Hydrodynamic response and free surface modes for two immiscible fluids

We consider a system consisting of two immiscible fluids and their interface. The equilibrium interface is assumed to be planar. The velocity fields in the fluids are described by the linearized Navier-Stokes equations with appropriate boundary conditions at the interface. Explicit expressions for the response of the system to arbitrary bulk and/or surface forces are derived. In particular, we consider the transmission and reflection of sound modes and conclude that ultrasonic techniques can be used to measure the coefficient of sliding friction between fluids. In addition, we obtain dispersion relations for the free surface modes.