where \(u,\ v\) is the velocity components in the \(x,\ y\) directions,\(\upsilon\) is the kinematic viscosity, \(\sigma\) is the fluid
electric conductivity, \(B_{0}\) is applied magnetic field strength,\(\rho\) is fluid density, \(g\) is gravitational acceleration,\(\beta\) is thermal expansion coefficient, \(\beta^{*}\) is
concentration expansion coefficient, \(T\) is temperature,\(T_{\infty}\) is free stream temperature, \(\kappa(T)\) is thermal
conductivity variation, \(q_{r}\) is radiative heat flux, \(q\) is
volumetric rate, \(D\) is mass diffusivity, \(k_{t}\) is thermal
diffusion ratio, \(c_{p}\) is specific heat at constant pressure,\(c_{s}\) is concentration susceptibility, \(C\) is concentration,\(C_{\infty}\) is free stream concentration, \(T_{m}\) is mean fluid
temperature. The Dufour and Soret effects are described by a second
order concentration and temperature derivatives respectively\(\left[\mathbf{12}\right]\) in Eq. (3) and Eq. (4). The
boundary conditions are given as