  This paper presents a model for quasi two-dimensional MHD flows between two
planes with small magnetic Reynolds number and constant transverse magnetic
field orthogonal to the planes. A method is presented that allows to take 3D
effects into account in a 2D equation of motion thanks to a model for the
transverse velocity profile. The latter is obtained by using a double
perturbation asymptotic development both in the core flow and in the Hartmann
layers arising along the planes. A new model is thus built that describes
inertial effects in these two regions. Two separate classes of phenomena are
thus pointed out : the one related to inertial effects in the Hartmann layer
gives a model for recirculating flows and the other introduces the possibility
of having a transverse dependence of the velocity profile in the core flow. The
''recirculating'' velocity profile is then introduced in the transversally
averaged equation of motion in order to provide an effective 2D equation of
motion. Analytical solutions of this model are obtained for two experimental
configurations : isolated vortices aroused by a point electrode and
axisymmetric parallel layers occurring in the MATUR (MAgneticTURbulence)
experiment. The theory is found to give a satisfactory agreement with the
experiment so that it can be concluded that recirculating flows are actually
responsible for both vortices core spreading and excessive dissipative behavior
of the axisymmetric side wall layers.
