Wave Propagation in Multicomponent Flow Models

Abstract

Abstract. We consider systems of hyperbolic balance laws governing flows of an arbitrary number of components equipped with general equations of state. The components are assumed to be immiscible.We compare two such models; one in which thermal equilibrium is attained trough a relaxation procedure, and a fully relaxed model in which equal temperatures are instantaneously imposed. We describe how the relaxation procedure may be made consistent with the second law of thermodynamics.Exact wave velocities for both models are obtained and compared. In particular, our formulation directly proves a general subcharacteristic condition: For an arbitrary number of components and thermodynamically stable equations of state, the mixture sonic velocity of the relaxed system can never exceed the sonic velocity of the relaxation system. subject classification. 76T30, 35L60, 35L65 key words. multicomponent flow, sonic waves, sound velocity, relaxation system

On the effect of temperature and velocity relaxation in two-phase flow models

“…• Energy balance: We observe that the df 5 model derived here is equivalent to the two-component version of the model considered in [17]. There, it was also shown that this two-component model, and hence the df 5 model, is equivalent to models studied in [28,35].…”

“…The N -component version of this model was the main focus of [17]. For our present case of N = 2, the following eigenvalues were found:…”

“…SC1 was proved in [17]. By inspection of (3.6), (3.8), (3.5), (3.69), and Definitions 1.1 and 4.2, we see that SC2-SC4 hold provided …”

“…This model has been extensively analysed by several authors [17,28,35], and the resulting eigenvalues are…”

An exponential time-differencing method for monotonic relaxation systems

“…Such systems model many relevant physical problems, such as two-phase flows which are locally not in thermodynamic equilibrium [7,8,32,37]. The limiting process → 0 in systems in the form (55) was extensively analysed by Liu [24] and Chen et al [4], with a particular focus on the relationship between stability and wave propagation.…”