Bothe, Dieter (2019)
Wellposedness of the discontinuous ODE associated with two-phase flows.
Report, Bibliographie
Kurzbeschreibung (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
| Typ des Eintrags: | Report |
|---|---|
| Erschienen: | 2019 |
| Autor(en): | Bothe, Dieter |
| Art des Eintrags: | Bibliographie |
| Titel: | Wellposedness of the discontinuous ODE associated with two-phase flows |
| Sprache: | Englisch |
| Publikationsjahr: | 11 Dezember 2019 |
| URL / URN: | http://arxiv.org/pdf/1905.04560 |
| Kurzbeschreibung (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 |
| Fachbereich(e)/-gebiet(e): | DFG-Sonderforschungsbereiche (inkl. Transregio) DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen > Projektbereich B: Modellierung und Simulation DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen > Projektbereich B: Modellierung und Simulation > B01: Modellierung und VOF-basierte Simulation der Multiphysik irreversibler thermodynamischer Transferprozesse an dynamischen Kontaktlinien |
| Hinterlegungsdatum: | 11 Dez 2019 12:34 |
| Letzte Änderung: | 05 Jun 2023 12:57 |
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