Three-dimensional instabilities of film flows

Two-dimensional (2-D) interfacial waves on flowing films are unstable with respect to both two- and three-dimensional instabilities. In this paper, several distinct three-dimensional instabilities that occur in different regions of the parameter space defined by the Reynolds number R and the frequency f of forced two-dimensional waves are discussed in detail. (a) A synchronous 3-D instability, in which spanwise deformations of adjacent wave fronts have the same transverse phase, appears over a wide range of frequency. These transverse modulations occur mainly along the troughs of the primary waves and eventually develop into sharp and nearly isolated depressions. The instability involves many higher harmonics of the fundamental 2-D waves. (b) A 3-D surbharmonic instability occurs for frequencies close to the neutral curve fc(R). In this case, the transverse modulations are out of phase for successive wave fronts, and herringbone patterns result. It is shown that this weakly nonlinear instability is due to the resonant excitation of a triad of waves consisting of the fundamental two-dimensional wave and two oblique waves. The evolution of wavy films after the onset of either of these 3-D instabilities is complex. However, sufficiently far downstream, large-amplitude solitary waves absorb the smaller waves and become dominant.