Rayleigh–Taylor instability in thin viscous films

Abstract

The behavior of a viscous fluid film bounded by a wall and a heavier overlying immiscible phase is examined in the limit of small Bond number. Evolution equations governing the behavior of the interface between the two fluids are derived for spatially periodic disturbances and studied numerically. The analysis shows that instability leads to a spectrum of steady-state interfacial shapes characterized by film rupture and formation of pendant drops, such that gravity and interfacial forces are in balance. An energy stability analysis reveals that one drop per wavelength is the most energetically favorable. An extension of the analysis, based on the boundary integral method, is described for arbitrary Bond number and creeping flow conditions, aimed at identifying a Bond number below which stable pendant drops are possible. Simulations are shown for Bond numbers in the range 0.4≤B≤10 for fluids of equal viscosity. When the Bond number is relatively large, the interface takes the form of a circular front that penetrates into the overlying fluid, followed by a trailing plume. A critical Bond number is found, which leads to steady-state shapes in the form of pendant drops, confirming the predictions of the asymptotic analysis.