The Riemann problem for a weakly hyperbolic two-phase flow model of a dispersed phase in a carrier fluid

Hantke, Maren; Matern, Christoph; Ssemaganda, Vincent; Warnecke, Gerald

doi:10.1090/qam/1556

Authors: Maren Hantke, Christoph Matern, Vincent Ssemaganda and Gerald Warnecke
Journal: Quart. Appl. Math. 78 (2020), 431-467
MSC (2010): Primary 35L65, 76T10, 76T15
DOI: https://doi.org/10.1090/qam/1556
Published electronically: November 19, 2019
MathSciNet review: 4100288
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Abstract: We consider Riemann problems for a two-phase isothermal flow model of a dispersed phase in a compressible carrier phase. It is a weakly hyperbolic system of conservative partial differential equations. This model is the conservation part of a more complete physical model involving phase transitions in case both phases are of the same material. The purpose of this paper is to better understand the mathematical properties of the simplified model. We investigate the characteristic structure of the Riemann problems and construct their exact solutions. Solutions may contain delta shocks or vaporless states. We give examples for initial data corresponding to a system of water bubbles dispersed in liquid water. The analysis is complicated considerably by the fact that a liquid such as water requires an affine equation of state.

References

Nikolai Andrianov and Gerald Warnecke, On the solution to the Riemann problem for the compressible duct flow, SIAM J. Appl. Math. 64 (2004), no. 3, 878–901. MR 2068446, DOI https://doi.org/10.1137/S0036139903424230
Nikolai Andrianov and Gerald Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model, J. Comput. Phys. 195 (2004), no. 2, 434–464. MR 2046106, DOI https://doi.org/10.1016/j.jcp.2003.10.006

M.R. Baer and J.W. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flows 12 (1986), 861–889.

F. Bouchut, On zero pressure gas dynamics, Advances in kinetic theory and computing, Ser. Adv. Math. Appl. Sci., vol. 22, World Sci. Publ., River Edge, NJ, 1994, pp. 171–190. MR 1323183
Shaozhong Cheng, Jiequan Li, and Tong Zhang, Explicit construction of measure solutions of Cauchy problem for transportation equations, Sci. China Ser. A 40 (1997), no. 12, 1287–1299. MR 1613902, DOI https://doi.org/10.1007/BF02876374

C. Crowe, M. Sommerfeld, and Y. Tsuji, Multiphase flows with droplets and particles, CRC Press, Boca Raton, 1998.

K. Davitt, E. Rolley, F. Caupin, A. Arvengas, and S. Balibar, Equation of state of water under negative pressure, Journal of Chemical Physics 133 (2010), no. 17, 1745071 –1745078.

Donald A. Drew and Stephen L. Passman, Theory of multicomponent fluids, Applied Mathematical Sciences, vol. 135, Springer-Verlag, New York, 1999. MR 1654261
Maren Hantke, Wolfgang Dreyer, and Gerald Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition, Quart. Appl. Math. 71 (2013), no. 3, 509–540. MR 3112826, DOI https://doi.org/10.1090/S0033-569X-2013-01290-X
Wolfgang Dreyer, Maren Hantke, and Gerald Warnecke, Bubbles in liquids with phase transition—part 2: on balance laws for mixture theories of disperse vapor bubbles in liquid with phase change, Contin. Mech. Thermodyn. 26 (2014), no. 4, 521–549. MR 3225536, DOI https://doi.org/10.1007/s00161-013-0316-7

J. Dymond and R. Malhotra, The Tait equation: 100 years on., International Journal of Thermodynamics 9(6) (1988), 941–951.

Weinan E, Yu. G. Rykov, and Ya. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys. 177 (1996), no. 2, 349–380. MR 1384139

M.H. Ernst, R.M. Ziff, and E.M. Hendriks, Coagulation processes with a phase transition, J. Colloid Interface Sci. 97 (1984), 266 –277.

M. Escobedo, S. Mischler, and B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys. 231 (2002), no. 1, 157–188. MR 1947695, DOI https://doi.org/10.1007/s00220-002-0680-9

L.C. Evans, Partial differential equations, Vol. 19, Amer. Math. Soc. Providence, Rhode Island, 1998.

Ee Han, Maren Hantke, and Gerald Warnecke, Exact Riemann solutions to compressible Euler equations in ducts with discontinuous cross-section, J. Hyperbolic Differ. Equ. 9 (2012), no. 3, 403–449. MR 2974765, DOI https://doi.org/10.1142/S0219891612500130
Maren Hantke and Ferdinand Thein, Why condensation by compression in pure water vapor cannot occur in an approach based on Euler equations, Quart. Appl. Math. 73 (2015), no. 3, 575–591. MR 3400760, DOI https://doi.org/10.1090/S0033-569X-2015-01393-9

M. Ishii, Thermal-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975.

Intae Jeon, Existence of gelling solutions for coagulation-fragmentation equations, Comm. Math. Phys. 194 (1998), no. 3, 541–567. MR 1631473, DOI https://doi.org/10.1007/s002200050368
Barbara Lee Keyfitz and Herbert C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451. MR 1330835, DOI https://doi.org/10.1006/jdeq.1995.1080
Jiequan Li and Hanchun Yang, Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics, Quart. Appl. Math. 59 (2001), no. 2, 315–342. MR 1827367, DOI https://doi.org/10.1090/qam/1827367

J. Li and T. Zhang, Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations, Nonlinear partial differential equations and related areas, 1999.

Tai Ping Liu, Transonic gas flow in a duct of varying area, Arch. Rational Mech. Anal. 80 (1982), no. 1, 1–18. MR 656799, DOI https://doi.org/10.1007/BF00251521
Tai-Ping Liu, Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 28 (1987), no. 11, 2593–2602. MR 913412, DOI https://doi.org/10.1063/1.527751
T. P. Liu and J. A. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv. in Appl. Math. 1 (1980), no. 4, 345–359. MR 603135, DOI https://doi.org/10.1016/0196-8858%2880%2990016-0

M. Mazzotti, Nonclassical composition fronts in nonlinear chromatography: Delta-shock, Ind. Eng. Chem. Res. 48 (2009), 7733 –7752.

M. Mazzotti, A. Tarafder, J. Cornel, F. Gritti, and G. Guiochon, Experimental evidence of a delta-shock in nonlinear chromatography, J. Chromatography A 1217 (2010), 2002 –2012.

Ralph Menikoff and Bradley J. Plohr, The Riemann problem for fluid flow of real materials, Rev. Modern Phys. 61 (1989), no. 1, 75–130. MR 977944, DOI https://doi.org/10.1103/RevModPhys.61.75
Ingo Müller and Tommaso Ruggeri, Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, vol. 37, Springer-Verlag, New York, 1998. With supplementary chapters by H. Struchtrup and Wolf Weiss. MR 1632151

R. Nigmatulin, Dynamics of multiphase media, Vol. 1, Hemisphere Publ., New York-Washington-Philadelphia-London, 1991.

Richard Saurel and Rémi Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150 (1999), no. 2, 425–467. MR 1684902, DOI https://doi.org/10.1006/jcph.1999.6187
S. F. Shandarin and Ya. B. Zel′dovich, The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys. 61 (1989), no. 2, 185–220. MR 989562, DOI https://doi.org/10.1103/RevModPhys.61.185
Wancheng Sheng and Tong Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137 (1999), no. 654, viii+77. MR 1466909, DOI https://doi.org/10.1090/memo/0654
Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
H. Bruce Stewart and Burton Wendroff, Two-phase flow: models and methods, J. Comput. Phys. 56 (1984), no. 3, 363–409. MR 768670, DOI https://doi.org/10.1016/0021-9991%2884%2990103-7
De Chun Tan, Tong Zhang, and Yu Xi Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), no. 1, 1–32. MR 1287550, DOI https://doi.org/10.1006/jdeq.1994.1093
Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, 3rd ed., Springer-Verlag, Berlin, 2009. A practical introduction. MR 2731357

W. Wagner and H.-J. Kretzschmar, International steam tables: Properties of water and steam based on the industrial formulation iapws-if97, Springer-Verlag Berlin Heidelberg, 2008.

Hanchun Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations 159 (1999), no. 2, 447–484. MR 1730728, DOI https://doi.org/10.1006/jdeq.1999.3629
Yuxi Zheng, Systems of conservation laws with incomplete sets of eigenvectors everywhere, Advances in nonlinear partial differential equations and related areas (Beijing, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 399–426. MR 1690841