Instability of a viscous interface under horizontal oscillation

The linear stability of superposed layers of viscous, immiscible fluids of different densities subject to horizontal oscillations, is analyzed with a spectral collocation method and Floquet theory. We focus on counterflowing layers, which arise when the horizontal volume-flux is conserved, resulting in a streamwise pressure gradient. This model has been shown to accurately predict the onset of the frozen wave observed experimentally [E. Talib, S. V. Jalikop, and A. Juel, J. Fluid Mech. 584, 45 (2007)]. The numerical method enables us to gain new insights into the Kelvin-Helmholtz (KH) mode usually associated with the frozen wave, and the harmonic modes of the parametric-resonant instability, by resolving the flow for an exhaustive range of vibrational to viscous forces ratios and viscosity contrasts. We show that the viscous model is essential to accurately predict the onset of each mode of instability. We characterize the evolution of the neutral curves from the multiple modes of the parametric-resonant instability to the single frozen wave mode encountered in the limit of practical flows. We find that either the KH or the first resonant mode may persist when the fluid parameters are varied toward this limit. Interestingly, these two modes exhibit opposite dependencies on the viscosity contrast, which are understood by examining the eigenmodes near the interface. (c) 2007 American Institute of Physics.