Consider the steady, laminar flow of an incompressible electrically conducting non-Newtonian nanofluid over a permeable flat surface. The power-law non-Newtonian fluid model is assumed for the water based nanofluid and power-law velocity slip condition is employed at the boundary. A uniform magnetic field is applied in the transverse direction to the flow. The viscosity and the thermal conductivity vary with temperature *T*. The porosity of the surface is considered as uniform and the uniform magnetic field of strength *B*_{o} is applied in the direction perpendicular to sheet. The induced magnetic field is considered negligible when compared to the applied magnetic field. In view of the above assumptions, as well as of the usual boundary layer approximations the governing equations are thus:

$$\begin{array}{}\frac{\mathrm{\partial}u}{\mathrm{\partial}x}+\frac{\mathrm{\partial}v}{\mathrm{\partial}y}=0,\end{array}$$(1)

$$\begin{array}{}{\displaystyle u\frac{\mathrm{\partial}u}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}u}{\mathrm{\partial}y}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =\frac{1}{{\rho}_{nf}}\frac{\mathrm{\partial}}{\mathrm{\partial}y}\left[{\mu}_{nf}(T)|\frac{\mathrm{\partial}u}{\mathrm{\partial}y}{|}^{n-1}\frac{\mathrm{\partial}u}{\mathrm{\partial}y}\right]}\\ & {\displaystyle -\frac{{\sigma}_{nf}}{{\rho}_{nf}}{B}_{0}^{2}(u-{U}_{\mathrm{\infty}}),}\end{array}$$(2)

and
$$\begin{array}{}{\displaystyle u\frac{\mathrm{\partial}T}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}T}{\mathrm{\partial}y}=\frac{1}{(\rho {C}_{p}{)}_{nf}}\frac{\mathrm{\partial}}{\mathrm{\partial}y}\left[{\kappa}_{nf}(T)\frac{\mathrm{\partial}T}{\mathrm{\partial}y}\right].}\end{array}$$(3)

Here *u* is a component of velocity along the direction of the flow and *v* is the velocity perpendicular to it. *μ*_{nf}(*T*), *ρ*_{nf}, *σ*_{nf}, (*C*_{p})_{nf}, *κ*_{nf}(*T*) refer to nanofluid dynamic viscosity, density, electrical conductivity, specific heat capacity of nanofluid at fixed pressure and thermal conductivity respectively. Moreover, *U*_{∞} is the velocity far away from the surface known as the free stream velocity and *n* is the power-law index.

The appropriate conditions for the modeled problem are:

$$\begin{array}{}{\displaystyle u(0)={L}_{1}(\frac{\mathrm{\partial}u}{\mathrm{\partial}y}{)}^{n},\phantom{\rule{1em}{0ex}}v(0)={V}_{w},\phantom{\rule{1em}{0ex}}{u}_{y\to \mathrm{\infty}}={U}_{\mathrm{\infty}}}\end{array}$$(4)

$$\begin{array}{}{\displaystyle T(0)={T}_{w}+{D}_{1}\frac{\mathrm{\partial}T}{\mathrm{\partial}y},\phantom{\rule{1em}{0ex}}{T}_{y\to \mathrm{\infty}}={T}_{\mathrm{\infty}},}\end{array}$$(5)

where *T*_{w} is the surface temperature and *T*_{∞} is the temperature outside the boundary layer, $\begin{array}{}{\displaystyle {L}_{1}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{L}_{0}\sqrt{{R}_{{e}_{x}}}}\end{array}$ and $\begin{array}{}{\displaystyle {D}_{1}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{D}_{0}\sqrt{{R}_{{e}_{x}}}}\end{array}$ are the velocity and thermal slip factors with *L*_{0} as initial velocity slip, *D*_{0} as initial thermal slip and $\begin{array}{}{\displaystyle {R}_{{e}_{x}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{{\rho}_{f}{x}^{n}}{{\mu}_{f}{U}_{\mathrm{\infty}}^{(n-2)}}}\end{array}$ is the local Reynolds number. Finally, *V*_{w} is the constant suction/injection velocity across the surface. Commonly used property relations for nanofluids are presented as (for details see [31, 32, 33]):

$$\begin{array}{}{\displaystyle {\mu}_{nf}(T)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{\mu}_{nf}^{\ast}\phantom{\rule{thinmathspace}{0ex}}\left[a+b({T}_{w}-T)\right],}\\ {\kappa}_{nf}(T)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{\kappa}_{nf}^{\ast}\phantom{\rule{thinmathspace}{0ex}}\left[1+\u03f5\frac{T\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{T}_{\mathrm{\infty}}}{{T}_{w}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{T}_{\mathrm{\infty}}}\right],\end{array}$$(6)

$$\begin{array}{}\phantom{\rule{2em}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\rho}_{{n}_{f}}=(1-\varphi ){\rho}_{f}+\varphi {\rho}_{s},\\ (\rho {C}_{p}{)}_{{n}_{f}}=(1-\varphi )(\rho {C}_{p}{)}_{f}+\varphi (\rho {C}_{p}{)}_{s},\end{array}$$(7)

$$\begin{array}{}{\displaystyle {\mu}_{{n}_{f}}^{\ast}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{{\mu}_{f}}{(1-\varphi {)}^{2.5}},}\\ {\displaystyle \frac{{\kappa}_{{n}_{f}}^{\ast}}{{\kappa}_{f}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{({\kappa}_{s}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}2{\kappa}_{f})\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}2\varphi ({\kappa}_{f}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{\kappa}_{s})}{({\kappa}_{s}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}2{\kappa}_{f})\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\varphi ({\kappa}_{f}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{\kappa}_{s})},}\end{array}$$(8)

and
$$\begin{array}{}{\displaystyle \frac{{\sigma}_{{n}_{f}}}{{\sigma}_{f}}=\left[1+\frac{3(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1)\varphi}{(\frac{{\sigma}_{s}}{{\sigma}_{f}}-2)-(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1)\varphi}\right].}\end{array}$$(9)

In Eqs. (6)-(9), $\begin{array}{}{\mu}_{nf}^{\ast}\end{array}$ is the effective dynamic viscosity of the nanofluid, $\begin{array}{}{\kappa}_{nf}^{\ast}\end{array}$ is a constant thermal conductivity with *ϵ* as a parameter, *a* and *b* are positive constants, *ϕ* is the volume fraction of solid nanoparticles in the base fluid, *ρ*_{f}, (*Cp*)_{f}, *μ*_{f}, *κ*_{f} and *σ*_{f} are the density, specific heat capacity, coefficient of viscosity, thermal conductivity and the electrical conductivity of the base fluid. Whereas *ρ*_{s}, (*Cp*)_{s}, *μ*_{s}, *κ*_{s} and *σ*_{s} are the density, specific heat capacity, coefficient of viscosity, thermal conductivity and electrical conductivity of the nanoparticles respectively.

In order to solve the governing boundary value problem (1)-(3), the stream function *ψ*(*x, y*) is introduced which identically satisfies Eq. (1) with

$$\begin{array}{}{\displaystyle u=\frac{\mathrm{\partial}\psi}{\mathrm{\partial}y}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}v=-\frac{\mathrm{\partial}\psi}{\mathrm{\partial}x}.}\end{array}$$(10)

Equations (2)-(3) after utilizing Eqs.

$$\begin{array}{}{\displaystyle}& \frac{\mathrm{\partial}\psi}{\mathrm{\partial}y}\frac{{\mathrm{\partial}}^{2}\psi}{\mathrm{\partial}x\mathrm{\partial}y}-\frac{\mathrm{\partial}\psi}{\mathrm{\partial}x}\frac{{\mathrm{\partial}}^{2}\psi}{\mathrm{\partial}{y}^{2}}=\frac{{\mu}_{nf}^{\ast}}{{\rho}_{nf}}\left[-b\frac{\mathrm{\partial}T}{\mathrm{\partial}y}|\frac{{\mathrm{\partial}}^{2}\psi}{\mathrm{\partial}{y}^{2}}{|}^{n-1}\frac{{\mathrm{\partial}}^{2}\psi}{\mathrm{\partial}{y}^{2}}\right]\\ & -\frac{{\sigma}_{nf}}{{\rho}_{nf}}{B}^{2}(\frac{\mathrm{\partial}\psi}{\mathrm{\partial}y}-{u}_{\mathrm{\infty}})+\frac{{\mu}_{nf}^{\ast}}{{\rho}_{nf}}\left[\left[a+b({T}_{w}-T)\right]\{\left(n\right.\right.\\ & \left.\left.-1\right)|\frac{{\mathrm{\partial}}^{2}\psi}{\mathrm{\partial}{y}^{2}}{|}^{n-1}(\frac{{\mathrm{\partial}}^{2}\psi}{\mathrm{\partial}{y}^{2}}{)}^{2}\frac{{\mathrm{\partial}}^{3}\psi}{\mathrm{\partial}{y}^{3}}+|\frac{{\mathrm{\partial}}^{2}\psi}{\mathrm{\partial}{y}^{2}}{|}^{n-1}\frac{{\mathrm{\partial}}^{3}\psi}{\mathrm{\partial}{y}^{3}}\}\right]\end{array}$$(11)

and
$$\begin{array}{}{\displaystyle}& \frac{\mathrm{\partial}\psi}{\mathrm{\partial}y}\frac{\mathrm{\partial}T}{\mathrm{\partial}x}-\frac{\mathrm{\partial}\psi}{\mathrm{\partial}x}\frac{\mathrm{\partial}T}{\mathrm{\partial}y}=\frac{{\kappa}_{nf}^{\ast}}{(\rho {C}_{p}{)}_{nf}}\left[\frac{\u03f5}{{T}_{w}-{T}_{\mathrm{\infty}}}\right]{\left(\frac{\mathrm{\partial}T}{\mathrm{\partial}y}\right)}^{2}\\ & +\frac{{\kappa}_{nf}^{\ast}}{(\rho {C}_{p}{)}_{nf}}\left[1+\u03f5\frac{T-{T}_{\mathrm{\infty}}}{{T}_{w}-{T}_{\mathrm{\infty}}}\right]\frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{y}^{2}}.\end{array}$$(12)

Boundary conditions (4) are likewise transformed into

$$\begin{array}{}{\displaystyle}& \frac{\mathrm{\partial}\psi}{\mathrm{\partial}y}={L}_{1}{\left(\frac{{\mathrm{\partial}}^{2}\psi}{\mathrm{\partial}{y}^{2}}\right)}^{n},\phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial}\psi}{\mathrm{\partial}x}=-{V}_{w},\\ & \text{at}\phantom{\rule{1em}{0ex}}y=0;\phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial}\psi}{\mathrm{\partial}y}\to 0\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{1em}{0ex}}y\to \mathrm{\infty}.\end{array}$$(13)

The following dimensionless similarity variable and similarity transformations are introduced into Eqs. (11)-(12)

$$\begin{array}{}{\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\eta \phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{\left(\frac{{R}_{e}}{x/L}\right)}^{\frac{1}{n+1}}\frac{y}{L},}\\ \psi (x,y)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}L{U}_{\mathrm{\infty}}(\frac{x/L}{{R}_{e}}{)}^{\frac{1}{n+1}}f(\eta ),\phantom{\rule{1em}{0ex}}\theta (\eta )\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{T-{T}_{\mathrm{\infty}}}{{T}_{w}-{T}_{\mathrm{\infty}}},\end{array}$$(14)

where *θ* is the dimensionless temperature and $\begin{array}{}{\displaystyle {R}_{e}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{{\rho}_{f}{L}^{n}}{{\mu}_{f}{U}_{\mathrm{\infty}}^{(n-2)}}}\end{array}$ is the generalized Reynolds number.

The system (11)-(12) is reduced into a self-similar system of ordinary differential equations

$$\begin{array}{}{\displaystyle}& n(a+A-A\theta )|{f}^{\u2033}{|}^{n-1}{f}^{\u2034}+({\varphi}_{2}/{\varphi}_{1})\left(\frac{1}{n+1}\right)f{f}^{\u2033}\\ & -A{\theta}^{\prime}|{f}^{\u2033}{|}^{n}-({\varphi}_{4}/{\varphi}_{1})M({f}^{\prime}-1)=0,\end{array}$$(15)

$$\begin{array}{}{\displaystyle (1+\u03f5\theta ){\theta}^{\u2033}+\frac{1}{n+1}({\varphi}_{3}/{\varphi}_{5}){P}_{r}f{\theta}^{\prime}+\u03f5{\theta}^{\prime 2}=0.}\end{array}$$(16)

Here $\begin{array}{}{\displaystyle M=\frac{{\sigma}_{f}{B}^{2}x}{{\rho}_{f}{U}_{\mathrm{\infty}}}}\end{array}$ is the magnetic parameter, *A* = *b*(*T*_{w} − *T*_{∞}) is the viscosity parameter,

$\begin{array}{}{\displaystyle}\\ {P}_{rx}=\frac{(\rho {C}_{p}{)}_{f}{L}^{2}{U}_{\mathrm{\infty}}}{{\kappa}_{f}x}\end{array}$ is the local Prandtl number and

$$\begin{array}{}{\displaystyle}& {\varphi}_{1}=(1-\varphi {)}^{2.5},\phantom{\rule{1em}{0ex}}{\varphi}_{2}=\left((1-\varphi )+\varphi \frac{{\rho}_{s}}{{\rho}_{f}}\right),\\ & {\varphi}_{3}=\left((1-\varphi )+\varphi \frac{(\rho {C}_{p}{)}_{s}}{(\rho {C}_{p}{)}_{f}}\right),\\ & {\varphi}_{4}=\left(1+\frac{3(r-1)\varphi}{(r-2)-(r-1)\varphi}\right),\\ & {\varphi}_{5}=\left(\frac{({k}_{s}+2{k}_{f})-2\varphi ({k}_{f}-{k}_{s})}{({k}_{s}+2{k}_{f})+\varphi ({k}_{f}-{k}_{s})}\right),\phantom{\rule{1em}{0ex}}r=\frac{{\sigma}_{s}}{{\sigma}_{f}}.\end{array}$$

Similarly, the boundary conditions in equation(13) are transformed to

$$\begin{array}{}{\displaystyle f(0)=\phantom{\rule{thinmathspace}{0ex}}S,\phantom{\rule{1em}{0ex}}{f}^{\prime}(0)=1+\delta ({f}^{\u2033}(0){)}^{n},}\\ \theta (0)=1+\gamma {\theta}^{\prime}(0),\end{array}$$(17)

$$\begin{array}{}{\displaystyle {f}^{\prime}(\eta )\to 1,\phantom{\rule{1em}{0ex}}\theta (\eta )\to 0,\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{1em}{0ex}}\eta \to \mathrm{\infty},}\end{array}$$(18)

where $\begin{array}{}{\displaystyle S=-\frac{(n+1){x}^{\frac{n}{n+1}}}{({U}_{\mathrm{\infty}}{)}^{\frac{2n-1}{n+1}}}(\frac{{\rho}_{f}}{{\mu}_{f}}{)}^{\frac{1}{n+1}}}\end{array}$ *V*_{w} is the suction/injection parameter, $\begin{array}{}{\displaystyle \delta =\frac{{L}^{\frac{n+2}{2}}}{({U}_{\mathrm{\infty}}{)}^{\frac{n(n-5)}{2n+2}}}(\frac{{\rho}_{f}}{{\mu}_{f}}{)}^{\frac{3n+1}{2n+2}}}\end{array}$ is the velocity slip parameter and $\begin{array}{}{\displaystyle \gamma \phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}D(\frac{{L}^{n(n+2)}}{x}{)}^{\frac{1}{n+1}}(\frac{{\rho}_{f}}{{\mu}_{f}}{)}^{\frac{n+3}{n+1}}}\end{array}$ is the thermal slip parameter.

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