MHD Carreau fluid slip flow over a porous stretching sheet with viscous dissipation and variable thermal conductivity

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 We2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{We}^{2}$\end{document}, magnetic parameter M2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M^{2}$\end{document}, 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.


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, glass-fiber 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 [] 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  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).

Mathematical formulation 2.1 Description of the problem
Consider two-dimensional 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 x-axis is taken in the direction of sheet motion, and the y-axis is normal to it. The sheet is stretched with velocity U w = U  (x + b) m , where U  is the reference velocity. Assume that the sheet is not flat, which is specified as 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 =  because for m = , the problem reduces to a flat sheet.

Governing equations and boundary conditions
The basic governing equations of continuity, boundary layer flow and heat transfer are where the velocity components u and v are along the x and y axes, ν, ρ and σ are the kinematic viscosity, fluid density and electrical conductivity, respectively. Other parameters, such as the acceleration due to gravity is g, T is the fluid temperature, κ is the thermal diffusivity, is the time constant, J is the magnetic field, k is the permeability of the porous medium, q r is the radiative heat flux, c p is the specific heat at constant pressure and n is the power law index. For n = , the Carreau model reduces to the Newtonian one.
The radiative heat flux q r is employed according to Rosseland approximation [] such that where σ * = . ·  - W m  K - is the Stefan-Boltzmann constant and k * is the mean absorption coefficient. Following Rapits [], we assume that the temperature differences within flow are sufficiently small such that T  may be expressed as a linear function of the temperature. Expanding T  in a Taylor series about T ∞ and neglecting higher order terms, we have The physical and mathematical advantage of the Rosseland formula () consists in the fact that it can be combined with Fourier's second law of conduction to an effective conduction-radiation flux q eff in the form is the effective thermal conductivity. So, the steady energy balance equation, including the net contribution of the radiation emitted from the hot wall and observed in the colder fluid, takes the form To obtain the similarity solutions, it is assumed that the permeability of the porous The corresponding equations are subjected to the boundary conditions where λ  is the slip coefficient having dimension of length. For similarity solutions, it is assumed that the slip coefficient λ  is of the form λ  = (x + b) -m  . The mathematical analysis of the problem is simplified by introducing the following dimensionless coordinates: where η is the similarity variable, ψ is the stream function defined as u = ∂ψ ∂y and v = -∂ψ ∂x and (η) is the dimensionless temperature. In this study, the equation for the dimensionless thermal conductivity κ is generalized for temperature dependence as follows: where κ ∞ is the ambient thermal conductivity and ξ is the thermal conductivity parameter. Using these variables, the boundary layer governing equations ()-() can be written in a non-dimensional form as follows: where ν is the parameter related to the sheet thickness, and η = α = A U  (m+) ν indicates the plate surface. In order to facilitate the computation, we introduce the function f (ζ ) = f (ηα) = F(η) and θ (ζ ) = θ (ηα) = (η). The similarity equations ()-() for f (ζ ) and the associated boundary conditions ()-() become, respectively, where the prime denotes differentiation with respect to ζ . Based on the variable transformation, the solution's domain will be fixed from  to ∞.
The physical quantity of interest in this study is the skin friction coefficient C f and the local Nusselt number Nu x , which are defined as where R e (x) = U w X ν is the local Reynolds number and X = x + b.

Solution by the homotopy analysis method
In this section we apply the HAM to obtain approximate analytical solutions of the 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. We select initial guesses and the linear operator for equations () and () as the above auxiliary linear operators have the following properties: where c i (i = -) are arbitrary constants. The zeroth order deformation problems can be obtained as where q is an embedding parameter, h f and h θ are the non-zero auxiliary parameters and N f , N θ are nonlinear operators.
For q =  and q = , we have As the embedding parameter q increases from  to , f (z; q) and θ (z; q) vary from their initial guesses f  and θ  to the exact solutions f (z) and θ (z), respectively.
Taylor's series expansion of these functions yields where Keep in mind that the above series depends on h f and h θ . On the assumption that the nonzero auxiliary parameters are chosen so that equations () converge at q = , we have Differentiating k-times the zeroth order deformation of Eqs. () and () one has the kth order deformation equations as where the boundary conditions () and () take the form Finally, the general solution may be written as follows: where f * k and θ * k are the special solutions.

Error analysis
Before starting analysis of the problem, we first analyze the accuracy of the HAM on this specific problem.

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 ther-      is true for m < . It is clear from figures that the thickness of the boundary layer becomes thicker for higher values of α when m >  and becomes thinner for higher values of α when m < .
The effect of the slip parameter λ on the velocity profile is shown in Figure . It is clear that the velocity profile decreases quickly throughout the fluid with the increase in the slip parameter λ. Boundary layer thickness decreases due to the effect of increasing the slip parameter λ while keeping other parameters fixed. The behavior of temperature distribution for the variation of the slip parameter λ is shown in Figure . It is obvious that the temperature profile increases with an increase in the slip parameter λ, while the other parameters are fixed.
The influence of velocity power index m on the velocity profile is displayed in Figure . The behavior of the velocity profile rises with a decrease in the values of velocity power index m. Thickness of boundary layer becomes thinner as m increases along the sheet   keeping the values of all other parameters fixed. Figure  depicts the velocity distribution for different values of M  . As the magnetic parameter increases, the velocity distribution gets decreased. Further the boundary layer thickness is decreased due to the influence of magnetic parameter M  while keeping other parameters fixed.
The variation of thermal conductivity parameter on the temperature profile is shown in Figure . From here we see that the temperature profile as well as the thickness of thermal boundary layer increase when the thermal boundary parameter is increased. The effect of radiation parameter R on the temperature profile θ (η) is plotted in Figure . It is depicted that the temperature field and the thermal boundary layer thickness increase with the increase in R. Figure  displays the effect of Prandtl number on the temperature profile. It can be seen that the behavior of temperature distribution decreases with an increase in the Prandtl number. The temperature distribution for various values of Eckert number   Ec is plotted in Figure . It is that the temperature distribution θ (η) increases with the increase in the value of Eckert number Ec keeping the other parameters fixed.
In order to investigate the accuracy of (HAM), we compared the values of skin friction -f () with those given in Eid et al. The quantitative comparison is shown in Tables  and . Analytical and numerical results are found to be in good agreement. Table  demonstrates the effects of power law index, magnetic parameter, Weissenberg number, porosity parameter, wall thickness parameter, slip velocity parameter, velocity power index parameter, radiation parameter, Eckert number, thermal conductivity parameter and Prandtl number on the skin friction coefficient and the local Nusselt number. It is noticed that the skin friction coefficient increased but the local Nusselt number reduced with increasing porosity parameter and velocity Table 3 Values of -f (0) for different values of m when n = 0.5, We = 0, λ = 0, M = 0, D = 0 and α = 0.5   Table 5 Values power index parameter. Skin friction coefficient increases with an increase in the wall thickness parameter for m = . and . The local Nusselt number increases with an increase in the Prandtl number. Moreover, it is clear that the local Nusselt number decreases with an increase in the Eckert number, the thermal conductivity parameter and the radiation parameter. The effect of slip parameter is to decrease the skin friction coefficient.

Conclusion
In this article, similarity solution of the steady boundary layer flow of MHD Carreau fluid slip flow with viscous dissipation and heat transfer is analyzed by taking the effects of thermal radiation over a stretching sheet embedded in a porous medium with variable thickness and variable thermal conductivity. The characteristics of velocity and temperature profiles are studied graphically. The main conclusion can be summarized as follows: • The increase in porosity parameter D, wall thickness parameter α, slip parameter λ and magnetic parameter M  leads to the decrease in velocity; on the other hand, velocity increases with an increase in velocity power index parameter.