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 , where λ is the velocity parameter. The flow is assumed to take place at , 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 and , respectively, while the free stream values are taken to be and , respectively. Following the nanofluid model proposed by Tiwari and Das  along with Boussinesq and boundary layer approximations, the governing equations for the present problem are:
where u and v are the velocity components along the x-axis and y-axis, respectively, is the thermal diffusivity of the fluid, ν is the kinematic viscosity coefficient, k is the thermal conductivity, is the heat flux, is the Brownian diffusion coefficient, is the thermophoresis diffusion coefficient, 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
The surface moving parameter corresponds to the downstream movement of the plate from the origin, while corresponds to the upstream movement of the plate.
By using the Rosseland diffusion approximation (Hossain et al. ) and following Raptis  among other researchers, the radiative heat flux is given by
where and 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 may be expressed as a linear function of temperature.
Using (7) and (8) in the last term of equation (3), we obtain
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:
where is the dimensionless stream function, η is the similarity variable, is the dimensionless temperature, 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 and . Substituting the similarity variables into equations (2)-(4) gives
subject to the boundary conditions
where is the radiation parameter, is the Prandtl number, is the Lewis number, is the local Hartman number, is the Brownian motion parameter and is the thermophoresis parameter.
The quantities of engineering interest are the skin-friction coefficient , the local Nusselt number , and the local Sherwood number . These quantities are defined as follows:
where , and are the shear stress, heat flux and mass flux at the surface, respectively. Upon using the similarity variables into the above expressions, we get
where is the local Reynolds number.