Stability of plane-parallel vibrational flow in a two-layer system

The stability of the interface separating two immiscible incompressible fluids of different densities and viscosities is considered in the case of fluids filling a cavity which performs horizontal harmonic oscillations. There exists a simple basic state which corresponds to the unperturbed interface and plane-parallel unsteady counter flows; the properties of this state are examined. A linear stability problem for the interface is formulated and solved for both (a) inviscid and (b) viscous fluids. A transformation is found which reduces the linear stability problem under the inviscid approximation to the Mathieu equation. The parametric resonant regions of instability associated with the intensification of capillary-gravity waves at the interface are examined and the results are compared to those found in the viscous case in a fully numerical investigation.