Solving Magnetohydrodynamic Free Convection Problems in Complex-shaped Cavities by the R-functions Method

A novel approach for solving magnetohydrodynamic free convection problems in complex-shaped cavities is proposed, which gives the possibility to avoid the necessity of constructing a complicated mesh in the neighborhood of the domain boundary. The technique is based on the R-function method combining the means of analytical geometry and variational techniques of mathematical physics, in particular the Galerkin method. The semi-analytical expressions in the form of expansions in certain bases are obtained for the temperature, vorticity, and stream functions, which satisfy boundary conditions exactly and approximate the fields inside the cavity with accuracy depending on the number of terms of the expansion. The effects of the Hartman number and the Rayleigh number on the fluid flow and heat transfer inside a square cavity are investigated.