×

zbMATH — the first resource for mathematics

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid-fluid transitions. (English) Zbl 1341.80023
Summary: On the example of two-phase continua experiencing stress-induced solid-fluid phase transitions, we explore the use of the Euler structure in the formulation of the governing equations. The Euler structure guarantees that solutions of the time evolution equations possessing it are compatible with mechanics and with thermodynamics. The former compatibility means that the equations are local conservation laws of the Godunov type, and the latter compatibility means that the entropy does not decrease during the time evolution. In numerical illustrations, in which the one-dimensional Riemann problem is explored, we require that the Euler structure is also preserved in the discretization.

MSC:
80A22 Stefan problems, phase changes, etc.
80A20 Heat and mass transfer, heat flow (MSC2010)
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74F05 Thermal effects in solid mechanics
Software:
LAPACK
PDF BibTeX Cite
Full Text: DOI
References:
[1] Euler L.: Principes généraux du mouvement des fluides. Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémories. 11 (1755)
[2] de Groot S.R., Mazur P.: Non-Equilibrium Thermodynamics. Dover, New York (1984) · Zbl 1375.82004
[3] Godunov, S.K., An interesting class of quasilinear systems, Sov. Math. Dokl., 2, 947-949, (1961) · Zbl 0125.06002
[4] Friedrichs, K.O.; Lax, P.D., Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68, 1686-1688, (1971) · Zbl 0229.35061
[5] Godunov, S.K., Symmetric form of the magnetohydrodynamic equation, Chislennye Metody Mekhaniki Sploshnoi Sredy, 3, 26-34, (1972)
[6] Friedrichs, K.O., Conservation equations and the laws of motion in classical physics, Commun. Pure Appl. Math., 31, 123-131, (1978) · Zbl 0379.35002
[7] Godunov S.K., Romenski E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer/Plenum Publishers, New York (2003) · Zbl 1031.74004
[8] Ruggeri, T., The entropy principle: from continuum mechanics to hyperbolic systems of balance laws, Bollettino dell’Unione Matematica Italiana, 8, 1-20, (2005) · Zbl 1150.80001
[9] Clebsch, A.: Über die Integration der hydrodynamische Gleichungen. J. Reine Angew. Math. 56, 1-10 (1895) · Zbl 0529.58011
[10] Dzyaloshinski, I.E.; Volovick, G.E., Poisson brackets in condense matter physics, Ann. Phys., 125, 67-97, (1980)
[11] Grmela, M., Particle and bracket formulations of kinetic equations, Contemp. Math., 28, 125-132, (1984) · Zbl 0558.58012
[12] Grmela, M., Bracket formulation of dissipative fluid mechanics equations, Phys. Lett. A., 102, 355, (1984)
[13] Kaufman, A.N., Dissipative Hamiltonian systems: a unifying principle, Phys. Lett. A., 100, 419-422, (1984)
[14] Morrison, P.J., Bracket formulation for irreversible classical fields, Phys. Lett. A., 100, 423-427, (1984)
[15] Beris A.N., Edwards B.J.: Thermodynamics of Flowing Systems. Oxford University Press, Oxford (1994)
[16] Grmela, M.; Oettinger, H.C., Dynamics and thermodynamics of complex fluids I: general formulation, Phys. Rev. E, 56, 6620-6633, (1997)
[17] Oettinger, H.C.; Grmela, M., Dynamics and thermodynamics of complex fluids II: illustration of the general folmalism, Phys. Rev. E., 56, 6633-6650, (1997)
[18] Grmela, M., Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering, Adv. Chem. Eng., 39, 75-128, (2010)
[19] Grmela, M., Role of thermodynamics in multiscale physics, Comput. Math. Appl., 65, 1457-1470, (2013) · Zbl 1342.82004
[20] Oettinger H.C.: Beyond Equilibrium Thermodynamics. Wiley, New York (2005)
[21] Putz, A.M.V.; Burghelea, T.I., The solid-fluid transition in a yield stress shear thinning physical gel, Rheol. Acta, 48, 673-689, (2009)
[22] Balmforth, N.J.; Frigaard, I.A.; Ovarlez, G., Yielding to stress: recent developments in viscoplastic fluid mechanics, Annu. Rev. Fluid Mech., 46, 121-146, (2014) · Zbl 1297.76008
[23] Friedrichs, K.O., Symmetric positive linear differential equations, Pure Appl. Math., 11, 333-418, (1958) · Zbl 0083.31802
[24] Godunov, S.K., Romensky, E.: Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media. In: Hafez, M., Oshima, K. (eds.) Computational Fluid Dynamics Review, pp. 19-31. Wiley, New York (1995) · Zbl 0875.73025
[25] Godunov, S.K.; Mikhailova, T.Y.; Romenski, E.I., Systems of thermodynamically coordinated laws of conservation invariant under rotations, Sib. Math. J., 37, 690-705, (1996) · Zbl 0891.73003
[26] Romensky, E., Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics, Math. Comput. Model., 28, 115-130, (1998) · Zbl 1076.74501
[27] Romensky, E.I.; Toro, E.F. (ed.), Thermodynamics and hyperbolic systems of balance laws in continuum mechanics, 745-761, (2001), New York · Zbl 1017.74004
[28] Callen H.B.: Thermodynamics. Wiley, New York (1960) · Zbl 0095.23301
[29] Cattaneo, C., Sulla conduzione del calore, Atti del Seminario Matematico e Fisico della Universita di Modena, 3, 83-101, (1948) · Zbl 0035.26203
[30] Jou D., Casas-Vàzquez J., Lebon G.: Extended Irreversible Thermodynamics. Springer, Berlin (2010) · Zbl 1185.74002
[31] Gouin, H.; Gavrilyuk, S., Hamilton’s principle and rankine-hugoniot conditions for general motions of mixtures, Meccanica, 34, 39-47, (1999) · Zbl 0947.76090
[32] Zhang, H.; Calderer, M.C., Incipient dynamics of swelling of gels, SIAM J. Appl. Math., 68, 1641-1664, (2008) · Zbl 1160.35468
[33] Trusdell C.: Rational Thermodynamics. Springer, New York (1984)
[34] Romenski, E.: Conservative formulation for compressible fluid flow through elastic porous media. In: Vazquez-Cendon, E., Hidalgo, A., Garcia-Navarro, P., Cea, L. (eds.) Numerical Methods for Hyperbolic Equations, pp. 193-199. Taylor & Francis Group, London (2013) · Zbl 0379.35002
[35] Gavrilyuk, S., Multiphase flow modeling via hamilton’s principle, CISM Courses Lect., 535, 163-210, (2011) · Zbl 1247.76085
[36] Favrie, N.; Gavrilyuk, S.L.; Saurel, R., Solid-fluid diffuse interface model in cases of extreme deformations, J. Comput. Phys., 228, 6037-6077, (2009) · Zbl 1280.74013
[37] Marsden J., E.; Ratiu, T.; Weinstein, A., Semidirect products and reduction in mechanics, Trans. Am. Math. Soc., 281, 147-177, (1984) · Zbl 0529.58011
[38] Hamiltonian, R.S., Fluid mechanics, Annu. Rev. Fluid Mech., 20, 225-256, (1988)
[39] Miller, G.H.; Colella, P., A high-order Eulerian Godunov method for elastic-plastic flow in solids, J. Comput. Phys., 167, 137-176, (2001) · Zbl 0997.74078
[40] Romenski, E.; Drikakis, D.; Toro, E., Conservative models and numerical methods for compressible two-phase flow, J. Sci. Comput., 42, 68-95, (2010) · Zbl 1203.76095
[41] Godunov, S.K.; Romenskii, E.I., Nonstationary equations of nonlinear elasticity theory in Eulerian coordinates, J. Appl. Mech. Tech. Phys., 13, 868-884, (1972)
[42] Godunov S.K.: Elements of Continuum Mechanics. Nauka, Moscow (1978)
[43] Grmela, M., Fluctuation in extended mass-action-law dynamics, Phys. D, 24, 976-986, (2012) · Zbl 1252.80013
[44] Romenski, E.; Toro, E.F., Compressible two-phase flows: two-pressure models and numerical methods, Comput. Fluid Dyn., 13, 403-416, (2004) · Zbl 1136.76702
[45] Steinmann, P., Views on multiplicative elastoplasticity and the continuum theory of dislocations, Int. J. Eng. Sci., 34, 1717-1735, (1996) · Zbl 0905.73060
[46] Powell, K.G.; Roe, P.L.; Linde, T.J.; Gombosi, T.I.; DeZeeuw, D.L., A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comput. Phys., 154, 284-309, (1999) · Zbl 0952.76045
[47] Munz, C.D.; Omnes, P.; Schnaider, R.; Sonnendrücker, E.; Voss, U., Divergence correction techniques for Maxwell solvers based on a hyperbolic model, J. Comput. Phys., 161, 484-511, (2000) · Zbl 0970.78010
[48] Babii, D.P.; Godunov, S.K.; Zhukov, V.T.; Feodoritova, O.B., On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics, Comput. Math. Math. Phys., 47, 427-441, (2007) · Zbl 1210.35168
[49] Bouchbinder, E.; Langer, J.S.; Procaccia, I., Athermal shear-transformation-zone theory of amorphous plastic deformation. I. basic principles, Phys. Rev. E., 75, 036107, (2007)
[50] Kosevich, A.M., Dynamical theory of dislocations, Physics-Uspekhi., 7, 837-854, (1965)
[51] Kosevich, A.M.: Crystal dislocations and the theory of elasticity. In: Nabarro, F.R.N. Dislocations in Solids, pp. 33-141. North-Holland, Amsterdam (1979) · Zbl 0171.46204
[52] Malygin, G.A., Dislocation self-organization processes and crystal plasticity, Physics-Uspekhi., 42, 887, (1999)
[53] Toro E.F.: Riemann Solvers and Numerical Methods in Fluid Dynamics. Springer, Berlin (1999) · Zbl 0923.76004
[54] Godunov, S.K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb., 47, 271-306, (1959) · Zbl 0171.46204
[55] LeVeque R.J.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002) · Zbl 1010.65040
[56] Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA (1999) · Zbl 0934.65030
[57] Moyers-Gonzales, M.; Burghelea, T.I.; Mak, J., Linear stability analysis for plane-Poiseuille flow of an elastoviscoplastic fluid with internal microstructure for large Reynolds numbers, J. Non-Newton. Fluid Mech., 166, 515-531, (2011) · Zbl 1282.76105
[58] El Afif, A.; Grmela, M., Non-Fickian mass transport in polymers, J. Rheol., 46, 591-628, (2002)
[59] Godunov, S.K.; Peshkov, I.M., Thermodynamically consistent nonlinear model of an elastoplastic Maxwell medium, Comput. Math. Math. Phys., 50, 1409-1426, (2010) · Zbl 1224.74017
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.