We consider the steady MHD boundary layer flow of a nanofluid past a moving semi-infinite flat plate in a uniform free stream in the presence of thermal radiation. We assume that the velocity of the uniform stream is *U* and that of the plate is {U}_{w}=\lambda U, where *λ* is the velocity parameter. The flow is assumed to take place at y\ge 0, with *y* being the coordinate measured normal to the moving surface. A uniform magnetic field is applied in the *y* direction. At moving surface, the temperature and the nanoparticles take constant values {T}_{w} and {C}_{w}, respectively, while the free stream values are taken to be {T}_{\mathrm{\infty}} and {C}_{\mathrm{\infty}}, respectively. Following the nanofluid model proposed by Tiwari and Das [21] along with Boussinesq and boundary layer approximations, the governing equations for the present problem are:

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

(1)

u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{\partial}^{2}u}{\partial {y}^{2}}-\frac{\sigma {B}_{0}^{2}}{{\rho}_{f}}u,

(2)

u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{\partial}^{2}T}{\partial {y}^{2}}-\frac{\alpha}{k}\frac{\partial {q}_{r}}{\partial y}+\tau [{D}_{B}\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}+\left(\frac{{D}_{T}}{{T}_{\mathrm{\infty}}}\right){\left(\frac{\partial T}{\partial y}\right)}^{2}],

(3)

u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={D}_{B}\frac{{\partial}^{2}C}{\partial {y}^{2}}+\frac{{D}_{B}}{{T}_{\mathrm{\infty}}}\frac{{\partial}^{2}T}{\partial {y}^{2}},

(4)

where *u* and *v* are the velocity components along the *x*-axis and *y*-axis, respectively, \alpha =k/{(\rho c)}_{f} is the thermal diffusivity of the fluid, *ν* is the kinematic viscosity coefficient, *k* is the thermal conductivity, {q}_{r} is the heat flux, {D}_{B} is the Brownian diffusion coefficient, {D}_{T} is the thermophoresis diffusion coefficient, {B}_{0} is the uniform magnetic field strength of the base fluid, *σ* is the electrical conductivity of the base fluid, *τ* is the ratio of the nanoparticle heat capacity and the base fluid heat capacity.

The boundary conditions are taken as

v=0,\phantom{\rule{2em}{0ex}}u=\lambda U,\phantom{\rule{2em}{0ex}}T={T}_{w},\phantom{\rule{2em}{0ex}}C={C}_{w}\phantom{\rule{1em}{0ex}}\text{at}y=0,

(5)

u\to U,\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}.

(6)

The surface moving parameter \lambda >0 corresponds to the downstream movement of the plate from the origin, while \lambda <0 corresponds to the upstream movement of the plate.

By using the Rosseland diffusion approximation (Hossain *et al.* [22]) and following Raptis [23] among other researchers, the radiative heat flux {q}_{r} is given by

{q}_{r}=-\frac{4{\sigma}^{\ast}}{3{K}_{s}}\frac{\partial {T}^{4}}{\partial y},

(7)

where {\sigma}^{\ast} and {K}_{s} are the Stefan-Boltzman constant and the Rosseland mean absorption coefficient, respectively. We assume that the temperature differences within the flow are sufficiently small such that {T}^{4} may be expressed as a linear function of temperature.

{T}^{4}\approx 4{T}_{\mathrm{\infty}}^{3}T-3{T}_{\mathrm{\infty}}^{4}.

(8)

Using (7) and (8) in the last term of equation (3), we obtain

\frac{\partial {q}_{r}}{\partial y}=-\frac{16{\sigma}^{\ast}{T}_{\mathrm{\infty}}^{3}}{3{K}_{s}}\frac{{\partial}^{2}T}{\partial {y}^{2}}.

(9)

### 2.1 Similarity transformations

In order to reduce the governing equations into a system of ordinary differential equations, we introduce the following local similarity variables:

\begin{array}{r}\psi ={(2U\nu x)}^{\frac{1}{2}}f(\eta ),\phantom{\rule{2em}{0ex}}\theta (\eta )=\frac{T-{T}_{\mathrm{\infty}}}{{T}_{w}-{T}_{\mathrm{\infty}}},\\ \varphi (\eta )=\frac{C-{C}_{\mathrm{\infty}}}{{C}_{w}-{C}_{\mathrm{\infty}}},\phantom{\rule{2em}{0ex}}\eta ={(U/2\nu x)}^{\frac{1}{2}}y,\end{array}

(10)

where f(\eta ) is the dimensionless stream function, *η* is the similarity variable, \theta (\eta ) is the dimensionless temperature, \varphi (\eta ) is the dimensionless nanoparticles concentration. It is worth mentioning that the continuity equation (1) is identically satisfied from our choice of the stream function with u=\frac{\partial \psi}{\partial y} and v=-\frac{\partial \psi}{\partial x}. Substituting the similarity variables into equations (2)-(4) gives

{f}^{\u2034}+f{f}^{\u2033}-\mathit{Ha}{f}^{\prime}=0,

(11)

\left(\frac{3+4R}{\mathit{Pr}}\right){\theta}^{\u2033}+f{\theta}^{\prime}+\mathit{Nb}{\varphi}^{\prime}{\theta}^{\prime}+\mathit{Nt}{\theta}^{\prime 2}=0,

(12)

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

(13)

subject to the boundary conditions

f(0)=0,\phantom{\rule{2em}{0ex}}{f}^{\prime}(0)=\lambda ,\phantom{\rule{2em}{0ex}}\theta (0)=1,\phantom{\rule{2em}{0ex}}\varphi (0)=1,

(14)

{f}^{\prime}\to 1,\phantom{\rule{2em}{0ex}}\theta \to 0,\phantom{\rule{2em}{0ex}}\varphi \to 0\phantom{\rule{1em}{0ex}}\text{as}\eta \to \mathrm{\infty},

(15)

where R=\frac{4\alpha \delta {T}_{\mathrm{\infty}}^{3}}{kk} is the radiation parameter, \mathit{Pr}=\frac{\nu}{\alpha} is the Prandtl number, \mathit{Le}=\frac{\nu}{{D}_{B}} is the Lewis number, \mathit{Ha}=\frac{2x{B}_{0}^{2}}{U\rho f} is the local Hartman number, \mathit{Nb}=\frac{{(\rho c)}_{p}{D}_{B}({\varphi}_{w}-{\varphi}_{\mathrm{\infty}})}{(\rho C)f\nu} is the Brownian motion parameter and \mathit{Nt}=\frac{{(\rho C)}_{p}{\rho}_{T}({T}_{w}-{T}_{\mathrm{\infty}})}{(\rho c)f{T}_{\mathrm{\infty}}\nu} is the thermophoresis parameter.

The quantities of engineering interest are the skin-friction coefficient {C}_{f}, the local Nusselt number {\mathit{Nu}}_{x}, and the local Sherwood number {\mathit{Sh}}_{x}. These quantities are defined as follows:

{C}_{f}=\frac{{\tau}_{w}}{\rho {u}^{2}},\phantom{\rule{2em}{0ex}}{\mathit{Nu}}_{x}=\frac{{q}_{w}}{k({T}_{w}-{T}_{\mathrm{\infty}})},\phantom{\rule{2em}{0ex}}{\mathit{Sh}}_{x}=\frac{x{q}_{m}}{{D}_{B}({C}_{w}-{C}_{\mathrm{\infty}})},

(16)

where {\tau}_{w}, {q}_{w} and {q}_{m} are the shear stress, heat flux and mass flux at the surface, respectively. Upon using the similarity variables into the above expressions, we get

{(2\mathit{Re})}^{\frac{1}{2}}{C}_{f}={f}^{\u2033}(0),\phantom{\rule{2em}{0ex}}{\left(\frac{{\mathit{Re}}_{x}}{2}\right)}^{-1}2{\mathit{Nu}}_{x}=-{\theta}^{\prime}(0),\phantom{\rule{2em}{0ex}}{\left(\frac{{\mathit{Re}}_{x}}{2}\right)}^{-1}2{\mathit{Sh}}_{x}=-{\varphi}^{\prime}(0),

(17)

where {\mathit{Re}}_{x}=Ux/\nu is the local Reynolds number.