TU Darmstadt ULB TUbiblio

# Wellposedness of the discontinuous ODE associated with two-phase flows

## 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 2019 Bothe, Dieter Wellposedness of the discontinuous ODE associated with two-phase flows English 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 DFG-Collaborative Research Centres (incl. Transregio)DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research CentresDFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres > CRC 1194: Interaction between Transport and Wetting ProcessesDFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres > CRC 1194: Interaction between Transport and Wetting Processes > Research Area B: Modeling and SimulationDFG-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 11 Dec 2019 12:34 http://arxiv.org/pdf/1905.04560 HTML CitationReference ManagerMultiline CSVDublin CoreMODSASCII CitationRDF+XMLJSONAtomSimple MetadataIBW_RDABibTeXEndNoteT2T_XMLEP3 XML TUfind oder in Google
 Send an inquiry

Options (only for editors)
 Show editorial Details