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Long waves on a thin layer of conducting fluid flowing down an inclined plane in an electromagnetic field. (English) Zbl 0934.76097
Summary: We study 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. The Navier-Stokes equations with electromagnetic force in the limit of low magnetic Reynolds number and subject to corresponding boundary conditions serve 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. We show that the electrical field has a destabilizing effect on the film flow, while the magnetic field stabilizers it. The strongest stabilizing effect of the magnetic field in the presence of electrical one can be achieved if the magnetic field is purely longitudinal. We also consider the application of the Kármán-Pohlhausen integral boundary-layer theory.
76W05Magnetohydrodynamics and electrohydrodynamics
76E25Stability and instability of magnetohydrodynamic and electrohydrodynamic flows