Time dependent magnetic field effects on the MHD flow and heat transfer in a rectangular duct

We consider the effect of time dependent magnetic field on the natural convection magnetohydrodynamic flow in a rectangular duct. The flow is two-dimensional, unsteady, laminar and the fluid is electrically conducting and incompressible. The heat transfer is taken into account using the Boussinesq approximation and the buoyancy force is included in the momentum equations for the thermal coupling. The flow is considered at low magnetic Reynolds number setting and hence, the induced magnetic field is neglected. The proposed model in which the governing equations are given in terms of stream function, vorticity and temperature, is approximated by using a Chebyshev spectral collocation method for the spatial discretisation coupled with a backward difference scheme that is unconditionally stable for the temporal integration. The flow and heat transfer characteristic are analysed for several definitions of applied time varying magnetic field. Increase of the Hartmann number and the time variation of the applied magnetic field, changes the flow behaviour, especially at transient time levels and at the steady-state.