Energetic variational approaches for multiphase flow systems with phase transition

We are interested in the motion of the fluid particles near the boundary between the atmosphere and the ocean. We call the boundary the air-water interface. Since evaporation and condensation phenomena occur at the interface, we have to study multiphase flow with phase transition in order to understand air-sea interaction. This paper considers the governing equations for the motion of the fluid particles in two moving domains and the interface from an energetic point of view. We employ an energetic variational approach to derive our multiphase flow systems with phase transition. Of course, this paper proposes our system as one of the models for phase transition.

Let us first introduce fundamental notations. Let $t\geq 0$ be the time variable, and $x(={}^{t}(x_{1},x_{2},x_{3}))\in\mathbb{R}^{3}$ the spatial variable. Fix $T>0$. Let $\Omega\subset\mathbb{R}^{3}$ be a bounded domain with a smooth boundary $\partial\Omega$. The symbol $n_{\Omega}=n_{\Omega}(x)={}^{t}(n^{\Omega}_{1},n^{\Omega}_{2},n^{\Omega}_{3})$ denotes the unit outer normal vector at $x\in\partial\Omega$. Let $\Omega_{A}(t)(=\{\Omega_{A}(t)\}_{0\leq t<T})$ be a bounded domain in $\mathbb{R}^{3}$ with a moving boundary $\Gamma(t)$. Assume that $\Gamma(t)(=\{\Gamma(t)\}_{0\leq t<T})$ is a smoothly evolving surface and is a closed Riemannian 2-dimensional manifold. The symbol $n_{\Gamma}=n_{\Gamma}(x,t)={}^{t}(n^{\Gamma}_{1},n^{\Gamma}_{2},n^{\Gamma}_{3})$ denotes the unit outer normal vector at $x\in\Gamma(t)$. For each $t\in[0,T)$, assume that $\Omega_{A}(t)\Subset\Omega$. Set $\Omega_{B}(t)=\Omega\setminus\overline{\Omega_{A}(t)}$. It is clear that $\Omega=\Omega_{A}(t)\cup\Gamma(t)\cup\Omega_{B}(t)$ (see Figure 1). Set

In this paper we assume that the fluid in $\Omega_{A,T}$ is an incompressible one, and that the fluid in $\Omega_{B,T}$ is a compressible one. Let us state physical notations. Let $\rho_{A}=\rho_{A}(x,t)$, $v_{A}=v_{A}(x,t)={}^{t}(v^{A}_{1},v^{A}_{2},v^{A}_{3})$, $\pi_{A}=\pi_{A}(x,t)$, and $\mu_{A}=\mu_{A}(x,t)$ be the density, the velocity, the pressure, and the viscosity of the fluid in $\Omega_{A}(t)$, respectively. Let $\rho_{B}=\rho_{B}(x,t)$, $v_{B}=v_{B}(x,t)={}^{t}(v^{B}_{1},v^{B}_{2},v^{B}_{3})$, $\pi_{B}=\pi_{B}(x,t)$, and $\mu_{B}=\mu_{B}(x,t)$, $\lambda_{B}=\lambda_{B}(x,t)$ be the density, the velocity, the pressure, and two viscosities of the fluid in $\Omega_{B}(t)$, respectively. Let $v_{S}=v_{S}(x,t)={}^{t}(v^{S}_{1},v^{S}_{2},v^{S}_{3})$ be the motion velocity of the evolving surface $\Gamma(t)$. The symbol $\rho_{0}>0$ denotes the density of the interface $\Gamma(t)$, and $\pi_{0}\in\mathbb{R}\setminus\{0\}$ denotes the surface tension (coefficient) at $x\in\Gamma(t)$. We assume that $\rho_{A}$, $v_{A}$, $\pi_{A}$, $\mu_{A}$, $\rho_{B}$, $v_{B}$, $\pi_{B}$, $\mu_{B}$, $\lambda_{B}$, $v_{S}$ are smooth functions in $\mathbb{R}^{4}$, and that $\rho_{0}$, $\pi_{0}$ are constants.

Let us explain the background and the ansatz of this study. Let us now consider the phase transition phenomenon on ice melting. There exists a layer between the ice and the air. The layer is called a quasi-liquid layer (see Kuroda-Lacmann [13], Furukawa-Yamamoto-Kuroda [6]). It is well-known that a quasi-liquid layer has both liquid and solid properties. Experiments in Sazaki-Zepeda-Nakatsubo-Yokoyama-Furukawa [16] showed that ice particles change into particles in the quasi-liquid layer and then the particles in the layer change into water vapor. A similar process occurs when ice forms. Therefore, we can consider a two-phase problem with a phase transition as a three-phase problem. In this paper, we admit the existence of a surface mass at the interface $\Gamma(t)$, and assume that the particles at the interface can change into both particles in $\Omega_{A}(t)$ and particles in $\Omega_{B}(t)$ (see Figure 1).

Let us explain the key restriction of mathematical modeling of multiphase flow systems with phase transition. We assume that

The condition ${\rm{div}}v_{A}=0$ means the incompressibility condition of the fluid in $\Omega_{A,T}$, and $v_{B}\cdot n_{\Omega}=0$ means that fluid particles do not go out of the domain $\Omega$. In general, we assume several jump conditions when we make models for multiphase flow with phase transition. This paper does not use assumptions on jump conditions to derive our systems. See Slattery-Sagis-Oh [19] for several jump conditions on interfacial phenomena.

This paper has three purposes. The first one is to make the following inviscid model for multiphase flow with phase transition:

with (1.2). Here $D_{t}^{A}f=\partial_{t}f+(v_{A}\cdot\nabla)f$, $D_{t}^{B}f=\partial_{t}f+(v_{B}\cdot\nabla)f$, $(v_{A}\cdot\nabla)f=v^{A}_{1}\partial_{1}f+v^{A}_{2}\partial_{2}f+v^{A}_{3}\partial_{3}f$, $(v_{B}\cdot\nabla)f=v^{B}_{1}\partial_{1}f+v^{B}_{2}\partial_{2}f+v^{B}_{3}\partial_{3}f$, ${\rm{div}}v_{A}=\nabla\cdot v_{A}$, ${\rm{div}}v_{B}=\nabla\cdot v_{B}$, ${\rm{grad}}f=\nabla f$, $\nabla={}^{t}(\partial_{1},\partial_{2},\partial_{3})$, $\partial_{i}=\partial/{\partial x_{i}}$, and $\partial_{t}=\partial/{\partial t}$. The symbol $H_{\Gamma}=H_{\Gamma}(x,t)$ denotes the mean curvature in the direction $n_{\Gamma}$ defined by $H_{\Gamma}=-{\rm{div}}_{\Gamma}n_{\Gamma}=-(\partial_{1}^{\Gamma}n_{1}^{\Gamma}+\partial_{2}^{\Gamma}n^{\Gamma}_{2}+\partial_{3}^{\Gamma}n^{\Gamma}_{3})$, where $\partial^{\Gamma}_{i}f:=\sum_{j=1}^{3}(\delta_{ij}-n^{\Gamma}_{i}n^{\Gamma}_{j})\partial_{j}f=\partial_{j}f-n_{j}^{\Gamma}(n_{\Gamma}\cdot\nabla)f$. More precisely, under the restriction (1.2) we apply an energetic variational approach to derive system (1.3). See subsection 4.2 for details. Remark that the motion velocity $v_{S}$ is given by

if $H_{\Gamma}\neq 0$ and $P_{\Gamma}v_{S}={}^{t}(0,0,0)$. See the proof of Theorem 2.2 in Section 3 for details.

The second one is to make the following viscous model for multiphase flow with phase transition:

with

where

Here $D(v_{A})=\{{}^{t}(\nabla v_{A})+(\nabla v_{A})\}/2$ and $D(v_{B})=\{{}^{t}(\nabla v_{B})+(\nabla v_{B})\}/2$. The symbol $I_{3\times 3}$ denotes the $3\times 3$ identity matrix, and $P_{\Gamma}=P_{\Gamma}(x,t)$ the orthogonal projection to a tangent space defined by $P_{\Gamma}=I_{3\times 3}-n_{\Gamma}\otimes n_{\Gamma}$, where $\otimes$ is the tensor product. More precisely, under the restriction (1.5) we apply an energetic variational approach to derive system (1.4). See subsection 4.1 for details. Remark that the motion velocity $v_{S}$ is given by

if $H_{\Gamma}\neq 0$ and $P_{\Gamma}v_{S}={}^{t}(0,0,0)$. See the proof of Theorem 2.2 in Section 3 for details. See the proof of Theorem 2.2 in Section 3 for details.

The third one is to investigate the conservation and energy laws, and the conservative form of our systems (1.3) and (1.4). In fact, any solution to system (1.4) with (1.5) satisfies that for $t_{1}<t_{2}$,

and

Moreover, any solution to system (1.3) with (1.2) satisfies (1.7) and (1.8) with $\mu_{A}\equiv\mu_{B}\equiv\lambda_{B}\equiv 0$. Here $|D(v_{A})|^{2}=D(v_{A}):D(v_{A})$, $|D(v_{B})|^{2}=D(v_{B}):D(v_{B})$, and $d\mathcal{H}^{2}_{x}$ denotes the 2-dimensional Hausdorff measure. We often call (1.7) and (1.8), the law of conservation of mass and the energy equality, respectively. We easily check that system (1.4) with ${\rm{div}}v_{A}=0$ satisfies the following conservative form:

Let us explain three key ideas of deriving our multiphase flow systems with phase transition. The first point is to acknowledge the existence of the interface, that is, we assume that the density of the interface is a positive constant $\rho_{0}$. The second point is to divide the condition $(v_{A}-v_{B})\cdot n_{\Gamma}=0$ into $v_{A}\cdot n_{\Gamma}=v_{S}\cdot n_{\Gamma}$ and $v_{B}\cdot n_{\Gamma}=v_{S}\cdot n_{\Gamma}$. The third point is to make use of an energetic variational approach. More precisely, we apply an energetic variational approach in order to look for functions $\Phi_{A}$ and $\Phi_{B}$ satisfying

and

An energetic variational approach is a method for deriving PDEs by using the forces derived from a variation of energies. Gyarmati [7] applied an energetic variational approach, which had been studied by Strutt [21] and Onsager [14, 15], to make several models for fluid dynamics in domains. Hyon-Kwak-Liu [8] made use of their energetic variational approach to study complex fluid in domains. Koba-Sato [12] applied their energetic variational approach to make their non-Newtonian fluid systems in domains. Koba-Liu-Giga [11] and Koba [9] employed their energetic variational approaches to derive their fluid systems on an evolving closed surface. However, these papers [8, 12, 11, 9] did not consider multiphase flow. This paper improves and modifies their methods in [8, 12, 11, 9] to derive our multiphase flow systems. See Section 4 for details.

Finally, we introduce the results related to this paper. Bothe-Prüss [3, 2] considered multiphase flow with interface effects. In [3], they made their models for multiphase flow with surface tension and viscosities by applying the Boussinesq-Scriven law. In [2], they made use of their jump conditions to make models for multi-component two-phase flow system with phase transition, and to study the individual mass densities of an isothermal mixture of $N$-species in a domain. Although this paper does not consider surface viscosity (surface flow), this paper considers interface effects such as surface tension and phase transition. Note that our models are different from the ones in [3, 2]. See also [11, 9] for models for surface flow.

The outline of this paper is as follows: In Section 2, we state the main results of this paper. In Section 3, we study the law of conservation of mass for multiphase flow with phase transition. In Section 4, we apply an energetic variational approach to make mathematical models for multiphase flow with phase transition. In Section 5, we investigate the conservation and energy laws of our systems. In Appendix, we provide two useful lemmas to derive our systems.

We first introduce the transport theorems. Then we state the main results.

Now we state the main results of this paper.

We prove Theorem 2.2 in Section 3 and Theorem 2.3 in Section 5. In Section 4, we derive our systems (1.3) and (1.4).

Let us immediately derive the one of the main results of this paper.