A model of discontinuous, incompressible two-phase flow

Lack of hyperbolicity is a recurring problem for models of two-phase flow assuming the form of systems of balance laws. In particular, smooth solutions occur only for very special initial data, and the standard results on the local structure of discontinuous weak solutions do not apply to such nonhyperbolic systems. A simple example is inviscid, incompressible two-fluid flow with a single pressure. We suggest that such an unattractive mathematical feature may result from the mathematical derivation of the model, rather than from the underlying physical assumptions. In particular, for the case described above we present an alternative treatment which leads to a consistent model for piecewise smooth, discontinuous solutions. We obtain admissibility conditions for the anticipated discontinuities by considering the limit of vanishing viscosity with a convenient dissipation term.