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**Bothe, Dieter** (2019):

*Wellposedness of the discontinuous ODE associated with two-phase flows.*

[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 |

URL / URN: | http://arxiv.org/pdf/1905.04560 |

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