2019/05/11 by Dieter Bothe, Bothe, Dieter
Mathematics · Physics and Astronomy · #34A36 (Primary) 34A60 #76A02 #76T10 (Secondary) #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Mathematical Physics (math-ph) #math-ph #math.AP #math.CA #math.MP #msc:34A36 #msc:34A60 #msc:76A02 #msc:76T10 #physics.flu-dyn
paper · pdf · doi:10.48550/arxiv.1905.04560
arxiv created 2019/05/11 · arxiv updated 2019/05/14
We consider the initial value problem x (t) = v(t,x(t)) for t∈ (a,b), x(t0)=x0 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) if x ∈ Ω+(t)
v-(t,x) if x ∈ Ω-(t) \endcases 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). 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^±:\rm gr(Ω^±)→ ℝn 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.