Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow

The two-fluid equations for two-phase flow, a model derived by averaging, analogy and experimental observation, have the property (in at least some commonly-occurring derivations) of losing hyperbolicity in their principal parts, those representing the chief entries in modeling conservation of mass and transfer of momentum and energy. Much attention has centered on reformulating details of the model to avoid this awkwardness. This paper takes a different approach: a study of the nonhyperbolic operator itself. The objective is to understand the nature of ill-posedness in nonlinear, as distinct from linearized, models. We present our initial study of the nonlinear operator that occurs in the two-fluid equations for incompressible two-phase flow. Our research indicates that one can solve Riemann problems for these nonlinear, nonhyperbolic equations. The solutions involve singular shocks, very low regularity solutions of conservation laws (solutions with singular shocks, however, are not restricted to nonhyperbolic equations). We present evidence, based on asymptotic treatment and numerical solution of regularized equations, that these singular solutions occur in the two-fluid model for incompressible two-phase flow. The Riemann solutions found using singular shocks have a reasonable physical interpretation.