Abstract
Single-component nonrelativistic dissipative fluids are treated independently of reference frames and flow-frames. First the basic fields and their balances are derived, then the related thermodynamic relations and the entropy production are calculated and the linear constitutive relations are given. The usual basic fields of mass, momentum, energy and their current densities, the heat flux, pressure tensor and diffusion flux are the time- and spacelike components of the third-order mass–momentum–energy density-flux four-tensor. The corresponding Galilean transformation rules of the physical quantities are derived. It is proved that the non-equilibrium thermodynamic frame theory, including the thermostatic Gibbs relation and extensivity condition and also the entropy production, is independent of the reference frame and also the flow-frame of the fluid. The continuity-Fourier–Navier–Stokes equations are obtained almost in the traditional form if the flow of the fluid is fixed to the temperature. This choice of the flow-frame is the thermo-flow. A simple consequence of the theory is that the relation between the total, kinetic and internal energies is a Galilean transformation rule.
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References
Weyl, H.: Raum-Zeit-Matterie. Julius Springer, Berlin (1918) (in German, English translation: Methuen and Co., Ltd., London, 1922)
Havas, P.: Four-dimensional formulations of Newtonian mechanics and their relation to the special and the general theory of relativity. Rev. Mod. Phys. 36, 938–965 (1964)
Friedman, M.: Foundations of Space–Time Theories (Relativistic Physics and Philosophy of Science). Princeton University Press, Princeton (1983)
Matolcsi, T.: A Concept of Mathematical Physics: Models in Mechanics. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest (1986)
Matolcsi, T.: Spacetime Without Reference Frames. Akadémiai Kiadó Publishing House of the Hungarian Academy of Sciences), Budapest (1993)
Fülöp, T.: Space is not absolute—spacetime as the consequence of the Galilean relativity principle. In: Fülöp, T., (ed.) Új eredmények a kontinuumfizikában, volume 8 of Mérnökgeológia-Kőzetmechanika Kiskönyvtár, Chapter 1, pp. 11–35. Műegyetemi Kiadó, Budapest (2008) (in Hungarian)
Matolcsi, T., Ván, P.: Absolute time derivatives. J. Math. Phys. 48, 053507–053519 (2007). arXiv:math-ph/0608065
Dicke, R.H.: Mach’s principle and invariance under transformation of units. Phys. Rev. 125(6), 2163 (1962)
Barenblatt, G.I.: Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press, Cambridge (1996)
László, A.: Conformal invariance without referring to metric. (2014) arXiv:1406.5888
Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2, 197–226, (1958/59)
Noll, W.: Space–time structures in classical mechanics. In: The Foundations of Mechanics and Thermodynamics (selected papers by Walter Noll), pp. 204–210. Springer, Berlin (1974). Originally: pp 28–34, Delaware Seminar in the Foundations of Physics, Berlin (1967)
Noll, W.: Five Contributions to Natural Philosophy (2004). www.math.cmu.edu/~wn0g/noll/FC
Noll, W.: A frame free formulation of elasticity. J. Elast. 83, 291–307 (2006)
Noll, W., Seguin, B.: Basic concepts of thermomechanics. J. Elast. 101, 121–151 (2010)
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin, 1965. Handbuch der Physik, III/3
Matolcsi, T., Ván, P.: Can material time derivative be objective? Phys. Lett. A 353, 109–112 (2006). arXiv:math-ph/0510037
Fülöp, T.: A new approach to the kinematics of continua. In: Fülöp T., (eds) Új eredmények a kontinuumfizikában, Mérnökgeológia-Kőzetmechanika Kiskönyvtár, Chapter 3, vol. 8, pp. 55–99. Műegyetemi Kiadó, Budapest (2008) (in Hungarian)
Fülöp, T., Ván, P.: Kinematic quantities of finite elastic and plastic deformations. Math. Methods Appl. Sci. 35, 1825–1841 (2012). arXiv:1007.2892v1
Fülöp, T.: Objective Thermomechanics (2015). arXiv:1510.08038
Eckart, C.: The thermodynamics of irreversible processes. IV. The theory of elasticity and anelasticity. Phys. Rev. 73(4), 373–382 (1948)
Bruhns, O.T., Xiao, H., Mayers, A.: Constitutive inequalities for an isotropic elastic strain energy function based on Hencky’s logarithmic strain tensor. Proc. R. Soc. Lond. A 457, 2207–2226 (2001)
Horgan, C.O., Murphy, J.G.: A generalization of Hencky’s strain-energy density to model the large deformations of slightly compressible solid rubbers. Mech. Mater. 79, 943–950 (2009)
Neff, P., Eidel, B., Osterbrink, F., Martin, R.: A Riemannian approach to strain measures in nonlinear elasticity. C. R. Acad. Sci. 342, 254–257 (2014)
Neff, P., Ghiba, I.-D., Lankeit, J.: The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity. J. Elast. 1–92 (2014). arXiv:1403.4675
Zaremba, S.: Sur une forme perfectionnée de la théorie de la relaxation. Bull. Int. Acad. Sci. Crac. 124, 594–614 (1903)
Jaumann, G.: Geschlossenes System physikalischer und chemischer Differentialgesetze (I. Mitteilung). Sitzungsberichte der kaiserliche Akademie der Wissenschaften in Wien CXVII(Mathematisch IIa), 385–528 (1911)
Oldroyd, J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541 (1949)
Müller, I.: On the frame dependence of stress and heat flux. Arch. Ration. Mech. Anal. 45, 241–250 (1972)
Edelen, D.G.B., McLennan, J.A.: Material indifference: a principle or a convenience. Int. J. Eng. Sci. 11, 813–817 (1973)
Bampi, F., Morro, A.: Objectivity and objective time derivatives in continuum physics. Found. Phys. 10(11/12), 905–920 (1980)
Murdoch, A.I.: On material frame-indifference, intrinsic spin and certain constitutive relations motivated by the kinetic theory of gases. Arch. Ration. Mech. Anal. 83, 185–194 (1983)
Ryskin, G.: Misconception which led to the "material frame indifference" controversy. Phys. Rev. E 32(2), 1239–1240 (1985)
Ryskin, G.: Reply to “comments on the ‘material frame indifference’ controversy". Phys. Rev. E 36(9), 4526 (1987)
Speziale, C.G.: Comments on the “material frame indifference" controversy. Phys. Rev. E 36(9), 4522–4525 (1987)
Speziale, C.G.: A review of material frame-indifference in mechanics. Appl. Mech. Rev. 51(8), 489–504 (1998)
Svendsen, B., Bertram, A.: On frame-indifference and form-invariance in constitutive theory. Acta Mech. 132, 195–207 (1999)
Bertram, A., Svendsen, B.: On material objectivity and reduced constitutive equations. Arch. Mech. 53, 653–675 (2001)
Massoudi, M.: On the importance of material frame-indifference and lift forces in multiphase flow. Chem. Eng. Sci. 57, 3687–3701 (2002)
Murdoch, A.I.: Objectivity in classical continuum physics: a rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favour of purely objective considerations. Contin. Mech. Thermodyn. 15, 309–320 (2003)
Liu, I.-S.: On Euclidean objectivity and the principle of material frame-indifference. Contin. Mech. Thermodyn. 16, 177–183 (2003)
Murdoch, A.I.: On criticism of the nature of objectivity in classical continuum physics. Contin. Mech. Thermodyn. 17, 135–148 (2005)
Liu, I.-S.: Further remarks on Euclidean objectivity and the principle of material frame-indifference. Contin. Mech. Thermodyn. 17, 125–133 (2005)
Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 042903 (2006)
Frewer, M.I.: More clarity on the concept of material frame-indifference in classical continuum mechanics. Acta Mech. 202(1–4), 213–246 (2009)
Mariano, P.M.: SO(3) invariance and covariance in mixtures of simple bodies. Int. J. Non-Linear Mech. 40, 1023–1030 (2005)
Mariano, P.M.: Geometry and balance of hyperstresses. Rendiconti dei Lincei Matematica Applicata 18, 311–331 (2007)
Mariano, P.M.: Cracks in complex bodies: covariance of tip balances. J. Nonlinear Sci. 18, 99–141 (2008)
Muschik, W.: Objectivity and frame indifference, revisited. Arch. Mech. 50, 541–547 (1998)
Muschik, W., Restuccia, L.: Changing the observer and moving materials in continuum physics: objectivity and frame-indifference. Technische Mechanik 22(3), 152–160 (2002)
Muschik, W., Restuccia, L.: Systematic remarks on objectivity and frame-indifference, liquid crystal theory as an example. Arch. Appl. Mech. 78(11), 837–854 (2008)
Muschik, W.: Is the heat flux density really non-objective? a glance back, 40 years later. Contin. Mech. Thermodyn. 24(24), 333–337 (2012)
Matolcsi, T., Gruber, T.: Spacetime without reference frames: An application to the kinetic theory. Int. J. Theor. Phys. 35(7), 1523–1539 (1996)
Brenner, H.: Kinematics of volume transport. Phys. A 349, 11–59 (2005)
Brenner, H.: Navier–Stokes revisited. Phys. A 349, 60–132 (2005)
Brenner, H.: Fluid mechanics revisited. Phys. A 370(2), 190–224 (2006)
Brenner, H.: Bi-velocity hydrodynamics: single-component fluids. Int. J. Eng. Sci. 47(9), 930–958 (2009)
Brenner, H.: Diffuse volume transport in fluids. Phys. A 389(19), 4026–4045 (2010)
Brenner, H.: Beyond Navier–Stokes. Int. J. Eng. Sci. 54, 67–98 (2012)
Brenner, H.: Steady-state heat conduction in a gas undergoing rigid-body rotation. Comparison of Navier–Stokes–Fourier and bivelocity paradigms. Int. J. Eng. Sci. 70, 29–45 (2013)
Brenner, H.: Conduction-only transport phenomena in compressible bivelocity fluids: diffuse interfaces and Korteweg stresses. Phys. Rev. E 89(4), 043020 (2014)
Bedeaux, D., Kjelstrup, S., Öttinger, H.C.: On a possible difference between the barycentric velocity and the velocity that gives translational momentum in fluids. Phys. A 371(2), 177–187 (2006)
Öttinger, H.C.: Weakly and strongly consistent formulations of irreversible processes. Phys. Rev. Lett. 99(13), 130602(4) (2007)
Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Pergamon Press, London (1959)
Dzyaloshinskii, I.E., Volovick, G.E.: Poisson brackets in condensed matter physics. Ann. Phys. 125(1), 67–97 (1980)
Klimontovich, Y.L.: On the need for and the possibility of a unified description of kinetic and hydrodynamic processes. Theor. Math. Phys. 92(2), 909–921 (1992)
Ván, P.: Generic stability of dissipative non-relativistic and relativistic fluids. J. Stat. Mech. Theory Exp. (2009). arXiv:0811.0257
Ván, P., Biró, T.: Dissipation flow-frames: particle, energy, thermometer. In: Pilotelli, M., Beretta, G. P. (eds) Proceedings of the 12th Joint European Thermodynamics Conference, Brescia, pp. 546–551 (2013). Cartolibreria SNOOPY. ISBN 978-88-89252-22-2, arXiv:1305.3190
Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. Contin. Mech. Thermodyn. 1(1), 3–20 (1989)
Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, vol 37, 2nd edn. Springer Tracts in Natural Philosophy. Springer, New York (1998)
Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer, Berlin (2015)
Bíró, T.S., Ván, P.: About the temperature of moving bodies. EPL 89, 30001 (2010). arXiv:0905.1650v1
Kostädt, P., Liu, M.: Three ignored densities, frame-independent thermodynamics, and broken Galilean symmetry. Phys. Rev. E 58, 5535 (1998)
Horváth, R.: A new interpretation of the kinetic energy concept. KLTE MFK Tudományos Közleményei 23, 29–33 (1997). (in Hungarian)
Prix, R.: Variational description of multifluid hydrodynamics: uncharged fluids. Phys. Rev. D 69(4), 043001 (2004)
Lange, L.: On the Law of Inertia. Eur. Phys. J. H 39(2), 251–262 (2014)
Pfister, H.: Ludwig Lange on the law of inertia. Eur. Phys. J. H 39(2), 245–250 (2014)
Penrose, R.: The Road to Reality. Jonathan Cape, London (2004)
Liboff, R.L.: Kinetic Theory (Classical, Quantum, and Relativistic Descriptions). Prentice Hall, Englewood Cliffs (1990)
Matolcsi, T.: On material frame-indifference. Arch. Ration. Mech. Anal. 91(2), 99–118 (1986)
Gyarmati, I.: Non-equilibrium thermodynamics. Field theory and variational principles. Springer, Berlin (1970)
Gallavotti, G.: Foundations of Fluid Dynamics, vol. 172. Springer, Berlin (2002)
Jou, D., Casas-Vázquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer Verlag, Berlin (1992). 3rd, revised edition, 2001
Matolcsi, T.: Ordinary Thermodynamics. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest (2005)
Truesdell, C., Bharatha, S.: Classical Thermodynamics as a Theory of Heat Engines. Springer, Berlin (1977)
Ván, P.: Kinetic equilibrium and relativistic thermodynamics. In: EPJ WEB of Conferences, vol. 13, p 07004 (2011). arXiv:1102.0323
Ván, P., Biró, T.S.: First order and generic stable relativistic dissipative hydrodynamics. Phys. Lett. B 709(1–2), 106–110 (2012). arXiv:1109.0985 [nucl-th]
Ván, P., Biró, T.S.: Thermodynamics and flow-frames for dissipative relativistic fluids. In: Chacón-Acosta, G., Garcí-Perciante, A.L., Sandoval-Villalbazo, A., (eds.) Plasma physics and relativistic fluids, vol. 1578. AIP Conference Proceedings, pp. 114–121, 2014. Proceedings of the V Leopoldo García–Colín Mexican Meeting on Mathematical and Experimental Physics, El Colegio Nacional, September 9–13, 2013. Mexico City. arXiv:1310.5976
Müller, I.: Thermodynamics. Pitman, Toronto (1985)
Matolcsi, T.: Models of Spacetime. ETTE (2015) (in Hungarian)
Öttinger, H.C., Struchtrup, H., Liu, M.: Inconsistency of a dissipative contribution to the mass flux in hydrodynamics. Phys. Rev. E 80(5), 056303 (2009)
Ván, P., Pavelka, M., Grmela, M.: Extra mass flux in fluid mechanics (2015) arXiv:1510.03900
Prigogine, I., Stengers, I.: La nouvelle alliance: métamorphose de la science. Gallimard, Paris (1986)
Matolcsi, T.: Dynamical laws in thermodynamics. Phys. Essays 5(3), 320–327 (1992)
Ván, P.: Asymptotic stability and the second law in extended irreversible thermodynamics. In: Rionero, S., Ruggeri, T. (eds.) 7th Conference on Waves and Stability in Continuous Media, Bologna, Italy. October 4–9. 1993, volume 23 of Series on Advances in Mathematics for Applied Sciences, pp. 384–389, Singapore-New Jersey-London-Hong Kong, October 1994. Quaderno CNR - Gruppo nazionale per la Fisica Matematica, World Scientific
Ván, P.: Other dynamic laws in thermodynamics. Phys. Essays 8(4), 457–465 (1995)
Ván, P., Bíró, T.S.: Relativistic hydrodynamics—causality and stability. Eur. Phys. J. Spec. Top. 155, 201–212 (2008). arXiv:0704.2039v2
Ván, P.: Internal energy in dissipative relativistic fluids. J. Mech. Mater. Struct. 3(6), 1161–1169 (2008). arXiv:0712.1437 [nucl-th]
Bíró, T.S., Molnár, E., Ván, P.: A thermodynamic approach to the relaxation of viscosity and thermal conductivity. Phys. Rev. C 78, 014909 (2008). arXiv:0805.1061 [nucl-th]
Matolcsi, T.: A Concept of Mathematical Physics: Models for SpaceTime. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest (1984)
Carter, B., Chamel, N.: Covariant analysis of newtonian multi-fluid models for neutron stars I: Milne–Cartan structure and variational formulation. Int. J. Mod. Phys. D 13(02), 291–325 (2004)
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Ván, P. Galilean relativistic fluid mechanics. Continuum Mech. Thermodyn. 29, 585–610 (2017). https://doi.org/10.1007/s00161-016-0545-7
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DOI: https://doi.org/10.1007/s00161-016-0545-7