Energy constraints in forced recirculating MHD flows

We are concerned here with forced steady recirculating flows which are lami nar, two-dimensional and have a high Reynolds number. The body force is con sidered to be prescribed and independent of the flow, a situation which ari ses frequently in magnetohydrodynamics. Such flows are subject to a strong constraint. Specifically, the body force generates kinetic energy throughou t the how field, yet dissipation is confined to narrow singular regions suc h as boundary layers. If the how is to achieve a steady state, then the kin etic energy which is continually generated within the bulk of the flow must find its way to the dissipative regions. Now the distribution of u(2)/2 is governed by a transport equation, in which the only cross-stream transport of energy is diffusion, nu del(2)(u(2)/2). It follows that there are only two possible candidates for the transport of energy to the dissipative regi ons: the energy could be diffused to the shear layers, or else it could be convected to the shear layers through entrainment of the streamlines. We in vestigate both options and show that neither is a likely candidate at high Reynolds number. We then describe numerical experiments for a model problem designed to resolve these issues. We show that, at least for our model pro blem, no stable steady solution exists at high Reynolds number. Rather, as soon as the Reynolds number exceeds a modest value of around 10, the flow b ecomes unstable via a supercritical Hopf bifurcation.