Long waves on a thin layer of conducting fluid flowing down an inclined plane in an electromagnetic field

Abstract

Wave formation on film flow is an intriguing hydrodynamic phenomenon with a variety of practical consequences, especially in heat and mass transfer. In this paper the propagation of weakly nonlinear waves over a flow of an electrically conducting viscous film flowing down an inclined plane under simultaneous action of electrical and magnetic fields is studied. The set of Navier-Stokes equations with electromagnetic force in the limit of low magnetic Reynolds number and subject to corresponding boundary conditions serves as a mathematical description of the problem. Long-wave expansions are carried out and an evolution equation of the Kuramoto-Sivashinsky type governing propagation of weak surface perturbations is derived. The critical values of the Reynolds number are determined explicitly and linear stability is investigated. It is shown that the electrical field provides a destabilizing effect on the film flow while the magnetic field stabilizes it. The strongest stabilizing effect of the magnetic field in the presence of the electrical one can be achieved if it is purely longitudinal. The Karman-Polhausen integral boundary-layer theory is considered for the MHD-approach and main results are discussed.