@misc{PototskyBestehorn, author = {Pototsky, Andrey and Bestehorn, Michael}, title = {Faraday instability of a two-layer liquid film with a free upper surface}, series = {Physical Review Fluids}, volume = {1}, journal = {Physical Review Fluids}, number = {2}, issn = {2469-990X}, doi = {10.1103/PhysRevFluids.1.023901}, pages = {023901-1 -- 023901-18}, abstract = {We study the linear stability of a laterally extended flat two-layer liquid film under the influence of external vertical vibration. The first liquid layer rests on a vibrating solid plate and is overlaid by a second layer of immiscible fluid with deformable upper surface. Surface waves, excited as the result of the Faraday instability, can be characterized by a time-dependent relative amplitude of the displacements of the liquid-liquid and the liquid-gas interfaces. The in-phase displacements are associated with a zigzag (barotropic) mode and the antiphase displacement corresponds to the varicose thinning mode. We numerically determine the stability threshold in the vibrated two-layer film and compute the dispersion relation together with the decay rates of the surface waves in the absence of vibration. The in-phase and the antiphase displacements are strongly coupled in the vibrated system. The interplay between the Faraday and the Rayleigh-Taylor instabilities in the system with heavier fluid on top of a lighter fluid is analyzed.}, language = {en} } @misc{BestehornPototsky, author = {Bestehorn, Michael and Pototsky, Andrey}, title = {Faraday instability and nonlinear pattern formation of a two-layer system: A reduced model}, series = {Physical Review Fluids}, volume = {1}, journal = {Physical Review Fluids}, number = {6}, issn = {2469-990X}, doi = {10.1103/PhysRevFluids.1.063905}, pages = {063905-1 -- 063905-24}, abstract = {Stability and pattern formation of a two-layer liquid system with large aspect ratio subjected to vertical harmonic oscillations is studied by means of an integrated boundary layer model. The lower layer rests on an oscillating solid substrate, the upper layer is separated by a deformable interface from the lower layer and bounded at the top with a second, free interface to the ambient passive air. The model is derived from the Navier-Stokes equations in long-wave approximation, including inertial terms. Applying a Floquet analysis, linear stability charts and dispersion relations are computed and compared with results from the full linearized Navier-Stokes equations and the long-wave approximation. Nonlinear Faraday patterns simultaneously occurring at the interface and at the film surface are studied by numerically solving the integrated boundary layer model in two and three spatial dimensions. For gravitationally stable two-layer films with a lighter fluid on top of the heavier fluid, we find squares, hexagons, quasiperiodic patterns with eightfold symmetry as well as localized states in the form of large scale depletion regions or finite depth holes, occurring at the interface and surface. For a Rayleigh-Taylor unstable combination (heavier fluid above the light one) we show that external vibration increases the lifetime of the film by delaying or completely suppressing the film rupture.}, language = {en} }