Continuous and jumping stationary transitions of a two-speed multiphase medium from one state to another

Abstract

A model of two-velocity multiphase medium with equal phase pressures was investigated. Two pulse equations were used where interphase force has a stabilizing term depending on flow parameters gradient. Phases were assumed incompressible. It was shown that this asssumption does not have any impact on the main conclusions while the model is significantly simplified. One does not need any energy equations because the movement does not depend on phase temperatures. Two differential equations describing relative phase movement were derived. Each of the two differential equations describing relative phase movement leads to its own wave velocity. For a steady-state stable transition when the mixture parameters do not change with time in some coordinate system (a steady-state or automodel flow) wave velocity must be equal for all the differential equations. This gives a condition for calculation of the stabilizing term. We defined conditions for continuous transition from one state to another as well as conditions when this transition can be only discontinuous. Analytical solutions helped us to find stabilizing term for interphase friction force (the only posssible in the corresponding class).