We are concerned here with forced steady recirculating flows which are laminar, two-dimensional and have a high Reynolds number. The body force is considered to be prescribed and independent of the flow, a situation which arises frequently in magnetohydrodynamics. Such flows are subject to a strong constraint. Specifically, the body force generates kinetic energy throughout the flow field, yet dissipation is confined to narrow singular regions such as boundary layers. If the flow is to achieve a steady state, then the kinetic energy which is continually generated within the bulk of the flow must find its way to the dissipative regions. Now the distribution of u2/2 is governed by a transport equation, in which the only cross-stream transport of energy is diffusion, v∇2 (u2/2). It follows that there are only two possible candidates for the transport of energy to the dissipative regions: 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 investigate 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 problem, 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 becomes unstable via a supercritical Hopf bifurcation.