Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).

[1]  Y. S. Lee,et al.  Bifurcation of coupled-mode responses by modal coupling in cubic nonlinear systems , 2015 .

[2]  P. Roe,et al.  A Solution-Adaptive Upwind Scheme for Ideal Magnetohydrodynamics , 1999 .

[3]  H. Struchtrup,et al.  Regularization of Grad’s 13 moment equations: Derivation and linear analysis , 2003 .

[4]  P. Lax,et al.  Systems of conservation equations with a convex extension. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[5]  A. Clebsch,et al.  Ueber die Integration der hydrodynamischen Gleichungen. , 1859 .

[6]  Michael Dumbser,et al.  Cell centered direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity , 2016 .

[7]  Dimitris Drikakis,et al.  An Eulerian finite‐volume scheme for large elastoplastic deformations in solids , 2010 .

[8]  E. Toro,et al.  CONSERVATIVE HYPERBOLIC MODEL FOR COMPRESSIBLE TWO-PHASE FLOW WITH DIFFERENT PHASE PRESSURES AND TEMPERATURES , 2004 .

[9]  Allan N. Kaufman,et al.  DISSIPATIVE HAMILTONIAN SYSTEMS: A UNIFYING PRINCIPLE , 1984 .

[10]  S. Godunov,et al.  Elements of Continuum Mechanics and Conservation Laws , 2003, Springer US.

[11]  J. Hirschfelder.  Kinetic Theory of Liquids. , 1956 .

[12]  I. Peshkov,et al.  Conservative formulation for compressible multiphase flows , 2014, 1405.3456.

[13]  J. Málek,et al.  Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids , 2016 .

[14]  J. Marsden,et al.  Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids , 1983 .

[15]  Alireza Mazaheri,et al.  A first-order hyperbolic system approach for dispersion , 2016, J. Comput. Phys..

[16]  Brian J. Edwards,et al.  Thermodynamics of flowing systems : with internal microstructure , 1994 .

[17]  P. Mazur,et al.  Non-equilibrium thermodynamics, , 1963 .

[18]  S. Godunov,et al.  Systems of thermodynamically coordinated laws of conservation invariant under rotations , 1996 .

[19]  Michael Dumbser,et al.  High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics , 2016, J. Comput. Phys..

[20]  Michael Dumbser,et al.  A unified hyperbolic formulation for viscous fluids and elastoplastic solids , 2016, 1705.02151.

[21]  H. Abarbanel,et al.  Hamiltonian formulation of inviscid flows with free boundaries , 1988 .

[22]  Arkadi Berezovski,et al.  Internal Variables in Thermoelasticity , 2017 .

[23]  Jaroslav Hron,et al.  On thermodynamics of incompressible viscoelastic rate type fluids with temperature dependent material coefficients , 2016, 1612.01724.

[24]  Philip J. Morrison,et al.  Bracket formulation for irreversible classical fields , 1984 .

[25]  Darryl D. Holm.  Hamiltonian dynamics of a charged fluid, including electro- and magnetohydrodynamics , 1986 .

[26]  Tommaso Ruggeri,et al.  Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics , 2005 .

[27]  H. C. Öttinger.  On the structural compatibility of a general formalism for nonequilibrium dynamics with special relativity , 1998 .

[28]  Václav Klika,et al.  Generalization of exergy analysis , 2015 .

[29]  Michael Dumbser,et al.  High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids , 2015, J. Comput. Phys..

[30]  Dima Bolmatov,et al.  Thermodynamic behaviour of supercritical matter , 2013, Nature Communications.

[31]  M. Grmela,et al.  A hierarchy of Poisson brackets in non-equilibrium thermodynamics , 2015, 1512.08010.

[32]  S. K. Godunov,et al.  THE PROBLEM OF A GENERALIZED SOLUTION IN THE THEORY OF QUASILINEAR EQUATIONS AND IN GAS DYNAMICS , 1962 .

[33]  Marián Fecko.  Differential Geometry and Lie Groups for Physicists: Frontmatter , 2006 .

[34]  Ilya Peshkov,et al.  On a pure hyperbolic alternative to the Navier-Stokes equations , 2014 .

[35]  Miroslav Grmela.  A framework for elasto-plastic hydrodynamics , 2003 .

[36]  Vito Antonio Cimmelli,et al.  Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview , 2016 .

[37]  H. Ch. Öttinger,et al.  Beyond Equilibrium Thermodynamics , 2005 .

[38]  M. Grmela,et al.  Hamiltonian coupling of electromagnetic field and matter , 2016, 1607.02023.

[39]  E. I. Romenskii.  Hyperbolic equations of Maxwell's nonlinear model of elastoplastic heat-conducting media , 1989 .

[40]  M. Torrilhon.  Modeling Nonequilibrium Gas Flow Based on Moment Equations , 2016 .

[41]  Kurt Friedrichs,et al.  [71-1] Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686–1688 , 1986 .

[42]  M. Grmela.  Bracket formulation of diffusion-convection equations , 1986 .

[43]  Ralf Deiterding,et al.  Eulerian adaptive finite-difference method for high-velocity impact and penetration problems , 2013, J. Comput. Phys..

[44]  Tosio Kato,et al.  The Cauchy problem for quasi-linear symmetric hyperbolic systems , 1975 .

[45]  Miroslav Grmela,et al.  Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism , 1997 .

[46]  Y. Cai,et al.  Unified phonon-based approach to the thermodynamics of solid, liquid and gas states , 2015, 1512.07191.

[47]  Tommaso Ruggeri,et al.  Main field and convex covariant density for quasi-linear hyperbolic systems : relativistic fluid dynamics , 1981 .

[48]  M. Pavelka,et al.  Gradient Dynamics and Entropy Production Maximization , 2016, 1610.05499.

[49]  Eleuterio F. Toro,et al.  Conservative Models and Numerical Methods for Compressible Two-Phase Flow , 2010, J. Sci. Comput..

[50]  Swen Kortig,et al.  Differential Geometry And Lie Groups For Physicists , 2016 .

[51]  Kurt Friedrichs,et al.  Symmetric positive linear differential equations , 1958 .

[52]  K. Kormann,et al.  GEMPIC: geometric electromagnetic particle-in-cell methods , 2016, Journal of Plasma Physics.

[53]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[54]  M. Hütter,et al.  Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling , 2012 .

[55]  Eleuterio F. Toro,et al.  Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures , 2007 .

[56]  Miroslav Grmela,et al.  Fluctuations in extended mass-action-law dynamics , 2012 .

[57]  Miroslav Grmela,et al.  Multiscale thermodynamics and mechanics of heat. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  Sylvie Benzoni-Gavage,et al.  Multidimensional hyperbolic partial differential equations : first-order systems and applications , 2006 .

[59]  Nicolas Favrie,et al.  Dynamics of shock waves in elastic-plastic solids , 2011 .

[60]  Ilya Peshkov,et al.  Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium , 2010 .

[61]  K. Rajagopal,et al.  A thermodynamic frame work for rate type fluid models , 2000 .

[62]  Miroslav Grmela,et al.  Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions , 2015 .

[63]  E. I. Romensky,et al.  Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics , 1998 .

[64]  Miroslav Grmela,et al.  Contact Geometry of Mesoscopic Thermodynamics and Dynamics , 2014, Entropy.

[65]  Martin Kröger,et al.  Automated symbolic calculations in nonequilibrium thermodynamics , 2010, Comput. Phys. Commun..

[66]  S. Gavrilyuk,et al.  A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves , 2017 .

[67]  E. M. Lifshitz,et al.  Electrodynamics of continuous media , 1961 .

[68]  Sylvie Benzoni-Gavage,et al.  Multi-dimensional hyperbolic partial differential equations , 2006 .

[69]  Jaroslav Hron,et al.  On thermodynamics of viscoelastic rate type fluids with temperature dependent material coefficients , 2016 .

[70]  Dochan Kwak,et al.  Computational Fluid Dynamics Review 2010 , 2010 .

[71]  M. Gurtin,et al.  An introduction to continuum mechanics , 1981 .

[72]  Marián Fecko,et al.  Differential Geometry and Lie Groups for Physicists: Index of (frequently used) symbols , 2006 .

[73]  J. Marchal,et al.  Loss of evolution in the flow of viscoelastic fluids , 1986 .

[74]  E. I. Romensky,et al.  Thermodynamics and Hyperbolic Systems of Balance Laws in Continuum Mechanics , 2001 .

[75]  S. Stoupin,et al.  Thermally triggered phononic gaps in liquids at THz scale , 2016, Scientific Reports.

[76]  Miroslav Grmela,et al.  Dynamics and thermodynamics of complex fluids. I. Development of a general formalism , 1997 .

[77]  I. Müller,et al.  Rational Extended Thermodynamics , 1993 .

[78]  V. Arnold.  Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits , 1966 .

[79]  Miroslav Grmela,et al.  Time reversal in nonequilibrium thermodynamics. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[80]  I︠A︡kov Ilʹich Frenkelʹ.  Kinetic Theory of Liquids , 1955 .

[81]  P. Morrison,et al.  Hamiltonian description of the ideal fluid , 1998 .

[82]  Dima Bolmatov,et al.  Revealing the Mechanism of the Viscous-to-Elastic Crossover in Liquids. , 2015, The journal of physical chemistry letters.

[83]  M. Hütter,et al.  Quasi-linear versus potential-based formulations of force–flux relations and the GENERIC for irreversible processes: comparisons and examples , 2013 .

[84]  Yu. D. Fomin,et al.  Two liquid states of matter: a dynamic line on a phase diagram. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[85]  I. Dzyaloshinskiǐ,et al.  Poisson brackets in condensed matter physics , 1980 .

[86]  N. I. Makarenko,et al.  Waves in Continuous Media , 2017 .

[87]  Tommaso Ruggeri,et al.  Rational Extended Thermodynamics beyond the Monatomic Gas , 2015 .

[88]  P. V'an,et al.  Guyer-Krumhansl–type heat conduction at room temperature , 2017, 1704.00341.

[89]  M. A. Peletier,et al.  On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion , 2013, 1312.7591.

[90]  S. K. Godunov,et al.  Elements of continuum mechanics , 1978 .

[91]  A. Bobylev,et al.  The Chapman-Enskog and Grad methods for solving the Boltzmann equation , 1982 .

[92]  E. I. Romenski,et al.  On modeling the frequency transformation effect in elastic waves , 2011 .

[93]  S. K. Godunov.  Symmetric form of the magnetohydrodynamic equation , 1972 .

[94]  S. K. Godunov,et al.  Nonstationary equations of nonlinear elasticity theory in eulerian coordinates , 1972 .