Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow
Department of Mathematics, University of Houston, Houston, Texas 77204-3008, United States, United States
Department of Mathematics, The Hebrew University, Jerusalem, Israel
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.
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