Magnetic damping of steady streaming vortices in an oscillatory viscous flow over a wavy wall

Abstract

The effect of a transverse magnetic field on the steady streaming vortices generated by an oscillatory viscous flow over a wavy wall is studied. The wall is assumed to be electrically insulated, and the axial magnetic field induced by the electric currents circulating in the fluid is neglected. A perturbative solution is developed assuming that the amplitude of fluid oscillation is very small compared with the wall wavelength and that the amplitude of the wavy wall is much smaller than the Stokes layer. Upper and lower steady regions of recirculation appear when the wavelength of the wall is much larger than the Stokes layer thickness. This steady streaming flow is found to be damped out by the action of the magnetic field. As the magnetic field strength increases, momentum diffuses out along the field lines causing the elongation of the vortices in the field direction. As a consequence of the flow two-dimensionality, the streaming vortices can be completely annihilated by magnetic braking. This behavior is confirmed through numerical calculations. In the absence of a magnetic field, a correction to the Rayleigh’s law of streaming is obtained.