A compressible mixture model with phase transition

Highlights

• We introduce a thermodynamically consistent model for multi-phase flows.

• The model allows for phase transitions and chemical reactions.

• We investigate physically admissible sharp interface limits.

• The limit is an Allen–Cahn/Euler system with admissible interfacial conditions.

• The model allows for two phase equilibria with surface tension in the leading order.

Abstract

We introduce a new thermodynamically consistent diffuse interface model of Allen–Cahn/Navier–Stokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen–Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satisfy, for instance, a Young–Laplace and a Stefan type law.

Introduction

In this study, we propose a model for chemically reacting viscous fluid mixtures that may develop a transition between a liquid and a vapor phase. The mixture consists of $N$ constituents and is described by $N$ partial mass balance equations and a single equation of balance for the barycentric momentum. We exclusively consider isothermal evolutions. To describe the phase transition, we introduce an artificial phase field indicating the present phase by assigning the values 1 and $-1$ to the liquid and the vapor phase, respectively. Within the transition layer between two adjacent phases, the phase field smoothly changes between 1 and $-1$. However, usually the transition layers are very thin leading to steep gradients of the phase field.

This model belongs to the class of diffuse interface models. An alternative model class, that likewise represents phase transitions in fluid mixtures, contains sharp interface models. From the modeling point of view, sharp interface models have a simpler physical basis than diffuse interface models. For this reason, there arises always the non-trivial question if the sharp interface limits of a given diffuse model lead to admissible sharp interface models. The main concern of this paper is a careful discussion of this problem.

While diffuse interface models solve partial differential equations in the transition region, sharp interface models deal with jump conditions across the interface between the phases. Sometimes the jump conditions are mixed with geometric partial differential equations.

For two phases without chemical reactions, our compressible model reduces to an Allen–Cahn/Navier–Stokes type model, which is quite similar to the model derived by Blesgen [1]. Blesgen’s model has been investigated analytically in [2], [3], where existence of strong local-in-time solutions and weak solutions has been shown.

We would like to emphasize that the thermodynamical approaches of Blesgen’s system and our derived model are different. For instance, the phase field variable $\chi$ in Blesgen’s model, which satisfies the Allen–Cahn equation, is a physical quantity, namely the local mass fraction of one phase, whereas in our proposed model $\chi$ simply indicates the present phase. In addition, the function $h$ that interpolates between the phases, which will be described later, is linear in Blesgen’s model. This means that, in contrast to our proposed model, in general the equilibria depend on the chosen interpolation function $h$, which implies that Blesgen’s system runs into different equilibria. Our approach with $\chi$ as an artificial phase field variable permits to obtain physical meaningful jump conditions at the interface agreeing with classical laws of thermodynamics.

A modified version of Blesgen’s model can be found in  [4]. In contrast to  [1] and our introduced model, Witterstein  [4] describes a mixture of two compressible fluids, which physically differ, exclusively by different Lamé coefficients which are assumed to depend on the phase field parameter and the mass density. Witterstein’s model  [4] also differs in the choice of the free energy, which contains two length scales. Moreover, the minima of the double well potential in the free energy have to be of different heights. This implies that the energy can only be controlled for transition regions with fixed width but not in the sharp interface limit.

Related to our work are diffuse interface models for incompressible and quasi-incompressible fluids. A diffuse interface model of Navier–Stokes–Cahn–Hilliard type for two incompressible, viscous Newtonian fluids, having the same densities, has been introduced by Hohenberg and Halperin in [5]. That model has been modified in several thermodynamically consistent ways such that different densities are allowed, see e.g. [6], [7], [8]. For existence results of strong local-in-time solutions and weak solutions, we refer to [9], [10], [11]. A diffuse interface model for two incompressible constituents which permits the transfer of mass between the phases due to diffusion and phase transitions has been proposed in [12], [13]. The densities of the fluids may be different, which leads to quasi-incompressibility of the mixture.

Our newly introduced diffuse interface model is given by the following system of PDEs for ($\rho ,{\rho }_{\alpha =1,\dots ,N-1},v,\chi$) in $[0,T_f)\times \Omega$, $\Omega \subset {\mathbb{R}}^d$:

$${\partial }_t\rho +\mathrm{div}\left(\rho v\right)=0\text{,}$$$${\partial }_t{\rho }_{\alpha }+\mathrm{div}\left({\rho }_{\alpha }v\right)-\mathrm{div}\left(\sum _{\beta =1}^{N-1}{M}_{\alpha \beta }\nabla ({\mu }_{\alpha }-{\mu }_N)\right)=\sum _{i=1}^{N_R}{\gamma }_{\alpha }^i{m}_{\alpha }{M}_r^i(1-exp\left(\frac{A^i}{kT}\right))\text{,}$$$${\partial }_t\left(\rho v\right)+\mathrm{div}(\rho v\otimes v)+\nabla p+\mathrm{div}(\gamma \nabla \chi \otimes \nabla \chi -{\sigma }_{NS})=0\text{,}$$$$\rho {\partial }_t\chi +\rho v\cdot \nabla \chi =-M_p(\frac{\partial \rho \psi }{\partial \chi }-\gamma \Delta \chi )\text{,}$$

where $p$ is the pressure, $T$ the temperature, $m_{\alpha }$ the atomic mass of constituent $\alpha$, $k$ the Boltzmann constant,

$$\rho \psi =W\left(\chi \right)+\frac{\gamma }{2}{\left|\nabla \chi \right|}^2+\rho f({\rho }_1,\dots ,{\rho }_N,\chi )\text{and}{\mu }_{\alpha }=\frac{\partial \left(\rho \psi \right)}{\partial {\rho }_{\alpha }}$$

with $\rho f({\rho }_1,\dots ,{\rho }_N,\chi )≔h\left(\chi \right)\rho {\psi }_L({\rho }_1,\dots ,{\rho }_N)+(1-h\left(\chi \right))\rho {\psi }_V({\rho }_1,\dots ,{\rho }_N)$. In addition, ${\gamma }_{\alpha }^i$ are the stoichiometric coefficients of $N_R$ possible chemical reactions, $A^i$ the affinities and $M_{\alpha \beta }$, $M_r^i$ and $M_p$ the mobilities.

The work is organized as follows. In the upcoming section we derive the thermodynamically consistent model for multi-component flows with phase transitions and chemical reactions. Section  3 is devoted to the non-dimensionalization, the introduction of two interesting scaling regimes of the system and the setting of asymptotic analysis. Finally, in Sections  4 Sharp interface limit of the dissipative regime, 5 Sharp interface limit of the non-dissipative regime, we determine the sharp interface limits for the two different scaling regimes introduced previously. We would like to emphasize that Section  4.2 contains a conjecture on the incapability of viscous diffuse models to generate viscous sharp models.