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Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. (English) Zbl 0753.76194
Summary: We propose an analytical solution to the equations of magnetohydrodynamics that can be used to model the effect of a transverse magnetic field on buoyancy driven convection in a two-dimensional cavity. In the high Hartmann number limit, the velocity gradient in the core is constant outside of the two Hartmann layers at the vicinity of the walls normal to the magnetic field. We show that this core solution is correct everywhere in the cavity except in a boundary layer of extent $H{a}^{-1/2}$ at the cold wall. The recirculating part of the flow is studied by means of a series expansion that allows the computation of the stream function. We also present the variation of both components of velocity as a function of $Ha$, along with a discussion of the validity of our assumptions.
##### MSC:
 76W05 Magnetohydrodynamics and electrohydrodynamics 76R10 Free convection (fluid mechanics) 35C10 Series solutions of PDE