Physical Review E
Multiphase flow -- Mathematical models, Fluid dynamics
Compared to single-pressure models, multipressure multiphase flow models require additional closure relations to determine the individual pressures of the different phases. These relations are often taken to be evolution equations for the volume fractions. We present a rigorous theoretical framework for constructing such equations for compressible multiphase mixtures in terms of submodels for the relative volumetric expansion rates ∆Ei of the phases. These quantities are essentially the rates at which the phases dynamically expand or contract in response to pressure differences, and represent the general tendency of the volume fractions to relax toward values that produce local pressure equilibrium. We present a simple provisional model of this type in which ∆Ei is proportional to pressure differences divided by the time required for sound waves to traverse an appropriate characteristic length. It is shown that the resulting approach to pressure equilibrium is monotonic rather than oscillatory, and occurs instantaneously in the incompressible limit.
Chang, C., & Ramshaw, J. (2008). Dynamical evolution of volume fractions in multipressure multiphase flow models. Physical Review. E, Statistical, Nonlinear, And Soft Matter Physics, 77(6 Pt 2), 066305.