Shallow fluid instability due to an electromagnetic force
An investigation is described of the instability of the horizontal free surface of a thin layer of liquid metal. The instability is due to the interaction of imposed electric and magnetic fields which are both initially horizontal. The interaction of fields produces a vertically upward magnetic force within the layer. Various two-dimensional theoretical models of the instability are described. A linear, viscous analysis of free surface instability is presented, and an analytical expression is obtained for the rate of growth of amplitude of a plane sinusoidal wave on the liquid free surface. A nonlinear model of a wave is also discussed on the assumption, later justified by experiment, that the waveform remains approximately sinusoidal providing the ratio of its amplitude and wavelength is small. An exact analysis is presented of a stationary wave in dynamical equilibrium. An experiment is described investigating the instability of a thin layer of liquid gallium. A new method for making instantaneous measurements of two-dimensional wave profiles is presented. Experimental observations are described of waves on an initially undisturbed free surface. It is confirmed that wave corrugations of the orientation predicted by linear theory to grow most quickly are observed to grow more quickly than other wave modes. Qualitative observations indicate the presence of secondary horizontal motions much larger than previous linear analyses have suggested. Quantative results indicate certain stability at non-zero amplitudes of linearly unstable waves if the Weber number is less than 0.12 and the Reynolds number is less than 75. Conversely, separation of gallium from the layer is observed invariably if the Weber number is greater than 0.31 and the Reynolds number is greater than 130.