1. 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

2. 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

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

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

6. Theoretical results

7. Numerical results
PhD Thesis Dennis Diehl

8. Jump conditions across the interface (static case)

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

14. Jump conditions: ??????
liquid
vapor

16. Multiply by a smooth testfunction ψ
Integration by parts

21. ?

22. =:R

27. Remember the assumptions:

28. Remember the assumptions:

29. Remember the assumptions:

30. Remember the assumptions:

31. Remember the assumptions:

35. liquid
vapor

36. liquid
vapor
localize

37. Jump conditions ????
liquid
localize
vapor

38. liquid
vapor

39. liquid
vapor

40. liquid
vapor

44. Summary

45. Summary
liquid
vapor
Jump conditions:

46. Summary
liquid
vapor
Jump conditions:

47. Summary
liquid
vapor
Jump conditions:
No curvature term !

48. Phase field like scaling
Different scaling to get the curvature term

53. d‘Alambert variation principle

54. d‘Alambert variation principle

55. d‘Alambert variation principle

56. d‘Alambert variation principle

57. Integration by parts:

60. ? ?

61. As before:

64. Lemma
Luckhaus
Modica
o(1)

65. Lemma
Luckhaus
Modica
o(1)
o(1)

66. ?
Lemma
Luckhaus
Modica
o(1)
O(ε)

67. We need several steps:

68. Summary

69. liquid
vapor
localize

70. liquid
localize
vapor