The relaxation limit of a homogeneous two-phase flow model: isothermal case

TL;DR

This paper rigorously analyzes the asymptotic behavior of a hyperbolic relaxation system modeling homogeneous two-phase flows, proving convergence to an equilibrium Euler system as the relaxation parameter approaches zero.

Contribution

It provides the first rigorous proof of the relaxation limit for a homogeneous two-phase flow model with an isothermal gas phase.

Findings

Solutions converge strongly to an entropy solution of the equilibrium Euler system as relaxation time vanishes.

The analysis employs entropy pairs, energy estimates, and compensated compactness techniques.

The work justifies the use of relaxation models in simulating two-phase flows.

Abstract

This paper investigates the asymptotic behavior of a hyperbolic relaxation system designed for homogeneous two-phase flows in the limit of vanishing relaxation time. The governing equations comprise conservation laws for mixture mass and momentum, supplemented by a transport equation for the gas phase mass that includes a stiff relaxation source term. This source term drives the system toward local thermodynamic equilibrium. Under the assumptions of constant liquid density and an ideal isothermal gas phase, we demonstrate that, as the relaxation parameter \(\epsilon \rightarrow 0\), a subsequence of solutions \((p^{\epsilon},u^{\epsilon})\) converges strongly in \(L_{\mathrm{loc}}^{1}\) to an entropy solution of the equilibrium Euler system. The proof integrates several analytical techniques: the construction of a suitable entropy pair and associated energy estimates, a transport equation approach for representing the error, commutator estimates, and the theory of compensated compactness. This work provides a rigorous justification of the relaxation limit for the homogeneous two-phase flows model.