Solutions for a hyperbolic model of multi-phase flow

ESAIM: PROCEEDINGS, Juillet 2013, Vol. 40, p. 1-15 C. Bourdarias, S. Gerbi, Editors SOLUTIONS FOR A HYPERBOLIC MODEL OF MULTI-PHASE FLOW Debora Amadori 1 and Andrea Corli 2 Abstract. We discuss a model for the flow of an inviscid fluid admitting liquid and vapor phases, as well as a mixture of them. The flow is modeled in one spatial dimension; the state variables are the specific volume, the velocity and the mass density fraction λ of vapor in the fluid. The equation governing the time evolution of λ contains a source term, which enables metastable states and drives the fluid towards stable pure phases. We first discuss, for the homogeneous system, the BV stability of Riemann solutions generated by large initial data and check the validity of several sufficient conditions that are known in the literature. Then, we review some recent results about the existence of solutions, which are globally defined in time, for λ close either to 0 or to 1 (corresponding to almost pure phases). These solutions possibly contain large shocks. Finally, in the relaxation limit, solutions are proved to satisfy a reduced system and the related entropy condition. Résumé. On discute un modèle pour l’écoulement d’un fluide non visqueux admettant phases liquides et de vapeur, ainsi qu’un mélange d’entre eux. L’écoulement est modélisé dans une dimension spatiale; les variables d’état sont le volume spécifique, la vitesse et la fraction de densité de masse λ de la vapeur dans le liquide. L’équation régissant l’évolution temporelle de λ contient un terme de source, ce qui permet des états métastables et conduit le fluide vers de phases stables pures. Nous discutons d’abord, pour le système homogène, la stabilité BV des solutions de Riemann générés par des grandes données initiales et vérifions la validité de plusieurs conditions suffisantes qui sont connues dans la littérature. Ensuite, nous passons en revue quelques résultats récents sur l’existence de solutions, qui sont definies pour tous les temps, pour λ soit près de 0 ou de 1 (correspondant à des phases presque pures). Ces solutions sont susceptibles de contenir des grands chocs. Enfin, dans la limite de la relaxation, les solutions sont prouvèes satisfaire un système réduit et la condition d’entropie. Introduction We consider here a simple hyperbolic model for the flow of a fluid capable of showing a liquid and a gaseous phase. A reaction term is also taken into consideration, which allows for metastable states. In Lagrangian variables the model is represented by the following system of balance laws:   vt − ux ut + p(v, λ)x  λt = 0, = 0, = τ1 (p − pe )λ(λ − 1). (1) 1 Department of Engineering, Computer Science and Mathematics, University of L’Aquila, Italy 2 Department of Mathematics and Computer Science, University of Ferrara, Italy c EDP Sciences, SMAI 2013 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201340001 2 ESAIM: PROCEEDINGS λ=0 stable liquid P λ=1 p 6 PP q   ) metastable vapor    pe metastable liquid   9 - stable vapor v Figure 1. Pressure curves. Here v > 0 is the specific volume and u the velocity. The state variable λ is the mass density fraction of the vapor in the fluid; as a consequence, we have that 0 ≤ λ ≤ 1. The value λ = 0 then refers to the liquid phase while the value λ = 1 to the vapor (gaseous) phase; intermediate values describe suitable mixtures of the previous pure phases. In the last equation, pe denotes a fixed equilibrium pressure and τ > 0 is a reaction time. An important feature of the model consists in the fact that the pressure p is allowed to depend not only on v but also on λ. More precisely, we assume that p = p(v, λ) = A(λ)π(v) , A(λ) > 0 , π(v) > 0 (2) where A, π are C 2 functions that satisfy A0 (λ) 6= 0 , πv < 0 , πvv > 0 . (3) Under the above assumptions, the homogeneous part of (1) turns out to be strictly hyperbolic. The eigenvalues √ are ± −pv and 0; the two former are genuinely nonlinear and the latter is linearly degenerate. A typical example for π is the usual γ–law for the isentropic (isothermal) gasdynamics, i.e., π(v) = 1/v γ with γ ≥ 1. For A(λ) a possible choice is a linear interpolation between the two pure phases; namely, A(λ) = k0 + λ(k1 − k0 ), with 0 < k0 < k1 . Of course, in the case that A is constant the model reduces to the standard p-system in gasdynamics. A motivation for the pressure to depend on λ can be given by considering a pressure law for real gases (the van der Waals pressure law, for instance) when the temperature is above the critical point. In such cases what is really observable are the two hyperbolic branches of the pressure associated to the liquid and to the vapor phase; in our model they are represented by A(0)π(v) and A(1)π(v), respectively. The pressure curves A(λ)π(v), for 0 < λ < 1, interpolate the curves of the pressure in the liquid and vapor phases in the case of mixtures. The model (1) was first introduced in [21]; indeed the system in [21] is much more complete, including the fluid viscosity, the diffusion of the vapor in the fluid, nucleation terms and thermal effects. As a consequence, the analytic study of that model is extremely hard and some simplifications are demanded, as it is the case for (1). We notice that system (1) is isothermal, an assumption that can be done when studying phase changes in retrograde fluids, as a consequence of the their large heat capacities [44]. The reaction term on the right-hand side of the third equation in (1) allows for metastable states, because of the presence of the equilibrium pressure pe : namely, metastable states are the vapor states lying above the line p = pe or the liquid states lying below it. The line p = pe , in the (v, p) plane, then plays the role of the equal-area Maxwell line in the standard theory of phase transitions for fluids [43]. A model analogous to (1) but also including a damping term −αu in the second equation to model a flow through a porous medium, has ESAIM: PROCEEDINGS 3 been recently studied in [16]. Such flows occur, for instance, in the process of oil recovery from porous rocks or in the operation of a polymer electrolyte fuel cell, where a gaseous phase may develop as a consequence of high temperatures. About that model we also refer to [32], where however a different pressure law is considered. A close system is also studied in [32,33]; in those papers the third equation includes a transport term and becomes λt + c1 λx = β(h(v) − λ)/τ , for β constant and a suitable function h. It is also interesting to compare (1) with a system proposed in [29] to model reacting flows. In that paper the term p−pe , which leads to metastability, is missing and the third equation in (1) turns out be λt = (λ(v)−λ)/τ , for a suitable positive and decreasing function λ(v), while the pressure can be written under the form (2) and satisfies (3). Also t (...truncated)