Models for Interfacial Capillary-Gravity Waves in the Long-Wave Limit

A matched asymptotic expansion is used to give a formal derivation of a system of partial differential equations modeling the evolution of the free interface in a two-fluid system. Under the assumption of one-way propagation, the system is reduced to a single equation. INTRODUCTION The object of this note is the systematic derivation of a number of model equations which are of use in the description of the evolution of long-crested internal waves in two-fluid systems. The equations put forward here are a system of two coupled evolution equations, similar to the Boussinesq equations describing waves at the surface of a fluid. The equations have the form wt + ηx + σ wwx − ρ2 ρ1 T1ηxx − μ ηxxx = 0, ηt + wx + σ (wη)x = 0. The quantities appearing in the equations will be defined momentarily. For now, note that μ is a parameter related to the interfacial tension and T1 is an integral operator. As it will turn out, this model is approximately valid for long interfacial waves, if one of the layers is thin, the other is deep, and the interface is subject to capillarity. Similar model equations have been obtained by Choi and Camassa (1996, 1999), Ostrovsky and Grue (2003), and Craig, Guyenne and Kalisch (2004). However, in these treatments capillarity was neglected. Since many of these last-mentioned model equations are ill-posed, i.e. unstable with respect to small perturbations, capillarity can be seen as a way to stabilize these equations. In fact, physical considerations indicate that capillarity should be included in the model if a sharp interface between two fluids is present, and this is the point of view taken here. The scientific interest in two-layer flows with a sharp interface originates from wave tank experiments as described in Koop and Butler (1981) and industrial coating processes (see Singh and Joseph). There have been previous studies of interfacial capillary-gravity waves, mostly in the context of the full Euler equations. For example, the papers by Sha and Vanden-Broeck (1997) and Laget and Dias (1997) contain studies leaning on numerical approximation of steady waves of the Euler equations and bifurcation analysis. The work of Christodoulides and 1Department of Mathematics, NTNU, 7491 Trondheim, Norway, [email protected]