A complete classification of shock waves in a van der Waals fluid is undertaken. This is in order to gain a theoretical understanding of those shock-related phenomena as observed in real fluids which cannot be accounted for by the ideal gas model. These relate to admissibility of rarefaction shock waves, shock-splitting phenomena, and shock-induced phase transitions. The crucial role played by the nature of the gaseous state before the shock (the unperturbed state), and how it affects the features of the shock wave are elucidated. A full description is given of the characteristics of shock waves propagating in a van der Waals fluid. The strength of these shock waves may range from weak to strong. The study is carried out by means of the theory of hyperbolic systems supported by numerical calculations.
REFERENCES
1.
L. D.
Landau
and E. M.
Lifshitz
, Course of Theoretical Physics, Fluid Mechanics
Vol. 6
(Pergamon
, London
1959
).2.
L. E.
Henderson
, “General laws for propagation of shock waves through matter
,” in Handbook of Shock Waves
, Vol. 1
– Theoretical, Experimental, and Numerical Techniques, edited by G.
Ben-Dor
, O.
Igra
, and T.
Elperin
(Academic
, San Diego
2001
) pp. 143
–183
.3.
M. S.
Cramer
, “Nonclassical dynamics of classical gases
,” in Nonlinear Waves in Real Fluids
, edited by A.
Kluwick
(Springer-Verlag
, New York
1991
) Chap. 5, pp. 91
–145
.4.
H. A.
Bethe
, “On the theory of shock waves for an arbitrary equation of state
,” Technical Report 545, Office of Scientific Research and Development (1942
).5.
Ya. B.
Zel’dovich
, “On the possibility of rarefaction shock waves
,” Zh. Eksp. Teor. Fiz.
4
, 363
(1946
).6.
P. A.
Thompson
and K. C.
Lambrakis
, “Negative shock waves
,” J. Fluid Mech.
60
, 187
(1973
).7.
Shock Wave Science and Technology Reference Library
, Multiphase Flows I, Vol. 1
, edited by M. E. H.
van Dongen
(Springer Verlag
, Berlin
2007
).8.
A. A.
Borisov
, A. L.
Borisov
, S. S.
Kutateladze
, and V. E.
Nakoyakov
, “Rarefaction shock wave near the thermodynamic critical point
,” J. Fluid Mech.
126
, 59
(1983
).9.
S. S.
Kutateladze
, V. E.
Nakoryakov
, and A. A.
Borisov
, “Rarefaction waves in liquid and gas-liquid media
,” Annu. Rev. Fluid Mech.
19
, 577
(1987
).10.
A.
Kluwick
, “Rarefaction shocks
,” in Handbook of Shock Waves
, Vol. 1
. Theoretical, Experimental, and Numerical Techniques, edited by G.
Ben-Dor
, O.
Igra
, and T.
Elperin
(Academic
, San Diego
2001
), Chap. 3 and 4, pp. 339
–411
.11.
C.
Zamfirescu
, A.
Guardone
, and P.
Colonna
, “Admissibility region for rarefaction shock waves in dense gases
,” J. Fluid Mech.
599
, 363
(2008
).12.
A.
Guardone
, C.
Zamfirescu
, and P.
Colonna
, “Maximum intensity of rarefaction shock waves for dense gases
,” J. Fluid Mech.
642
, 127
(2010
).13.
P. A.
Thompson
and Y.-G.
Kim
, “Direct observation of shock splitting in a vapor-liquid system
,” Phys. Fluids
26
, 3211
(1983
).14.
P. A.
Thompson
, H.
Chaves
, G. E. A.
Maier
, Y.-G.
Kim
, and H.-D.
Speckmann
, “Wave splitting in a fluid of large heat-capacity
,” J. Fluid Mech.
185
, 385
(1987
).15.
M. S.
Cramer
, “Shock splitting in single-phase gases
,” J. Fluid Mech.
199
, 281
(1989
).16.
J. W.
Bates
and D. C.
Montgomery
, “Some numerical studies of exotic shock wave behavior
,” Phys. Fluids
11
, 462
(1999
).17.
H.
Hattori
, “The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion–Nonisothermal case
,” J. Differ. Equations
65
, 158
(1986
).18.
P. A.
Thompson
, G. A.
Carofano
, and Y.-G.
Kim
, “Shock waves and phase changes in a large heat capacity fluid emerging from a tube
,” J. Fluid Mech.
166
, 57
(1986
).19.
P. A.
Thompson
, “Liquid-vapor adiabatic phase changes and related phenomena
,” in Nonlinear Waves in Real Fluids
, edited by A.
Kluwick
(Springer
, New York
1991
) Chap. 6, pp. 147
–213
.20.
N.
Zhao
, M.
Sugiyama
, and T.
Ruggeri
, “Phase Transition induced by a shock wave in hard-sphere and hard-disc systems
,” J. Chem. Phys.
129
, 054506
(2008
).21.
S.
Taniguchi
, A.
Mentrelli
, N.
Zhao
, T.
Ruggeri
, and M.
Sugiyama
, “Shock-induced phase transition in systems of hard spheres with internal degrees of freedom
,” Phys. Rev. E
81
, 066307
(2010
).22.
Y.
Zheng
, N.
Zhao
, T.
Ruggeri
, M.
Sugiyama
, and S.
Taniguchi
, “Non-polytropic effect on shock-induced phase transitions in a hard-sphere system
,” Phys. Lett. A
374
, 3315
(2010
).23.
G. E. A.
Meier
, “Liquefaction shock waves
”, in Shock Wave Science and Technology Reference Library
, Vol. 1
– Multiphase Flows I, edited by M. E. H.
van Dongen
(Springer Verlag
, Berlin
2007
) Chap. 7, pp. 231
–267
.24.
R.
Menikoff
and B. J.
Plohr
, “The Riemann problem for fluid flow of real materials
,” Rev. Mod. Phys.
61
, 75
(1989
).25.
Nonlinear Waves in Real Fluids
, edited by A.
Kluwick
(Springer
, New York
1991
).26.
Handbook of Shock Waves
, Vol. 1
– Theoretical, Experimental, and Numerical Techniques, edited by G.
Ben-Dor
, O.
Igra
, and T.
Elperin
(Academic
, San Diego
2001
).27.
Ya. B.
Zel’dovich
and Yu. P.
Reizer
, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena
(Dover
, Minola
2002
).28.
W.
Dahmen
, S.
Müller
, and A.
Voß
, “Riemann problem for the Euler equation with non-convex equation of state including phase transitions
,” in Analysis and Numerics for Conservation Laws
, edited by G.
Warnecke
(Springer
, Berlin
2005
) pp. 137
–162
.29.
M.
Slemrod
, “Admissibility criteria for propagating phase boundaries in a van der Waals fluid
,” Arch. Ration. Mech. Anal.
81
, 301
(1983
).30.
H.
Hattori
, “The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion–Isothermal case
,” Arch. Ration. Mech. Anal.
92
, 247
(1986
).31.
L.
Quartapelle
, L.
Castelletti
, A.
Guardone
, and G.
Quaranta
, “Solution of the Riemann problem of classical gasdynamics
,” J. Comput. Phys.
190
, 118
(2003
).32.
S.
Müller
, and A.
Voß
“The Riemann problem for the Euler equations with nonconvex and nonsmooth equation of state: Constructon of wave curves
,” SIAM J. Sci. Comput.
28
, 651
(2006
).33.
J. L.
Lebowitz
and O.
Penrose
, “Rigorous treatment of the van der Waals-Maxwell theory of liquid-vapor transition
,” J. Math. Phys.
7
, 98
(1966
).34.
T.-P.
Liu
, “The entropy condition and the admissibility of shocks
,” J. Math. Anal. Appl.
53
, 78
(1976
).35.
T.-P.
Liu
, “Admissible solutions of hyperbolic conservation laws
,” Mem. Am. Math. Soc.
240
, 12
(1981
).36.
P. D.
Lax
, “Hyperbolic systems of conservation land the mathematical theory of shock waves
,” in CBMS-NSF, Regional Conference Series in Applied Mathematics
Vol. 11
(SIAM
, Philadelphia
1973
).37.
E. E.
Michaelides
and K. L.
Zissis
, “Velocity of sound in two-phase mixtures
,” Int. J. Heat Fluid Flow
4
, 79
(1983
).38.
D. J.
Picard
and P. R.
Bishnoi
, “Calculation of the thermodynamic sound velocity in two-phase multicomponent fluids
,” Int. J. Multiphase Flow
13
, 295
(1987
).39.
G.
Boillat
, “Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systémes hyperboliques
”, Acad. Sci., Paris, C. R. A
278
, 909
(1974
).40.
T.
Ruggeri
and A.
Strumia
, “Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics
,” Ann. Inst. Henri Poincare, Sect. A
34
, 65
(1981
).41.
G.
Boillat
and T.
Ruggeri
, “Hyperbolic principal subsystems: Entropy convexity and subcharacteristic conditions
,” Arch. Ration. Mech. Anal.
137
, 305
(1997
).42.
C. M.
Dafermos
, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der mathematischen Wissenschaften 325
, 3rd ed. (Springer
, Berlin Heidelberg
, 2010
) Chap. 8.43.
44.
T.-P.
Liu
and T.
Ruggeri
, “Entropy production and admissibility of shocks
,” Acta Math. Appl. Sin. English Ser.
19
, 1
(2003
).45.
T.
Ruggeri
, A.
Muracchini
, and L.
Seccia
, “Continuum approach to phonon gas and shape changes of second sound via shock waves theory
”, Nuovo Cimento
16
, 15
(1994
).46.
P. A.
Thompson
, “A fundamental derivative in gas dynamics
,” Phys. Fluids
14
, 1843
(1971
).47.
S.
Taniguchi
, A.
Mentrelli
, T.
Ruggeri
, M.
Sugiyama
, and N.
Zhao
, “Prediction and simulation of compressive shocks with lower perturbed density for increasing shock strength in real gases
,” Phys. Rev. E
82
, 036324
(2010
).48.
M. S.
Cramer
and R.
Sen
, “Exact solutions for sonic shocks in van der Waals gases
,” Phys. Fluids
30
, 377
(1987
).49.
P. A.
Thompson
and D. A.
Sullivan
, “On the possibility of complete condensation shock waves in retrograde fluids
,” J. Fluid Mech.
70
, 639
(1975
).50.
L.
Pareschi
, G.
Puppo
, and G.
Russo
, “Central Runge-Kutta schemes for conservation laws
,” SIAM J. Sci. Comput.
26
, 979
(2005
).51.
A.
Mentrelli
and T.
Ruggeri
, “Asymptotic behavior of Riemann and Riemann with structure problems for a 2 × 2 hyperbolic dissipative system
,” Suppl. Rend. Circ. Mat. Palermo
II/78
, 201
(2006
).52.
A.
Mentrelli
, T.
Ruggeri
, M.
Sugiyama
, and N.
Zhao
, “Interaction between a shock and an acceleration wave in a perfect gas for increasing shock strength
,” Wave Motion
45
, 498
(2008
).53.
As usual in the literature of hyperbolic systems, we call weak shock a shock such that and strong shock a shock such that , where ξ0 is the value of the parameter corresponding to the null shock (u1 → u0 when ξ → ξ0). The weak shock limit and strong shock limit are obtained, respectively, when ξ → ξ0 and |ξ| → ∞.
54.
Since, as explained in Sec. IV, we consider without any loss of generality v0 = 0, the unperturbed Mach number is proportional to the shock velocity: M0 = s/c0. The quantities s and λ appearing in the selection rules given in Sec. V A are thus connected to the quantities M0 and λ* through the same constant of proportionality.
55.
This is easily seen considering that (see Eqs. (22) and (25)) the Hugoniot locus has the vertical asymptote which is, for any meaningful value of both and δ, always on the left, in the plane, of the asymptote of the critical isotherm (see Eq. (8)2), thus guaranteeing that the states belonging to the Hugoniot locus are, at least in the strong shock limit, always is the gas phase.
56.
This is proven considering that any Hugoniot locus for passes above the critical point, in the plane, in accordance with the definition of the curve. Since the (reduced) temperature is always increasing, along the Hugoniot locus, as the (reduced) density increases (see Eqs. (8) and (22)), it follows that the Hugoniot locus can never cross the critical isotherm on the right of the critical point.
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