MHD convection flow in a constricted channel

We consider the steady, laminar, convection flow in a long channel of 2D rectangular constricted cross-section under the influence of an applied magnetic field. The Navier-Stokes equations including Lorentz and buoyancy forces are coupled with the temperature equation and are solved by using linear radial basis function (RBF) approximations in terms of the velocity, pressure and the temperature of the fluid. RBFs are used in the approximation of the particular solution which becomes also the approximate solution of the problem. Results are obtained for several values of Grashof number (Gr), Hartmann number (M) and the constriction ratios (CR) to see the effects on the flow and isotherms for fixed values of Reynolds number and Prandtl number. As M increases, the flow is flattened. An increase in Gr increases the magnitude of the flow in the channel. Isolines undergo an inversion at the center of the channel indicating convection dominance due to the strong buoyancy force, but this inversion is retarded with the increase in the strength of the applied magnetic field. When both Hartmann number and constriction ratio are increased, flow is divided into more loops symmetrically with respect to the axes