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Wellposedness of the discontinuous ODE associated with two-phase flows

Bothe, Dieter (2019):
Wellposedness of the discontinuous ODE associated with two-phase flows.
[Online-Edition: http://arxiv.org/pdf/1905.04560],
[Report]

Abstract

We consider the initial value problem \[ \dot x (t) = v(t,x(t)) \;\mbox for t\in (a,b), \;\; x(t_0)=x_0 \] which determines the pathlines of a two-phase flow, i.e.\ $v=v(t,x)$ is a given velocity field of the type \[ v(t,x)= \begincases v\^+(t,x) &\text if x \in Ømega\^+(t) v\^-(t,x) &\text if x \in Ømega\^-(t) \endcases \] with $Ømega\^± (t)$ denoting the bulk phases of the two-phase fluid system under consideration. The bulk phases are separated by a moving and deforming interface $Σ (t)$. Since we allow for flows with phase change, these pathlines are allowed to cross or touch the interface. Imposing a kind of transversality condition at $Σ (t)$, which is intimately related to the mass balance in such systems, we show existence and uniqueness of absolutely continuous solutions of the above ODE in case the one-sided velocity fields $v\^±:øverline\rm gr(Ømega\^±)\to \mathbbR\^n$ are continuous in $(t,x)$ and locally Lipschitz continuous in $x$. Note that this is a necessary prerequisite for the existence of well-defined co-moving control volumes for two-phase flows, a basic concept for mathematical modeling of two-phase continua

Item Type: Report
Erschienen: 2019
Creators: Bothe, Dieter
Title: Wellposedness of the discontinuous ODE associated with two-phase flows
Language: English
Abstract:

We consider the initial value problem \[ \dot x (t) = v(t,x(t)) \;\mbox for t\in (a,b), \;\; x(t_0)=x_0 \] which determines the pathlines of a two-phase flow, i.e.\ $v=v(t,x)$ is a given velocity field of the type \[ v(t,x)= \begincases v\^+(t,x) &\text if x \in Ømega\^+(t) v\^-(t,x) &\text if x \in Ømega\^-(t) \endcases \] with $Ømega\^± (t)$ denoting the bulk phases of the two-phase fluid system under consideration. The bulk phases are separated by a moving and deforming interface $Σ (t)$. Since we allow for flows with phase change, these pathlines are allowed to cross or touch the interface. Imposing a kind of transversality condition at $Σ (t)$, which is intimately related to the mass balance in such systems, we show existence and uniqueness of absolutely continuous solutions of the above ODE in case the one-sided velocity fields $v\^±:øverline\rm gr(Ømega\^±)\to \mathbbR\^n$ are continuous in $(t,x)$ and locally Lipschitz continuous in $x$. Note that this is a necessary prerequisite for the existence of well-defined co-moving control volumes for two-phase flows, a basic concept for mathematical modeling of two-phase continua

Divisions: DFG-Collaborative Research Centres (incl. Transregio)
DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres
DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres > CRC 1194: Interaction between Transport and Wetting Processes
DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres > CRC 1194: Interaction between Transport and Wetting Processes > Research Area B: Modeling and Simulation
DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres > CRC 1194: Interaction between Transport and Wetting Processes > Research Area B: Modeling and Simulation > B01: Modelling and VOF based Simulation of the Multiphysics of Irreversible Thermodynamic Transfer Processes at Dynamic Contact Lines
Date Deposited: 11 Dec 2019 12:34
Official URL: http://arxiv.org/pdf/1905.04560
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