Mathematical Properties of Nonhyperbolic Models for Incompressible Two-Phase Flow

Many models for multi-fluid flow result in equations which fail to be hyperbolic. One example is one-dimensional flow of an incompressible two-phase fluid. In the simplest model, the principal part of the differential operator has characteristics with nonzero imaginary part for any state of the fluid which contains both phases. Thus, the linearized equations are catastrophically unstable. This fact has caused distrust of the equations and concern about the modeling processes. However, these nonlinear equations behave very differently from their linearizations. Although states which are linearly unstable are also unstable in the nonlinear equations, nonlinear theory predicts jump transitions, via stable shocks, from unstable to stable states. Furthermore, the nonlinear theory eliminates both infinitegrowth modes and high-frequency oscillations. The solution depends continuously on the data except at certain values where threshold or bifurcation phenomena occur. This overall stability is not affected by viscous or drag terms in the system.