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Co-moving volumes and the Reynolds transport theorem for two-phase flows

Dieter Bothe, Matthias Köhne

Abstract

We consider the local kinematics at fluid interfaces in two-phase flows within the sharp interface framework. In the considered case with phase change and slip at the interface, the governing velocity field is discontinuous at the phase boundary with possible jumps of the normal and the tangential components. This causes the associated initial value problems for the kinematic differential equation, governing the motion of fluid elements, to be ill-posed in general. Motivated by a corresponding example, where the velocity field is physically consistent regarding the balance of mass and momentum as well as the entropy inequality, we employ concepts from differential inclusions, to rigorously define co-moving sets within this framework. Based on this general two-phase flow setting, we proof a natural extension of the Reynolds transport theorem to this case.

Co-moving volumes and the Reynolds transport theorem for two-phase flows

Abstract

We consider the local kinematics at fluid interfaces in two-phase flows within the sharp interface framework. In the considered case with phase change and slip at the interface, the governing velocity field is discontinuous at the phase boundary with possible jumps of the normal and the tangential components. This causes the associated initial value problems for the kinematic differential equation, governing the motion of fluid elements, to be ill-posed in general. Motivated by a corresponding example, where the velocity field is physically consistent regarding the balance of mass and momentum as well as the entropy inequality, we employ concepts from differential inclusions, to rigorously define co-moving sets within this framework. Based on this general two-phase flow setting, we proof a natural extension of the Reynolds transport theorem to this case.
Paper Structure (6 sections, 10 theorems, 133 equations, 4 figures)

This paper contains 6 sections, 10 theorems, 133 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Sharp-Interface Modeling Framework
  3. Two-Phase Flow Kinematics
  4. Two-phase Flow Map
  5. Two-Phase Transport Theorem for Co-Moving Volumes
  6. Appendix

Key Result

Theorem 1.1

Let $J=(a,b)\subset {\rm I R}$, $\Omega \subset{\rm I R}^3$ be open and $\mathbf{v} \in L^\infty (J\times \Omega ; {\rm I R}^3)$ be Lipschitz continuous in $\mathbf{x}$, uniformly for $t\in J$, i.e. with some $L\geq 0$. Let $t_0\in J$ and $G_0\subset \Omega$ be a compact Lipschitz domain. Let $r>0$ be such that the unique solutions of the initial value problems (DBeq:kinematic-ODE1) exist on $J_0

Figures (4)

  • Figure 1: Graph ${\Omega}_0$ of a family of co-moving volumes $G(t)$, employed as a domain for the divergence theorem in ${\rm I R}\times {\rm I R}^3$.
  • Figure 2: Simple two-phase co-moving volumes. Time proceeds from left to right. Top row: zero normal velocity with interfacial slip; middle row: phase change without slip; bottom row: phase change with slip.
  • Figure 3: Co-moving volumes associated with the velocity field from (\ref{['eq-counter-ex']}). Initial set is $G(0)=G_0:=[-1,0]\times [0,1/2]$ (top). Depicted is the kinematically transported set at the time instances $t=0.25$ (middle) and $t=0.5$ (bottom). Note that a non-trivial part of the boundary emanates from the single corner point $(0,0)$ of $G(0)$. This is the signature of the non-uniqueness of the initial value problem.
  • Figure 4: Geometric construction for the interior approximation property.

Theorems & Definitions (18)

  • Theorem 1.1: Reynolds transport theorem
  • proof
  • Definition 2.1
  • Definition 2.2: Speed of normal displacement
  • Lemma 4.1: Two-phase kinematic differential inclusion
  • Definition 4.2: Two-phase flow map and co-moving sets
  • Proposition 4.3
  • Example 4.4
  • Proposition 5.1
  • Theorem 5.2: Transport theorem for co-moving volumes in two-phase flows
  • ...and 8 more