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The Riemann problem for fluid flow of real materials. (English) Zbl 1129.35439

Summary: The Riemann problem for fluid flow of real materials is examined. An arbitrary equation of state is allowed, subject only to the physical requirements of thermodynamics. The properties of the isentropes and the shock Hugoniot loci that follow from conditions imposed on the equation of state are reviewed systematically. Important properties of these wave curves are determined by three dimensionless variables characterizing the equation of state: the adiabatic exponent \(\gamma\), the Grüneisen coefficient \(\Gamma\), and the fundamental derivative \(G\). Standard assumptions on these variables break down near phase transitions. The result is an anomalous wave structure: either shock waves split into multiple waves, or composite waves form. Additional questions related to shock stability and nonuniqueness of the solution of the Riemann problem are discussed.

MSC:

35L65 Hyperbolic conservation laws
76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
76L05 Shock waves and blast waves in fluid mechanics
76N15 Gas dynamics (general theory)
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[1] Adb-El-Fattah, A., J. Fluid Mech. 76 pp 157– (1976) · Zbl 0345.76037
[2] Adamson, T., Jr., J. Aerosp. Sci. 26 pp 16– (1959)
[3] Bdzil, J., J. Fluid Mech. 171 pp 1– (1986) · Zbl 0607.76067
[4] Bukiet, B., SIAM J. Appl. Math. (1989)
[5] Chaves, H., in: Lecture Notes in Physics (1985)
[6] Chern, I-L., J. Comput. Phys. 62 pp 83– (1985) · Zbl 0577.76068
[7] Chorin, A., J. Comput. Phys. 22 pp 517– (1976) · Zbl 0354.65047
[8] Colella, P., SIAM J. Sci. Stat. Comput. 6 pp 104– (1985) · Zbl 0562.76072
[9] Colella, P., J. Comput. Phys. 59 pp 264– (1985) · Zbl 0581.76079
[10] Collella, P., J. Comput. Phys. 54 pp 174– (1984) · Zbl 0531.76082
[11] Courant, R., in: Supersonic Flow and Shock Waves (1948) · Zbl 0041.11302
[12] Cowan, R., J. Fluid Mech. 3 pp 531– (1958) · Zbl 0081.41503
[13] Cowperthwaite, M., J. Franklin Inst. 285 pp 275– (1968)
[14] Cramer, M., Phys. Fluids 30 pp 377– (1987) · Zbl 0617.76072
[15] Davis, W., in: Eighth International Detonation Symposium (1985)
[16] Dukowicz, J., J. Comput. Phys. 61 pp 119– (1985) · Zbl 0629.76074
[17] Duvall, G., Rev. Mod. Phys. 49 pp 523– (1977)
[18] D’yakov, S., Zh. Eksp. Teor. Fiz. 27 pp 288– (1954)
[19] Erpenbeck, J., Phys. Fluids 5 pp 1181– (1962) · Zbl 0111.38403
[20] Erpenbeck, J., Phys. Fluids 6 pp 1368– (1963) · Zbl 0119.20203
[21] Fickett, W., in: Detonation (1979)
[22] Fowles, G., Phys. Fluids 19 pp 227– (1976) · Zbl 0336.76027
[23] Fowles, G., Phys. Fluids 24 pp 220– (1981) · Zbl 0464.76049
[24] Fowles, G., Phys. Fluids 27 pp 1982– (1984) · Zbl 0546.76088
[25] Gardner, C., Phys. Fluids 6 pp 1366– (1963) · Zbl 0119.20202
[26] Gelfand, I., Usp. Mat. Nauk 14 pp 87– (1959)
[27] Gilbarg, D., Am. J. Math. 73 pp 256– (1951) · Zbl 0044.21504
[28] Gilmore, R., in: Catastrophe Theory for Scientists and Engineers (1981) · Zbl 0497.58001
[29] Glass, I., J. Fluid Mech. 84 pp 55– (1978)
[30] Glimm, J., Commun. Pure Appl. Math. XVIII pp 697– (1965) · Zbl 0141.28902
[31] Glimm, J., Adv. Appl. Math. 6 pp 259– (1985) · Zbl 0631.76068
[32] Glimm, J., Found. Phys. 16 pp 125– (1986)
[33] Godunov, S., Mat. Sb. 47 pp 271– (1959)
[34] Griffiths, R., J. Phys. D 8 pp 1681– (1975)
[35] Guderley, G., Luftfahrt-Forsch. 19 pp 302– (1942)
[36] Hayes, in: Fundamentals of Gas Dynamics (1958)
[37] Henderson, L., J. Fluid Mech. 26 pp 607– (1966)
[38] Henderson, L., J. Fluid Mech. 30 pp 699– (1967) · Zbl 0153.28902
[39] Hornung, H., Annu. Rev. Fluid Mech. 18 pp 33– (1986)
[40] Isaacson, E., in: Construction of Nonlinear Waves for Conservation Laws (1988)
[41] Isaacson, E., SIAM J. Appl. Math. 48 pp 1009– (1988) · Zbl 0688.35056
[42] Isaacson, E., SIAM J. Appl. Math. (1988)
[43] Isaacson, E., SIAM J. Appl. Math. (1988)
[44] Jones, D., Proc. R. Soc. London Ser. A 209 pp 238– (1951) · Zbl 0044.21603
[45] Jones, J., Adv. Appl. Math. (1989)
[46] Kontorovich, V., Zh. Eksp. Teor. Fiz. 33 pp 1525– (1958)
[47] Kutateladze, S., Annu. Rev. Fluid Mech. 19 pp 577– (1987)
[48] Landau, L., in: Statistical Physics (1958) · Zbl 0080.19702
[49] Landau, L., in: Fluid Mechanics (1959)
[50] Lax, P., Commun. Pure Appl. Math. X 537 (1957)
[51] Lax, P., in: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1972)
[52] Le Floch, P., Ann. Inst. Henri Poincaré (1988)
[53] Lieb, Elliott H., Rev. Mod. Phys. 48 pp 553– (1976)
[54] Liu, T.-P., J. Diff. Eq. 18 pp 218– (1975) · Zbl 0297.76057
[55] Liu, T.-P., J. Math. Anal. Appl. 53 pp 78– (1976) · Zbl 0332.76051
[56] Liu, T.-P., Commun. Math. Phys. 68 pp 141– (1979) · Zbl 0435.35054
[57] Liu, T.-P., Commun. Pure Appl. Math. 39 pp 565– (1986) · Zbl 0617.76069
[58] Majda, A., Mem. Am. Math. Soc. 41 pp 275– (1983)
[59] Majda, A., Proc. Natl. Acad. Sci. U.S.A. 84 pp 6011– (1987) · Zbl 0631.76073
[60] Majda, A., SIAM J. Appl. Math. 43 pp 1310– (1983) · Zbl 0544.76135
[61] Majda, A., Stud. Appl. Math. 71 pp 117– (1984) · Zbl 0584.76075
[62] Marsh, S., in: LASL Shock Hugoniot Data (1980)
[63] Meier, G., in: Lecture Notes in Physics (1985)
[64] Mock, M., J. Diff. Eq. 38 pp 176– (1980) · Zbl 0417.34053
[65] Moler, C., Arch. Ration. Mech. Anal. 37 pp 309– (1970) · Zbl 0192.44901
[66] More, R., Phys. Fluids 31 pp 3059– (1988) · Zbl 0654.76042
[67] Oleĭnik, O., Usp. Mat. Nauk 14 pp 165– (1959)
[68] Oleĭnik, O., in: Seminari 1962/63 Analisi, Algebra, Geometria e Topologia, Vol. 1 (1965)
[69] Pego, R., Trans. Am. Math. Soc. 282 pp 749– (1984)
[70] Pego, R., Arch. Ration. Mech. Anal. 94 pp 165– (1986) · Zbl 0652.76047
[71] Plohr, B., AIAA J. 26 pp 470– (1988) · Zbl 0665.76094
[72] Plohr, B., Adv. Appl. Math. (1989)
[73] Rabie, R., Phys. Fluids 22 pp 422– (1979)
[74] Reichl, L., in: A Modern Course in Statistical Physics (1980) · Zbl 1334.00084
[75] Rice, M., in: Solid State Physics, Vol. 6 (1958)
[76] Riemann, B., in: Collected Works of Bernhard Riemann (1953)
[77] Russo, G., Phys. Fluids 30 pp 2406– (1987) · Zbl 0663.76143
[78] Shearer, M., Proc. R. Soc. Edinburgh, Sec. A 93 pp 233– (1983)
[79] Shearer, M., in: IMA Volumes in Mathematics and its Applications (1988)
[80] Shearer, M., Arch. Ration. Mech. Anal. 97 pp 299– (1987) · Zbl 0673.35072
[81] Sidorenko, A., Prikl. Mat. Mekh. 46 pp 241– (1982)
[82] Sidorenko, A., J. Appl. Math. Mech. 46 pp 180– (1982) · Zbl 0536.76064
[83] Slemrod, M., Arch. Ration. Mech. Anal. 81 pp 303– (1983) · Zbl 0505.76082
[84] Smith, R., Trans. Am. Math. Soc. 249 pp 1– (1979)
[85] Smoller, J., in: Shock Waves and Reaction-Diffusion Equations (1982) · Zbl 0508.35002
[86] Strehlow, R., Astronaut. Acta 15 pp 345– (1970)
[87] Tang, Z., Int. J. Eng. Sci. 25 pp 1343– (1987) · Zbl 0637.73020
[88] Thompson, P., Phys. Fluids 14 pp 1843– (1971) · Zbl 0236.76053
[89] Thompson, P., in: Compressible-Fluid Dynamics (1972) · Zbl 0251.76001
[90] Thompson, P., J. Fluid Mech. 166 pp 57– (1986) · Zbl 0601.76075
[91] van Leer, B., J. Comput. Phys. 32 pp 101– (1979) · Zbl 1364.65223
[92] Van Moorhem, W., J. Fluid Mech. 68 pp 97– (1975) · Zbl 0303.76038
[93] Wendroff, B., J. Math. Anal. Appl. 38 pp 454– (1972) · Zbl 0264.76054
[94] Weyl, H., Commun. Pure Appl. Math. II pp 103– (1949) · Zbl 0035.42004
[95] Whitham, G., in: Linear and Nonlinear Waves (1974) · Zbl 0373.76001
[96] Zel’dovich, Ya., in: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (1966)
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