Abstract
In his monograph Thermodynamics, I. Müller proves that for incompressible media the volume does not change with the temperature. This Müller paradox yields an incompatibility between experimental evidence and the entropy principle. This result has generated much debate within the mathematical and thermodynamical communities as to the basis of Boussinesq approximation in fluid dynamics. The aim of this paper is to prove that for an appropriate definition of incompressibility, as a limiting case of quasi-thermal-incompressible body, the entropy principle holds for pressures smaller than a critical pressure value. The main consequence of our result is the physically obvious one that for very large pressures, no body can be perfectly incompressible. The result is first established in the fluid case. In case of hyperelastic media subject to large deformations, the approach is similar, but with a suitable definition of the pressure associated with a convenient stress tensor decomposition.
Similar content being viewed by others
References
Klainerman S., Majda S.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34, 481–524 (1981)
Klainerman S., Majda S.: Compressible and incompressible fluids. Comm. Pure Appl. Math. 35, 629–653 (1982)
Lions P.L., Masmoudi N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)
Desjardins B., Grenier E., Lions P.L., Masmoudi N.: Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999)
Bechtel S.E., Cai M., Rooney F.J., Wang Q.: Investigation of simplified thermal expansion models for compressible Newtonian fluids applied to nonisothermal plane Couette and Poiseuille flows. Phys. Fluids. 16, 3955–3974 (2004)
Müller I.: Thermodynamics. Pitman, London (1985)
Rajagopal K.R., Ruzika M., Srinivasa A.R.: On the Oberbeck-Boussinesq approximation. Math. Models Methods Appl. Sci. 6(8), 1157–1167 (1996)
Rajagopal K.R., Saccomandi G., Vergori L.: On the Oberbeck-Boussinesq approximation for fluids with pressure dependent viscosities. Nonlinear Anal. Real World Appl. 10, 1139–1150 (2009)
Weast, R.C., Astle, M.J., Beyer, W.H. (eds.): CRC Handbook of Chemistry and Physics. CRC Press, Boca Raton (1988)
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability, ch. 2, sect. 8, p. 16, Dover, New York (1981) (reedited from Oxford University Press 1961)
Straughan B.: The Energy Method, Stability and Nonlinear Convection, ch 3, sect. 3.2, 2nd edn, pp. 48–49. Springer, New York (2004)
Fine R.A., Millero F.J.: Compressibility of water as a function of temperature and pressure. J. Chem. Phys. 59(10), 5529–5536 (1973)
Wang C.C., Truesdell C.: Introduction to Rational Elasticity. Noordhoff Int. Publ., Léyden, The Netherlands (1973)
Germain P.: Cours de mécanique des milieux continus. Masson, Paris (1973)
Ruggeri T.: Introduzione Alla Termomeccanica Dei Continui. Monduzzi, Bologna (2007)
Truesdell C., Toupin R.A.: Principles of Classical Mechanics and Field Theory, Encyclopedia of Physics, vol. III/1. Springer, Berlin (1960)
Flory P.J.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57, 829–838 (1961)
Gouin, H., Debiève, J.F.: Variational principle involving the stress tensor in elastodynamics. Int. J. Eng. Sci. 24, 1057–1066 (1986). http://arXiv:0807.3454
Flory P.J.: Principles of Polymer Chemistry. Cornell University Press, Ithaca, New York (1953)
Manacorda T.: Onde Elementari Nella Termoelasticità di Solidi Incomprimibili, pp. 503–509. Atti Accademia Scienze, Torino (1967)
Scott N.H.: Linear dynamical stability in constrained thermoelasticity: II. Deformation-entropy constraints. Q. J. Mech. Appl. Math. 45, 651–662 (1992)
Scott N.H.: Thermoelasticity with thermomechanical constraints. Int. J. Non-Linear Mech. 36, 549–564 (2001)
Chadwick P., Scott N.H.: Linear dynamical stability in constrained thermoelasticity: I. Deformation-temperature constraints. Q. J. Mech. Appl. Math. 45, 641–650 (1992)
Leslie D.J., Scott N.H.: Incompressibility at uniform temperature or entropy in isotropic thermoelasticity. Q. J. Mech. Appl. Math. 51, 191–211 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Manuel Torrilhon.
Dedicated to Professor Ingo Müller for his 75th birthday.
Rights and permissions
About this article
Cite this article
Gouin, H., Muracchini, A. & Ruggeri, T. On the Müller paradox for thermal-incompressible media. Continuum Mech. Thermodyn. 24, 505–513 (2012). https://doi.org/10.1007/s00161-011-0201-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-011-0201-1
Keywords
- Incompressible fluids and solids
- Entropy principle for incompressible materials
- Boussinesq approximation