Mixed convection MHD flow in a vertical channel: effects of Joule heating and viscous dissipation

Combined forced and free flow in a vertical channel with an adiabatic wall and an isothermal wall is investigated. The laminar, parallel and fully developed regime is considered. A uniform horizontal magnetic field is assumed to be applied to the fluid. The local balance equations are written in a dimensionless form and solved by taking into account the effects of Joule heating and viscous dissipation. The solutions are obtained both analytically by a power series method and numerically. The dimensionless governing parameters affecting the velocity and temperature profiles are the Hartmann number and the ratio between the Grashof number and the Reynolds number. Dual solutions are shown to exist for every value of the Hartmann number within a bounded range of the ratio between the Grashof number and the Reynolds number. Outside this range, no parallel flow solutions of the problem exist.