An Oil Sheet on the Wall Rising to a Heavy Fluid under Gravity

Abstract

When a heavy fluid is superposed over a light oil sheet adhering to a horizontal solid wall, the gravity will cause the lower fluid to rise to the upper fluid. In such a case, whether or not there is a stable stationary state is studied theoretically. It is assumed that the flow is two-dimensional, the fluids are incompressible, the upper fluid is ideal and the lower one is viscous. To the fourth order for wave number and wave amplitude, the height equation of the interface is derived from the long wave expansion. It is found from this equation that nonlinearities of the governing equation have a stabilizing effect and there is a stable equilibrium state with amplitude of about 0.4 times the height of the lower liquid layer.