The hydrodynamic stability of a thin film of liquid in uniform shearing motion

Abstract

The stability problem for a thin film of liquid having a linear mean-velocity profile and bounded by a fixed wall and free surface is solved asymptotically for large values of the Reynolds number R. The analysis is similar to that for plane Couette flow, but instability occurs for sufficiently large values of R in accordance with Heisenberg's criterion that neutral disturbances having finite wave numbers and phase velocities for R = ∞ are necessarily unstable as R → ∞. It is found that a sufficient condition for stability is W < 3, where W is the Weber number based on the mean speed at the free surface and the depth of the film. The minimum critical Reynolds number, also based on free surface speed and film depth, is found to be R = 203. This last figure is in order-of-magnitude agreement with observation, but there remains considerable uncertainty as to whether the observed instability corresponds to that considered here. Neutral stability curves are presented in an R vs α (= wave-number) plane with W as the family parameter. Brief consideration also is given to the time-rate-of-growth of unstable disturbances and to the lighter fluid that, in actual configurations, is responsible for the shear in the film. An appendix gives extended and more accurate results for the function $\cal F$(z), introduced and calculated previously by Tietjens (1925) and Lin (1955).