Linear and nonlinear stability of magnetohydrodynamic Couette and Hartmann shear flows

The problem of stability for classical Couette and Poiseuille flows (flows of a horizontal layer of fluid between plates in relative motion) is still open. Recently, Falsaperla et al. obtained values for critical linear and nonlinear energy Reynolds numbers which are in good agreement with the experiments of Prigent et al. and with the numerical simulation of Barkley and Tuckerman. Such critical values are computed for tilted perturbations which have the same inclination of the secondary motions appearing in the experiments. In this paper we consider the same problem when the fluid is electrically conducting and subject to a magnetic field orthogonal to the layer. We investigate the stability of the stationary solution called magnetic Couette and Hartmann flows and we show that such flows are nonlinearly stable if the Reynolds number Re is less then R̄θ=ReOrr(m)(2π∕(λsinθ))∕sinθ,when the perturbations are rolls inclined by an angle θ with respect to the direction of the fluid motion. In the expression above ReOrr(m)(μ) is the magnetic Orr–Reynolds number evaluated at the wavenumber μ=2π∕(λsinθ), where λ is the wavelength of the perturbation. We conjecture that in an experiment instability will emerge with secondary states that are rolls with inclination and wavelength related to the critical Reynolds number by the above formula