Reflection of shock fronts in a van der Waals fluid

Abstract

In this paper, the reflection phenomenon of a vapor shock front (both sides of the front are in the vapor phase) in a van der Waals fluid is considered. Both the 1-dimensional case and the multidimensional case are investigated. The authors find that under certain conditions, the reflected wave can be a single shock, or a single subsonic phase boundary, or one weak shock together with one subsonic phase boundary, which depends on the strength of the incident shock. This is different from the known result for the reflection of shock fronts in a gas dynamical system due to Chen in 1989.

Keywords

Shock / Subsonic phase boundary / Reflection / Boundary value problem

References

[1] Alinhac S. Le problème de goursat hyperbolique en dimension deux. Comm. Part. Diff. Eq., 1976, 1: 231-281

[2] Benzoni-Gavage S. Stability of multi-dimensional phase transitions in a van der Waals fluid. Nonlinear Analysis, T. M. A., 1998, 31: 243-263

[3] Benzoni-Gavage S. Stability of subsonic planar phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal., 1999, 150: 23-55

[4] Chen S. On reflection of multidimensional shock front. J. Diff. Eq., 1989, 80: 199-236

[5] Chen S. On the initial-boundary value problems for quasilinear symmetric hyperbolic system and the applications (in Chinese). Chin. Ann. Math., 1980, 1(3–4): 511-521

[6] Fan H. Traveling waves, Riemann problems and computations of a model for the dynamics of liquad/vaper phase transitions. J. Diff. Eq., 1998, 150: 385-437

[7] Fan H. Symmetry breaking, ring formation and other phase boundary structures in shock tube experiments on retrograde fluids. J. Fluid Mech., 2004, 513: 47-75

[8] Fan H, Slemrod M. Dynamic flows with liquid/vapor phase transitions, Handbook of Mathematical Fluid Dynamics, Vol. I, 2002, Amsterdam: North-Holland 373-420

[9] Freistühler H. Some results on the stability of non-classical shock waves. J. Part. Diff. Eq., 1998, 11: 25-38

[10] Guès O. Problème mixte hyperbolique quasi-linéaire caractéistique. Comm. Part. Diff. Eq., 1990, 15: 401-446

[11] Lax P D. Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math., 1957, 10: 537-467

[12] LeFloch P. Propagating phase boundaries: formulation of the problem and existence via the Glimm method. Arch. Rational Mech. Anal., 1993, 123: 153-197

[13] Majda A. The stability of multi-dimensional shock fronts. Mem. Amer. Math. Soc., 1983, 275: 1-95

[14] Majda A. The existence of multi-dimensional shock fronts. Mem. Amer. Math. Soc., 1983, 281: 1-93

[15] Métivier G. Interaction de deux chocs pour un systéme de deux lois conservation, en dimension deux d’espace. Trans. Amer. Math. Soc., 1986, 296: 431-479

[16] Shearer M. Nonuniqueness of admissible solutions of Riemann initial value problem for a system of conservation laws of mixed type. Arch. Rational Mech. Anal., 1986, 93: 45-59

[17] Slemrod M. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal., 1983, 81: 301-315

[18] Wang Y G, Xin Z. Stability and existence of multidimensional subsonic phase transitions. Acta Math. Appl. Sinica, 2003, 19: 529-558

[19] Zhang S Y, Wang Y G. Existence of multidimensional shock waves and phase boundaries. J. Diff. Eq., 2008, 244: 1571-1602