Magnetohydrodynamic steady flow computations in three dimensions

A complete three‐dimensional mathematical model has been developed governing the steady, laminar flow of an incompressible fluid subjected to a magnetic field and including internal heating due to the Joule effect, heat transfer due to conduction, and thermally induced buoyancy forces. The thermally induced buoyancy was accounted for via the Boussinesq approximation. The entire system of eight partial differential equations was solved by integrating intermittently a system of five fluid flow equations and a system of three magnetic field equations and transferring the information through source‐like terms. An explicit Runge‐Kutta time‐stepping algorithm and a finite difference scheme with artificial compressibility were used in the general non‐orthogonal curvilinear boundary‐conforming co‐ordinate system. Comparison of computational results and known analytical solutions in two and three dimensions demonstrates high accuracy and smooth monotone convergence of the iterative algorithm. Results of test cases with thermally induced buoyancy demonstrate the stabilizing effect of the magnetic field on the recirculating flows.