Abstract
We consider the initial value problem which determines the pathlines of a two-phase flow, i.e. v = v(t, x) is a given velocity field of the type with Ω±(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) at which v can have jump discontinuities. Since flows with phase change are included, the 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± are continuous in (t, x) and locally Lipschitz continuous in x on their respective domain of definition. A main step in proving this result, also interesting in itself, is to freeze the interface movement by means of a particular coordinate transform which requires a tailor-made extension of the intrinsic velocity field underlying a -family of moving hypersurfaces.
Highlights
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± are continuous in (t, x) and locally Lipschitz continuous in x on their respective domain of definition
We consider a specific class of discontinuous ODE-systems which appear naturally in the study of two-phase flows, where the underlying situation is as follows: a domain Ω ⊂ IRn is occupied by two immiscible fluid phases, denoted as Ω±
While this approach via directional continuity is very helpful for instance to prove the existence and study qualitative properties of a.c. solutions for differential inclusions with lower semicontinuous right-hand side, it does not apply to general two-phase flow velocity fields, since the cone Kα is not related to the discontinuity surface Σ
Summary
We consider a specific class of discontinuous ODE-systems which appear naturally in the study of two-phase flows, where the underlying situation is as follows: a domain Ω ⊂ IRn is occupied by two immiscible fluid phases, denoted as Ω±. F(t, x) := conv f (Bδ((t, x)) ∩ [J × Ω]) for t ∈ J, x ∈ Ω δ>0 would work in this situation While this approach via directional continuity is very helpful for instance to prove the existence and study qualitative properties of a.c. solutions for differential inclusions with lower semicontinuous right-hand side (see [9, 11]), it does not apply to general two-phase flow velocity fields, since the cone Kα is not related to the discontinuity surface Σ. Before giving a brief overview of available results, explaining why they do not cover the ODE-system associated to two-phase flows, recall that local Lipschitz continuity of f in x is sufficient for local existence of a unique solution to (4) due to the Picard–Lindelöf theorem In the latter classical result, f is assumed to be jointly continuous, which can be relaxed to mere measurability in t. In order to motivate our assumptions and to state as well as to prove our main result, some background on the physical model as well as some auxiliary results on moving hypersurfaces are required
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