| The rest state of a liquid/liquid or liquid/gas interface is usually unstable. Linearly unstable waves then arise from the rest state and saturate in amplitude. The saturated waves can further suffer modulations in their relative phase and destabilize by a secondary instability mechanism. A generic modulation instability theory which includes the coupling between a phase mode and a flow inducing mass mode is developed. This instability is distinct from and much stronger than the classical modulation instability where only the phase mode is excited. Due to the strong nonlinear interaction between the phase and mass modes, jumps in the mean flow thickness and in the wavenumber gradient are generated self-similarly in a finite time. These jumps serve as wave sinks to dilate the wave field. For near periodic waves, the final selected wavenumber is determined by the relaxation of the wave sinks. For the longer and more localized solitary waves, they interact individually and coalesce. A statistical model is formulated to describe this binary wave interaction and coalescence. It is also possible that there is no equilibrium of longer periodic wave states to go to after the primary waves suffer modulational instability. When this occurs, unsteady waves are observed on the interface. The predictions from our general modulation instability theory agree well with experiments and numerical simulations.;Another secondary instability of saturated interfacial structure we explore is the fingering phenomenon. As a constant volume liquid flows down an inclined plane, a regular or irregular fingering pattern may be developed. For small inclination angles, the liquid front is initially stable to disturbances on the plane. However, as the liquid flows further downstream, the front thins and the gravitational stabilizing force weakens. At a critical location down the plane, the front becomes linearly unstable and fingering occurs. The disturbance propagates from the solid plane by a unique transient growth mechanism, which selects specific transverse wavelength, before fingering ensues. We develop a weighted spectral theory for the front that captures both its long-range receptivity to disturbances and its local instability. Experiments and numerical simulation of fingering quantitatively confirm the predicted fingering position. |
