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An effective two-dimensional model for MHD flows with transverse magnetic field

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 three-dimensional effects to be taken into account in a two-dimensional equation of motion thanks to a model for the transverse velocity profile. This model 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 constructed that describes inertial effects in these two regions. Two separate classes of phenomena are found: 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 two-dimensional equation of motion. Analytical solutions of this model are obtained for two experimental configurations: isolated vortices generated 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 vortex core spreading and excessive dissipative behaviour of the axisymmetric sidewall layers.