The Faraday resonance of interfacial waves in weakly viscous fluids

The Faraday resonance of interfacial waves in a two-layer, weakly-viscous system in a rectangular domain is presented. A perturbation analysis is pursued and, at the second-order, the scaling of the viscosity results in boundary layer corrections at the solid walls and at the interface. Special attention is paid to the damping in the meniscus region where the interface contacts the side-walls. As a result of the presence of both destabilizing effects ~vertical oscillation! and stabilizing effects ~viscosity!, a threshold condition for instability is determined. The derived analytic results are quite general and prove useful in elucidating the influences of the various boundary layers, as well as the threshold for growth. In an effort to describe the maximum amplitude attained by the resonated wave, a third-order analysis is then presented for the idealized case of equal-depth, inviscid layers, with a rigid-lid condition at the free surface. A balance between cubic nonlinearity and the vertical shaking yields a Landau-type equation for the interfacial wave amplitude. Comparisons with some existing experimental data are made at both orders and indicate very good agreement. © 2002 American Institute of Physics. @DOI: 10.1063/1.1425846#