Closure conditions for non-equilibrium multi-component models. (English) Zbl 1355.76069
Summary: A class of non-equilibrium models for compressible multi-component fluids in multi-dimensions is investigated taking into account viscosity and heat conduction. These models are subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure, temperature and chemical potentials. Sufficient conditions are derived for these quantities that ensure meaningful physical properties such as a non-negative entropy production, thermodynamical stability, Galilean invariance and mathematical properties such as hyperbolicity, subcharacteristic property and existence of an entropy-entropy flux pair. For the relaxation of chemical potentials, a two-component and a three-component models for vapor-water and gas-water-vapor, respectively, are considered.
MSC:
| 76Txx | Multiphase and multicomponent flows |
Keywords:
multi-component flows; entropy; relaxation; phase transition; closure conditions; hyperbolicity; subcharacteristic property; Galilean invarianceReferences:
| [1] | Abgrall R., Karni S.: A comment on the computation of non-conservative products. J. Comput. Phys. 229, 2759-2763 (2010) · Zbl 1188.65134 · doi:10.1016/j.jcp.2009.12.015 |
| [2] | Ambroso A., Chalons C., Coquel F., Galié T., Godlewski E., Raviart P., Seguin N.: The drift-flux asymptotic limit of barotropic two-phase two-pressure models. Commun. Math. Sci. 6(2), 521-529 (2008) · Zbl 1141.76065 · doi:10.4310/CMS.2008.v6.n2.a13 |
| [3] | Andrianov, N.: Analytical and numerical investigation of two-phase flows. Ph.D. thesis, Otto-von-Guericke University, Magdeburg (2003) |
| [4] | Baer M., Nunziato J.: A two-phase mixture theory of the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flows 12, 861-889 (1986) · Zbl 0609.76114 · doi:10.1016/0301-9322(86)90033-9 |
| [5] | Coquel F., Gallouët T., Helluy P., Hérard J.M., Hurisse O., Seguin N.: Modelling compressible multiphase flows. Proc. ESAIM 40, 34-50 (2013) · Zbl 1329.76349 · doi:10.1051/proc/201340003 |
| [6] | Coquel F., Hérard J.M., Saleh K., Seguin N.: Two properties of two-velocity two-pressure models for two-phase flows. Commun. Math. Sci. 12(3), 593-600 (2014) · Zbl 1303.35069 · doi:10.4310/CMS.2014.v12.n3.a10 |
| [7] | Drew D.: Mathematical modeling of two-phase flow. Ann. Rev. Fluid Mech. 15, 261-291 (1983) · Zbl 0569.76104 · doi:10.1146/annurev.fl.15.010183.001401 |
| [8] | Drew D., Passman S.: Theory of Multicomponent Fluids, Applied Mathematical Sciences, vol. 135. Springer, Berlin (1999) · Zbl 0919.76003 · doi:10.1007/b97678 |
| [9] | Dreyer W., Duderstadt F., Hantke M., Warnecke G.: Bubbles in liquids with phase transition. Part 1: on phase change of a single vapor bubble in liquid water. Contin. Mech. Thermodyn. 24, 461-483 (2012) · Zbl 1258.76160 · doi:10.1007/s00161-011-0225-6 |
| [10] | Embid P., Baer M.: Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn. 4, 279-312 (1992) · Zbl 0760.76096 · doi:10.1007/BF01129333 |
| [11] | Flåtten T., Lund H.: Relaxation two-phase flow models and the subcharacteristic condition. Math. Models Methods Appl. Sci. 21(12), 2379-2407 (2011) · Zbl 1368.76070 · doi:10.1142/S0218202511005775 |
| [12] | Gallouët T., Hérard J.M., Seguin N.: Numerical modelling of two-phase flows using the two-fluid two-pressure approach. Math. Models Methods Appl. Sci. 14(5), 663-700 (2004) · Zbl 1177.76428 · doi:10.1142/S0218202504003404 |
| [13] | Goatin P., LeFloch P.: The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Inst. Henri Poincare 21(6), 881-902 (2004) · Zbl 1086.35069 · doi:10.1016/j.anihpc.2004.02.002 |
| [14] | Godlewski, E., Raviart, P.A.: Hyperbolic systems of conservation laws. Mathematics and Applications, Ellipses, Paris (1991) · Zbl 0768.35059 |
| [15] | Godlewski E., Raviart P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, New York, NY (1996) · Zbl 0860.65075 · doi:10.1007/978-1-4612-0713-9 |
| [16] | Guillemaud, V.: Modélisation et simulation numérique des éqoulements diphasiques par une approche bifluide à deux pressions. Ph.D. thesis, Aix Marseille Université (2007). https://tel.archives-ouvertes.fr/tel-0169178 |
| [17] | Han, E., Hantke, M., Müller, S.: Modeling of multi-component flows with phase transition and application to collapsing bubbles. IGPM Preprint 409, RWTH Aachen University (2014) · Zbl 1184.76268 |
| [18] | Hérard J.M.: A three-phase flow model. Math. Comput. Model. 45, 732-755 (2007) · Zbl 1165.76382 · doi:10.1016/j.mcm.2006.07.018 |
| [19] | Hérard J.M., Hurisse O.: A fractional step method to compute gas-liquid flows. Comput. Fluids 55, 57-69 (2012) · Zbl 1291.76217 · doi:10.1016/j.compfluid.2011.11.001 |
| [20] | Kapila A., Menikoff R., Bdzil J., Son S., Stewart D.: Two-phase modelling of DDT in granular materials: reduced equations. Phys. Fluid 13, 3002-3024 (2001) · Zbl 1184.76268 · doi:10.1063/1.1398042 |
| [21] | Kato T.: The cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58(3), 181-205 (1975) · Zbl 0343.35056 · doi:10.1007/BF00280740 |
| [22] | Lallemand M., Chinnayya A., Metayer O.L.: Pressure relaxation procedures for multi-phase compressible flows. Int. J. Numer. Methods Fluids 49(1), 1-56 (2005) · Zbl 1073.76056 · doi:10.1002/fld.967 |
| [23] | Liu, I., Sampaio, R.: On objectivity and the principle of material frame-indifference. In: Cardona, A., Kohan, P., Quinteros, R., Storti, M. (eds.) Mecánica Computacional vol XXXI, pp. 1553-1569. Salta, Argentina, 13-16 November 2012 (2012) · Zbl 0633.35049 |
| [24] | Liu T.P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys 108, 153-175 (1987) · Zbl 0633.35049 · doi:10.1007/BF01210707 |
| [25] | Menikoff R., Plohr B.: The riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75-130 (1989) · Zbl 1129.35439 · doi:10.1103/RevModPhys.61.75 |
| [26] | Métayer O.L., Massoni J., Saurel R.: Dynamic relaxation processes in compressible multiphase flows. Application to evaporation phenomena. Proc. ESAIM 40, 103-123 (2013) · Zbl 1330.76133 · doi:10.1051/proc/201340007 |
| [27] | Müller I.: Thermodynamics. Pitman, London (1985) · Zbl 0637.73002 |
| [28] | Müller I., Müller W.: Fundamentals of Thermodynamics and Applications. Springer, Berlin (2009) · Zbl 1168.80001 |
| [29] | Müller, S., Hantke, M., Richter, P.: Closure conditions for non-equilibrium multi-component models. IGPM Preprint 414, RWTH Aachen University (2014) · Zbl 1355.76069 |
| [30] | Parés C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44(1), 300-321 (2006) · Zbl 1130.65089 · doi:10.1137/050628052 |
| [31] | Rodio, M., Abgrall, R.: An innovative phase transition modeling for reproducing cavitation through a five-equation model and theoretical generalization to six and seven-equation models. Int. J. Heat Mass Transf. 89, 1386-1401 (2015). doi:10.1016/j.ijheatmasstransfer.2015.05.008 · Zbl 1188.65134 |
| [32] | Saleh, K.: Analyse et simulation numérique par relaxation d’écoulements diphasiques compressibles. Ph.D. thesis, Université Pierre et Marie Curie, Paris (2012). https://tel.archives-ouvertes.fr/tel-00761099 |
| [33] | Saurel R., Abgrall R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150(2), 425-467 (1999) · Zbl 0937.76053 · doi:10.1006/jcph.1999.6187 |
| [34] | Saurel R., Petitpas F., Abgrall R.: Modelling phase transition in metastable liquids: application to cavitation and flashing flows. J. Fluid Mech. 607, 313-350 (2008) · Zbl 1147.76060 · doi:10.1017/S0022112008002061 |
| [35] | Söhnholz, H., Kurz, T.: Thermal effects upon collapse of laser-induced cavitation bubbles. In: Proceedings of 8th International Symposium on Cavitation, Singapore, 13-16 August 2012 (2012). doi:10.3850/978-981-07-2826-7_137 · Zbl 0633.35049 |
| [36] | Whitham G.: Linear and Nonlinear Waves. Wiley, Hoboken (1974) · Zbl 0373.76001 |
| [37] | Zein, A.: Numerical methods for multiphase mixture conservation laws with phase transition. Ph.D. thesis, Otto-von-Guericke University, Magdeburg (2010) · Zbl 1368.76070 |
| [38] | Zein A., Hantke M., Warnecke G.: Modeling phase transition for compressible two-phase flows applied to metastable liquids. J. Comput. Phys. 229(8), 2964-2998 (2008) · Zbl 1307.76079 · doi:10.1016/j.jcp.2009.12.026 |
| [39] | Zein A., Hantke M., Warnecke G.: On the modelling and simulation of a laser-induced cavitation bubble. Int. J. Numer. Methods. Fluids. 73(2), 172-203 (2013) · Zbl 1455.76195 · doi:10.1002/fld.3796 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
