Dietmar Kröner, Freiburg Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations Dietmar Kröner, Freiburg Strasbourg, April 15, 2010 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

Outline Introduction: Modelling of phase transition by Navier-Stokes Korteweg Jump conditions across the interface for NSK, static case Jump conditions in the low Mach number limit for the NSK system Jump conditions for the NSK system in the dynamical case Jump conditions for Phase field like scaling

Double well ρψ(ρ) Pressure p β1 β2 ρ β1 β2 ρ

Double well ρψ(ρ) Pressure p β1 β2 ρ β1 β2 ρ

Theoretical results

Numerical results PhD Thesis Dennis Diehl

Jump conditions across the interface (static case)

Stationary case: (Luckhaus, Modica, Dreyer, Kraus)

Jump conditions: ?????? liquid vapor

Multiply by a smooth testfunction ψ Integration by parts

?

=:R

Remember the assumptions:

Remember the assumptions:

Remember the assumptions:

Remember the assumptions:

Remember the assumptions:

liquid vapor

liquid vapor localize

Jump conditions ???? liquid localize vapor

liquid vapor

liquid vapor

liquid vapor

Summary

Summary liquid vapor Jump conditions:

Summary liquid vapor Jump conditions:

Summary liquid vapor Jump conditions: No curvature term !

Phase field like scaling Different scaling to get the curvature term

d‘Alambert variation principle

d‘Alambert variation principle

d‘Alambert variation principle

d‘Alambert variation principle

Integration by parts:

? ?

As before:

Lemma Luckhaus Modica o(1)

Lemma Luckhaus Modica o(1) o(1)

? Lemma Luckhaus Modica o(1) O(ε)

We need several steps:

Summary

liquid vapor localize

liquid localize vapor