sheet.
2.2 Governing equations and boundary conditions
3 Solution by the homotopy analysis method
4 Error analysis
Optimal value of convergence control parameters for different orders of approximations
Order of approximation  \(\boldsymbol{\hbar}_{\boldsymbol{f}}\)  \(\boldsymbol{\hbar}_{\boldsymbol{\theta}}\)  \(\boldsymbol{\varepsilon_{m}^{t}}\) 

2  −1.50039  −0.52479  7.16404 × 10^{−4} 
3  −1.14981  −0.42060  6.90592 × 10^{−4} 
4  −1.19560  −1.16693  5.15041 × 10^{−4} 
5  −1.24825  −0.30060  5.55759 × 10^{−4} 
6  −1.31412  −1.28875  5.01547 × 10^{−4} 
7  −1.35632  −1.31419  4.05433 × 10^{−4} 
8  −1.38483  −1.36406  2.18914 × 10^{−4} 
Individual averaged squared residual errors using optimal values of auxiliary parameters
m  \(\boldsymbol{\varepsilon_{m}^{f}}\)  \(\boldsymbol{\varepsilon_{m}^{\theta}}\) 

2  3.28588 × 10^{−5}  4.15975 × 10^{−4} 
4  2.10586 × 10^{−5}  6.47816 × 10^{−5} 
6  1.97175 × 10^{−5}  3.97772 × 10^{−5} 
8  1.92867 × 10^{−5}  2.39297 × 10^{−5} 
10  1.91356 × 10^{−5}  1.47931 × 10^{−5} 
12  1.90796 × 10^{−5}  9.87921 × 10^{−6} 
14  1.90562 × 10^{−5}  7.34606 × 10^{−6} 
16  1.90439 × 10^{−5}  6.07003 × 10^{−6} 
18  1.90350 × 10^{−5}  5.42235 × 10^{−6} 
20  1.90271 × 10^{−5}  5.07083 × 10^{−6} 
5 Results and discussion
In this article, the steady boundary layer flow of MHD Carreau fluid slip flow with viscous dissipation and heat transfer is studied by taking the effects of thermal radiation over a stretching sheet embedded in a porous medium with variable thickness and variable thermal conductivity. The system of nonlinear ordinary differential equations (16)(17) with boundary conditions (18) and (19) is solved analytically by HAM. The effects of Weissenberg number, power law index, magnetic parameter, velocity power index parameter, porous parameter, wall thickness parameter, slip velocity parameter, thermal conductivity parameter, radiation parameter, Eckert number and Prandtl number on the velocity and temperature fields are analyzed with the help of graphical aids and numerical results.
In order to investigate the accuracy of (HAM), we compared the values of skin friction \(f''({0})\) with those given in Eid et al. [28] and Fang et al. [22] for the case \(\alpha=0.5\), \(\alpha=0.25\) when \(\mathit{We}=0\), \(M=0\), \(\lambda=0\), \(n=0.5\) and \(D=0\), respectively, for different values of velocity power index m.
Values of \(\pmb{f''({0})}\) for different values of m when \(\pmb{n=0.5}\) , \(\pmb{\mathit{We}=0}\) , \(\pmb{\lambda=0}\) , \(\pmb{M=0}\) , \(\pmb{D=0}\) and \(\pmb{\alpha=0.5}\)
Values of \(\pmb{f''({0})}\) for different values of m when \(\pmb{n=0.5}\) , \(\pmb{\mathit{We}=0}\) , \(\pmb{\lambda=0}\) , \(\pmb{M=0}\) , \(\pmb{D=0}\) and \(\pmb{\alpha=0.25}\)
Values of \(\pmb{f''({0})}\) and \(\pmb{\theta'({0})}\) for various values of m , D , α , λ , ξ , R , Ec , Pr
D  α  m  ξ  R  Pr  Ec  λ  n  We  M  \(\boldsymbol{f''({0})}\)  \(\boldsymbol{\theta'({0})}\) 

0.0  0.2  0.5  0.1  0.5  1.0  0.2  0.2  0.5  0.3  0.5  0.825155  0.475184 
0.5                      0.974741  0.456129 
1.0                      1.099356  0.438155 
0.5  0.0  0.5  0.1  0.5  1.0  0.2  0.2  0.5  0.3  0.5  0.957396  0.444435 
  0.25                    0.979058  0.459059 
  0.0  1.0                  1.008661  0.438441 
  0.25                    1.005863  0.436451 
0.5  0.2  0.0  0.1  0.5  1.0  0.2  0.2  0.5  0.3  0.5  0.914802  0.491316 
    0.5                  0.974741  0.456129 
    1.0                  1.006404  0.431844 
0.5  0.2  0.5  0.0  0.5  1.0  0.2  0.2  0.5  0.3  0.5  0.974915  0.469772 
      0.2                0.974546  0.444876 
      0.5                0.972809  0.334618 
0.5  0.2  0.5  0.1  0.5  1.0  0.2  0.2  0.5  0.3  0.5  0.974741  0.454128 
        0.7              0.974199  0.448591 
        1.0              0.973378  0.440501 
0.5  0.2  0.5  0.1  0.5  0.7  0.2  0.2  0.5  0.3  0.5  0.974907  0.436983 
          1.0            0.974740  0.456128 
          3.0            0.977835  0.631131 
0.5  0.2  0.5  0.1  0.5  1.0  0.0  0.2  0.5  0.3  0.5  0.974738  0.498102 
            0.5          0.974749  0.393482 
            1.0          0.974630  0.210871 
0.5  0.2  0.5  0.1  0.5  1.0  0.2  0.0  0.5  0.3  0.5  1.300149  0.449268 
              0.5        0.724172  0.362024 
              1.0        0.514419  0.447731 
6 Conclusion

The increase in porosity parameter D, wall thickness parameter α, slip parameter λ and magnetic parameter \(M^{2}\) leads to the decrease in velocity; on the other hand, velocity increases with an increase in velocity power index parameter.

The increase in porous parameter D and Prandtl number Pr leads to the decrease in heat transfer.

The increase in slip parameter λ, velocity power index parameter m, magnetic parameter \(M^{2}\), thermal conductivity parameter ε radiation parameter R and Eckert number Ec leads to the increase in heat transfer.

The increase in porosity parameter D, wall thickness parameter α and velocity power index parameter m leads to the increase in skin friction, while the skin friction coefficient decreases with an increase in slip parameter.

The increase in Prandtl number Pr leads to the increase in local Nusselt number. The increase in porosity parameter D, velocity power index m, radiation parameter R, thermal conductivity parameter ε and Eckert number Ec leads to the decrease in local Nusselt number.
Declarations
Acknowledgements
The authors would like to thank the reviewers for their constructive comments and valuable suggestions to improve the quality of the paper. This paper is self supported by authors in respect of funding and technically supported by Islamia Collogue University, KP, Peshawar, Pakistan.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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