A Hierarchy of Models for Two-Phase Flows

We derive a hierarchy of models for gas-liquid two-phase flows in the limit of infinite density ratio, when the liquid is assumed to be incompressible. The starting model is a system of nonconservative conservation laws with relaxation. At first order in the density ratio, we get a simplified system with viscosity, while at the limit we obtain a system of two conservation laws, the system of pressureless gases with constraint and undetermined pressure. Formal properties of this constraint model are provided, and sticky blocks solutions are introduced. We propose numerical methods for this last model and the results are compared with the two previous models. Key-words: two-phase flow – conservation laws with relaxation – conservation laws with constraint – pressureless gas – sticky particles and blocks Mathematics Subject Classification: 76T05, 35L65, 35L85 (1) Département de Mathématiques et Applications, Ecole Normale Supérieure et CNRS, UMR 8553, 45, rue d’Ulm, 75230 Paris cedex 05, France. e-mail: [email protected] (2) Laboratoire d’Analyse Numérique, UMR 7598, Université P. et M. Curie, B.C. 187, Tour 55-65, 5ème étage, 4, place Jussieu, 75252 Paris, France. e-mail: [email protected] (3) CMLA UMR-8536, Ecole Normale Supérieure de Cachan, Département de Mathématiques, 61, avenue du président Wilson, 94235 Cachan cedex, France. e-mail: [email protected] (4) Mathématiques Appliquées de Bordeaux UMR 5466, Université de Bordeaux 1 et CNRS, 351 cours de la libération, 33405 Talence cedex, France. e-mail: [email protected]