We consider the steady two-dimensional boundary layer of a non-Newtonian nanofluid over a vertical impermeable wall embedded in a porous medium. The *x*-coordinate is taken along the plate and *y*-coordinate is normal to it. Temperature *T* and nanoparticle fraction *C* are taken as constant values {T}_{w} and {C}_{w}, respectively. The ambient values of *T* and *C* are denoted by {T}_{\mathrm{\infty}} and {C}_{\mathrm{\infty}}. The Oberbeck-Boussinesq approximation was employed and homogeneity and local thermal equilibrium in the porous medium was assumed. Using the standard boundary layer approximations, we can write the governing equations as follows:

\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,

(1)

n{u}^{n-1}\frac{\partial u}{\partial y}=\frac{(1-{C}_{\mathrm{\infty}})Kg\beta \rho {f}_{\mathrm{\infty}}}{\mu}\frac{\partial T}{\partial y}-\frac{({\rho}_{p}-\rho {f}_{\mathrm{\infty}})Kg}{\mu}\frac{\partial C}{\partial y},

(2)

u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{\partial}{\partial y}\left({\alpha}_{e}\frac{\partial T}{\partial y}\right)+\tau \{{D}_{B}\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+\frac{{D}_{T}}{{T}_{\mathrm{\infty}}}{\left(\frac{\partial T}{\partial y}\right)}^{2}\},

(3)

\frac{1}{\epsilon}(u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y})={D}_{B}\frac{{\partial}^{2}C}{\partial {y}^{2}}+\left(\frac{{D}_{T}}{{T}_{\mathrm{\infty}}}\right)\frac{{\partial}^{2}T}{\partial {y}^{2}},

(4)

where

{\alpha}_{e}={\alpha}_{m}+\gamma du,\phantom{\rule{2em}{0ex}}{\alpha}_{m}=\frac{k}{{(\rho c)}_{f}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\tau =\frac{\epsilon {(\rho c)}_{p}}{{(\rho c)}_{f}}.

(5)

The boundary conditions for equations (1)-(4) are given in the form:

\begin{array}{r}v=0,\phantom{\rule{2em}{0ex}}T={T}_{w},\phantom{\rule{2em}{0ex}}C={C}_{w},\phantom{\rule{1em}{0ex}}\text{at}y=0,\\ u\to 0,\phantom{\rule{2em}{0ex}}T\to {T}_{\mathrm{\infty}},\phantom{\rule{2em}{0ex}}C\to {C}_{\mathrm{\infty}},\phantom{\rule{1em}{0ex}}\text{as}y\to \mathrm{\infty}.\end{array}

(6)

Here *u*, *v* are velocity components in vertical (*x*) and horizontal (*y*) directions, *n* is the power law index, *K* is the modified permeability of the porous medium, *g* is the gravitational acceleration, *β* is the volumetric expansion of the base fluid, {\rho}_{f} is the density of the base fluid, *μ* is the consistency index of the power law fluid, *T* is the local temperature, *C* is the nanoparticle volume fraction, {\rho}_{p} is the density of nanoparticle, {\alpha}_{e} is the effective thermal diffusivity of the porous medium that can be written as {\alpha}_{e}={\alpha}_{m}+\gamma du, {\alpha}_{m} is the molecular thermal diffusivity, \gamma du represent the thermal diffusivity, *γ* is the mechanical thermal dispersion coefficient. This value lies between 1/7 and 1/3 (see Kairi *et al.* [4]), *d* is the pore diameter, *τ* is ratio of the effective heat capacity of the nanoparticle material and the heat capacity of the fluid, {D}_{B} is the Brownian motion coefficient, {D}_{T} is the thermophoretic diffusion coefficient, *ε* is the porosity of the porous medium, respectively, *k* is the effective thermal conductivity of the porous medium, {(\rho c)}_{f} is the heat capacity of the nanofluid and {(\rho c)}_{p} is the effective heat capacity of the nanoparticle material.

The modified permeability of the porous medium *K* of the non-Newtonian power law fluid is defined as

K=\frac{1}{2{c}_{t}}{\left(\frac{n\epsilon}{3n+1}\right)}^{n}{\left(\frac{50{k}^{\ast}}{3\epsilon}\right)}^{\frac{n+1}{2}},\phantom{\rule{1em}{0ex}}{k}^{\ast}=\frac{{\epsilon}^{3}{d}^{2}}{150{(1-\epsilon )}^{2}}

and

{c}_{t}=\{\begin{array}{cc}\frac{25}{12}\hfill & \text{[11]},\hfill \\ \frac{2}{3}{\left(\frac{8n}{9n+3}\right)}^{n}\left(\frac{10n-3}{6n+1}\right){\left(\frac{75}{16}\right)}^{3(10n-3)/(10n+11)}\hfill & \text{[12]}.\hfill \end{array}

For n=1, {c}_{t}=25/12. Equation (1) is satisfied by introducing a stream function \psi (x,y) such that

u=\frac{\partial \psi}{\partial y}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}v=-\frac{\partial \psi}{\partial x},

(7)

where \psi ={\alpha}_{m}{\mathrm{Ra}}_{x}^{\frac{1}{2}}f(\eta ), f(\eta ) is the dimensionless stream function, \eta =(y/x){\mathrm{Ra}}_{x}^{\frac{1}{2}} and {\mathrm{Ra}}_{x} is the local Rayleigh number given by

{\mathrm{Ra}}_{x}=\frac{x}{{\alpha}_{m}}{\left(\frac{(1-{C}_{\mathrm{\infty}})Kg\beta \rho {f}_{\mathrm{\infty}}\mathrm{\Delta}T}{\mu}\right)}^{\frac{1}{n}}.

The velocity components are given by

u=\left(\frac{{\alpha}_{m}}{x}\right){\mathrm{Ra}}_{x}{f}^{\prime}(\eta )\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}v=\left(\frac{{\alpha}_{m}}{2x}\right){\mathrm{Ra}}_{x}^{\frac{1}{2}}[\frac{y}{x}{\mathrm{Ra}}_{x}^{\frac{1}{2}}{f}^{\prime}(\eta )-f(\eta )].

(8)

The temperature and concentration are presented as

T={T}_{\mathrm{\infty}}+\mathrm{\Delta}T\theta (\eta )\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}C={C}_{\mathrm{\infty}}+\mathrm{\Delta}C\varphi (\eta ),

(9)

where \theta (\eta ) and \varphi (\eta ) are the dimensionless temperature and dimensionless concentration. On using equations (8) and (9), equations (2), (3), (4) and (6) transform into the following two-point boundary value problem:

n{\left({f}^{\prime}\right)}^{n-1}{f}^{\u2033}-{\theta}^{\prime}+\mathrm{Nr}{\varphi}^{\prime}=0,

(10)

{\theta}^{\u2033}+\frac{1}{2}f{\theta}^{\prime}+\gamma {\mathrm{Ra}}_{d}({f}^{\prime}{\theta}^{\u2033}+{f}^{\u2033}{\theta}^{\prime})+\mathrm{Nb}{\theta}^{\prime}{\varphi}^{\prime}+\mathrm{Nt}{\theta}^{\prime 2}=0,

(11)

{\varphi}^{\u2033}+\frac{1}{2}\mathrm{Le}f{\varphi}^{\prime}+\frac{\mathrm{Nt}}{\mathrm{Nb}}{\theta}^{\u2033}=0,

(12)

subject to the boundary conditions

f(0)=0,\phantom{\rule{2em}{0ex}}{f}^{\prime}(\mathrm{\infty})\to 0,

(13)

\theta (0)=1,\phantom{\rule{2em}{0ex}}\theta (\mathrm{\infty})\to 0,

(14)

\varphi (0)=1,\phantom{\rule{2em}{0ex}}\varphi (\mathrm{\infty})\to 0,

(15)

where the primes denote differentiation with respect to *η*. The non-dimensional constants in equations (10)-(12) are the buoyancy ratio parameter Nr, the coefficient of thermal dispersion *γ*, the pore diameter dependent Rayleigh number {\mathrm{Ra}}_{d}, the Brownian motion parameter Nb, the thermophoresis parameter Nt and the Lewis number Le. These are defined as

\begin{array}{r}\mathrm{Nr}=\frac{({\rho}_{p}-\rho {f}_{\mathrm{\infty}})\mathrm{\Delta}C}{(1-{C}_{\mathrm{\infty}})\beta \rho {f}_{\mathrm{\infty}}\mathrm{\Delta}T},\phantom{\rule{2em}{0ex}}{\mathrm{Ra}}_{d}=\frac{d}{{\alpha}_{m}}{\left(\frac{(1-{C}_{\mathrm{\infty}})Kg\beta \rho {f}_{\mathrm{\infty}}\mathrm{\Delta}T}{\mu}\right)}^{\frac{1}{n}},\\ \mathrm{Nb}=\frac{\tau {D}_{B}\mathrm{\u25b3}C}{{\alpha}_{m}},\phantom{\rule{2em}{0ex}}\mathrm{Nt}=\frac{\tau {D}_{T}\mathrm{\u25b3}T}{{T}_{\mathrm{\infty}}{\alpha}_{m}},\phantom{\rule{2em}{0ex}}\mathrm{Le}=\frac{{\alpha}_{m}}{\epsilon {D}_{B}}.\end{array}

(16)