A uniqueness theorem for magnetohydrodynamic duct flows

Abstract

AbstractWe analyse the steady, fully developed, laminar flows of uniform, electrically conducting, incompressible fluids along arbitrarily shaped ducts of constant cross-section under the action of transverse magnetic fields to show that these flows are uniquely specified by certain parameters, namely: either the pressure drop along the duct per unit length or the volume flow rate and either the electric current leaving the walls of the duct at each point where the duct is connected to an external electrical circuit or the electric potential at these points. The most important, and novel, feature of the analysis is the proof that no secondary flow can exist in these fully developed flows. The analysis includes the special cases where the walls of the duct are non-conducting, and where the fluid is non-conducting (e.g. Poiseuille flow); in the latter case the uniqueness of the flow has not previously been demonstrated, at least in print.