A two-fluid four-equation model with instantaneous thermodynamical equilibrium
Alexandre Morin and Tore Flåtten
Abstract: We analyse a four-equation version of a common two-fluid model for pipe flow, containing one mixture mass equation and one mixture energy equation. The motivation is to obtain a fluid-dynamical model where the mixture is in thermodynamical equilibrium at all time. We start from a five-equation model with instantaneous thermal equilibrium, to which we add phase relaxation terms. An interfacial velocity appears, for which we give an expression based on the second law of thermodynamics. We then derive the limit of this model when the relaxation becomes instantaneous. The time derivatives appearing in this process are subsequently transformed into spatial derivatives to be able to use numerical methods for conservation laws. The Jacobian matrix of the fluxes can then be evaluated, and the system be put into quasilinear form. From the Jacobian matrix, we are able to extract the sound speed intrinsic to the model. By comparison to the sound speed in other two-phase flow models, we extend some previous results showing that the effect of relaxation on sound speed is independent of the order in which the variables are relaxed. We also check the subcharacteristic condition and place the model in a hierarchy of two-phase flow models. Finally, this model requires a regularisation term to be hyperbolic. With the help of a perturbation method, we find an expression for this term that makes the model conditionally hyperbolic.