### 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 \(U_{w}= U_{0}(x+b)^{m}\), where \(U_{0}\) is the reference velocity. Assume that the sheet is not flat, which is specified as \(y=A(x+b)^{\frac{1-m}{2}}\), 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.

### Governing equations and boundary conditions

The basic governing equations of continuity, boundary layer flow and heat transfer are

$$\begin{aligned} &\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \end{aligned}$$

(1)

$$\begin{aligned} &u \frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}= \nu\frac{\partial^{2} u}{\partial y^{2}}- \frac{\mu}{\rho k}u+3\nu \frac{(n-1)}{2}\Gamma^{2}\biggl( \frac{\partial u}{\partial y}\biggr)^{2}\frac{\partial^{2} u}{\partial y^{2}} -\frac{\sigma J^{2}}{\rho}u, \end{aligned}$$

(2)

$$\begin{aligned} &\rho c_{p}\biggl(u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}\biggr)=\frac{\partial}{\partial y}\biggl(\kappa\frac{\partial T}{\partial y} \biggr)-\frac{\partial q_{r}}{\partial y}+\mu \biggl(\frac{\partial u}{\partial y}\biggr)^{2}, \end{aligned}$$

(3)

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, Γ 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 \(n=1\), the Carreau model reduces to the Newtonian one.

The radiative heat flux \(q_{r}\) is employed according to Rosseland approximation [36] such that

$$ q _{r}=\frac{4 \sigma^{*}}{3 k^{*}}\frac{\partial T^{4}}{\partial y}, $$

(4)

where \(\sigma^{*} = 5.6696\cdot 10^{-8} \mbox{ W m}^{2} \mbox{K}^{-4}\) is the Stefan-Boltzmann constant and \(k^{*}\) is the mean absorption coefficient. Following Rapits [37], we assume that the temperature differences within flow are sufficiently small such that \(T^{4}\) may be expressed as a linear function of the temperature. Expanding \(T^{4}\) in a Taylor series about \(T_{\infty}\) and neglecting higher order terms, we have

$$ T^{4} \cong 4T_{\infty}^{3}T-3T_{\infty}^{4}. $$

(5)

The physical and mathematical advantage of the Rosseland formula (4) consists in the fact that it can be combined with Fourier’s second law of conduction to an effective conduction-radiation flux \(q_{\mathrm{eff}}\) in the form

$$ q_{\mathrm{eff}}=-\biggl(\kappa+\frac{16\sigma^{*} T_{\infty}^{3}}{3k^{*}}\biggr) \frac{\partial T}{\partial y}= -\kappa_{\mathrm{eff}}\frac{\partial T}{\partial y}, $$

(6)

where \(\kappa_{\mathrm{eff}}=(\kappa+\frac{16\sigma^{*} T_{\infty}^{3}}{3k^{*}})\) 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

$$ \rho c_{p}\biggl(u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}\biggr)=\frac{\partial}{\partial y}\biggl(\kappa_{\mathrm{eff}} \frac{\partial T}{\partial y}\biggr)+\mu{\biggl(\frac{\partial u}{\partial y}\biggr)^{2}}. $$

(7)

To obtain the similarity solutions, it is assumed that the permeability of the porous medium \(k(x)\) is of the form \(k(x)= k_{0}(x+b)^{1-m}\), where \(k_{0}\) is the permeability parameter.

The corresponding equations are subjected to the boundary conditions

$$\begin{aligned} & \begin{aligned} &u(x,y)=U_{0}(x+b)^{m}+ \lambda_{1}\biggl(\frac{\partial u}{\partial y}\biggr) ,\qquad v(x,y)=0, \\ &T(x,y)=T_{w} \quad \mbox{at }y=A(x+b)^{\frac{1-m}{2}}, \end{aligned} \end{aligned}$$

(8)

$$\begin{aligned} &u(x,y)=0 ,\qquad T(x,y)=0 \quad \mbox{at } y\rightarrow\infty, \end{aligned}$$

(9)

where \(\lambda_{1}\) is the slip coefficient having dimension of length. For similarity solutions, it is assumed that the slip coefficient \(\lambda_{1}\) is of the form \(\lambda_{1}=(x+b)^{\frac{1-m}{2}}\).

The mathematical analysis of the problem is simplified by introducing the following dimensionless coordinates:

$$ \begin{aligned} &\eta=y\sqrt{U_{0}\biggl( \frac{m+1}{2}\biggr){\frac{(x+b)^{m-1}}{\nu}}}, \\ &\psi(x,y)=\sqrt {\nu U_{0}\biggl(\frac{2}{m+1}\biggr) (x+b)^{m+1}}F({\eta}), \\ &\Theta(\eta)=\frac{T-T_{\infty}}{T_{w}-T_{\infty}}, \end{aligned} $$

(10)

where *η* is the similarity variable, *ψ* is the stream function defined as \(u=\frac{\partial \psi}{\partial y}\) and \(v=-\frac{\partial \psi}{\partial x}\) and \(\Theta(\eta)\) is the dimensionless temperature.

In this study, the equation for the dimensionless thermal conductivity *κ* is generalized for temperature dependence as follows:

$$ \kappa=\kappa_{\infty}(1+\xi \Theta), $$

(11)

where \(\kappa_{\infty}\) is the ambient thermal conductivity and *ξ* is the thermal conductivity parameter.

Using these variables, the boundary layer governing equations (1)-(3) can be written in a non-dimensional form as follows:

$$\begin{aligned} &F'''+FF''-DF'+ \frac{3}{4}(n-1) (m+1)\mathit{We}^{2}F'''F^{\prime\prime2}-M^{2}F'- \frac{2m}{m+1}F^{\prime2}=0, \end{aligned}$$

(12)

$$\begin{aligned} &\biggl(\frac{1+R}{\mathit{Pr}}\biggr) \bigl((1+\xi\Theta) \Theta''+\xi\Theta^{\prime2}\bigr)+F \Theta'+EcF^{\prime2}=0, \end{aligned}$$

(13)

where \(\mathit{We}^{2}=\frac{\Gamma^{2}U_{0}^{3}(x+b)^{3m-1}}{\nu}\) is the Weissenberg number, \(D=\frac{2\nu}{k_{0}U_{0}(m+1)}\) is the porosity parameter, \(M^{2}=\frac{\sigma^{2}J_{0}^{2}}{\rho U_{0}}\) is the magnetic parameter, \(\mathit{Pr}=\frac{\mu c_{p}}{\kappa_{\infty}}\) is the Prandtl number, \(\mathit{Ec}=\frac{U_{w}^{2}}{c_{p}(T_{w}-T_{\infty})}\) is the Eckert number and \(R=\frac{16\sigma^{*}T_{\infty}^{3}}{3k^{*}\kappa_{\infty}}\) is the radiation parameter.

Boundary conditions (8)-(9) will be transformed

$$\begin{aligned} &F(\alpha)=\alpha\biggl(\frac{1-m}{1+m}\biggr)\bigl[1+\lambda F''(\alpha)\bigr],\qquad F'(\alpha)=1+ \lambda F''(\alpha),\qquad \Theta(\alpha)=1, \end{aligned}$$

(14)

$$\begin{aligned} &F'(\infty)=0,\qquad \Theta(\infty)=0, \end{aligned}$$

(15)

where \(\lambda=\frac{\sqrt {U_{0}(m+1)}}{\sqrt{2\nu}}\) is the slip velocity parameter, \(\alpha=A\sqrt{\frac{U_{0}(m+1)}{2\nu}}\) is the parameter related to the sheet thickness, and \(\eta=\alpha=A\sqrt{\frac{U_{0}(m+1)}{2\nu}}\) indicates the plate surface. In order to facilitate the computation, we introduce the function \(f({\zeta})=f({\eta-\alpha})=F({\eta})\) and \(\theta({\zeta})=\theta({\eta-\alpha})=\Theta({\eta})\). The similarity equations (12)-(13) for \(f({\zeta})\) and the associated boundary conditions (14)-(15) become, respectively,

$$\begin{aligned} &f'''+ff''-Df'+ \frac{3}{4}(n-1) (m+1)\mathit{We}^{2}f'''f^{\prime\prime2}-M^{2}f'- \frac{2m}{m+1}f^{\prime2}=0, \end{aligned}$$

(16)

$$\begin{aligned} &\biggl(\frac{1+R}{\mathit{Pr}}\biggr) \bigl((1+\xi\theta) \theta''+\xi\theta^{\prime2}\bigr)+f \theta'+Ecf^{\prime\prime2}=0, \end{aligned}$$

(17)

$$\begin{aligned} &f(0)=\alpha\biggl(\frac{1-m}{1+m}\biggr)\bigl[1+\lambda f''(0)\bigr],\qquad f'(0)=1+\lambda f''(0),\qquad \theta(0)=1, \end{aligned}$$

(18)

$$\begin{aligned} &f'(\infty)=0,\qquad \theta(\infty)=0, \end{aligned}$$

(19)

where the prime denotes differentiation with respect to *ζ*. Based on the variable transformation, the solution’s domain will be fixed from 0 to ∞.

The physical quantity of interest in this study is the skin friction coefficient \(C_{f}\) and the local Nusselt number \(\mathit{Nu}_{x}\), which are defined as

$$ C_{f}=-2\sqrt{\frac{m+1}{2}}\mathit{Re}_{x}^{\frac{-1}{2}}f''(0),\qquad \mathit{Nu}=-\sqrt {\frac{m+1}{2}}\mathit{Re}_{x}^{\frac{1}{2}}\theta'(0), $$

(20)

where \(R_{e}({x})=\frac {U_{w}X}{\nu}\) is the local Reynolds number and \(X=x+b\).