Long-Wave Interfacial Instabilities in Viscous Liquid Films

Abstract

One well-known instability that can occur in a viscous liquid film produces a wavelike deformation of the free surface over a wavelength that is much larger than the thickness of the film. We shall examine this motion and explain the dynamics of the instability mechanism by carefully interpreting the mathematical approximations used to solve the stability problem.

Several different geometries will be considered. The most fundamental is the isothermal liquid film on an inclined plane. Here, we explain how the small effects of liquid inertia lead to the unstable interfacial motion. Next, we consider a concentric arrangement of two fluids flowing in a vertical pipe. Here, the unstable behavior of the interface between the two fluids is modified by a large lubrication pressure that appears because of the rigid pipe wall. Thermal effects are explored by cooling an inclined liquid layer from below. In this case, both buoyancy forces and direct liquid expansion have important effects. Finally, a horizontal layer driven by thermocapillarity is considered to demonstrate the effect surface-tension gradients have on the instability.