Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T18:40:16.073Z Has data issue: false hasContentIssue false
  • English
  • Français

A second-order continuum theory of fluids

Published online by Cambridge University Press:  09 May 2018

S. Paolucci Open the ORCID record for S. Paolucci [Opens in a new window]  and
C. Paolucci Open the ORCID record for C. Paolucci [Opens in a new window]
S. Paolucci*
Affiliation:
Aerospace & Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
C. Paolucci
Affiliation:
Chemical & Biomolecular Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
*
†Email address for correspondence: paolucci@nd.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We use continuum conservation equations in conjunction with the entropy inequality to obtain a theoretical model that is accurate for hypersonic flows. This is accomplished by extending the constitutive equations to second order in strain rate and gradients of density and temperature. The accuracy of the extended theory is demonstrated by computing the shock structure in argon and nitrogen gases over a large range of Mach numbers. The solutions indicate that our constitutive equations are more accurate than the classical linear ones and are also more robust and accurate than equations obtained through expansions of the Boltzmann equation or equations arising from diffusion of volume arguments.

JFM classification

Compressible Flows: Compressible Flows Mathematical Foundations: Mathematical Foundations Compressible Flows: Shock waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 686 - 710
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alsmeyer, H. 1976 Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech. 74 (3), 497–513.Google Scholar
Balakrishnan, R. 2004 An approach to entropy consistency in second-order hydrodynamic equations. J. Fluid Mech. 503, 201–245.CrossRefGoogle Scholar
Balakrishnan, R. & Agarwal, R. K. 1999 A comparative study of several higher-order kinetic formulations beyond Navier–Stokes for computing the shock structure. In 37th AIAA Aerospace Sciences Meeting and Exhibit, AIAA-99-0224. American Institute of Aeronautics and Astronautics.Google Scholar
Bird, G. A. 1970 Aspects of the structure of strong shock waves. Phys. Fluids 13 (5), 1172–1177.CrossRefGoogle Scholar
Brau, C. A. 1967 Kinetic theory of polyatomic gases: models for the collision processes. Phys. Fluids 10, 48–55.Google Scholar
Brenner, H. 2005a Kinematics of volume transport. Physica A 349 (1–2), 11–59.CrossRefGoogle Scholar
Brenner, H. 2005b Navier–Stokes revisited. Physica A 349 (1–2), 60–132.Google Scholar
Camac, M. 1965 Argon and nitrogen shock thicknesses. In Proceedings of the Fourth International Symposium on Rarefied Gas Dynamics (ed. de Leeuw, J. H.), vol. 1, p. 240. Academic Press.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Chen, J. 2013 Extension of nonlinear Onsager theory of irreversibility. Acta Mechanica 224, 31533158.CrossRefGoogle Scholar
Coleman, B. D., Markovitz, H. & Noll, W. 1966 Viscometric Flows of Non-Newtonian Fluids. Springer.Google Scholar
Coleman, B. D. & Noll, W. 1961 Foundations of linear viscoelasticity. Rev. Mod. Phys. 33 (2), 239–249.Google Scholar
Comeaux, K. A., Chapman, D. R. & MacCormack, R. W. 1995 An analysis of the Burnett equations based on the second law of thermodynamics. In 33rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA-95-0415. American Institute of Aeronautics and Astronautics.Google Scholar
Eringen, A. C. 1980 Mechanics of Continua. Krieger.Google Scholar
Eringen, A. C. 1999 Microcontinuum Field Theories I: Foundations and Solids. Springer.Google Scholar
Eringen, A. C. 2001 Microcontinuum Field Theories II: Fluent Media. Springer.Google Scholar
Garen, W., Synofzik, R. & Frohn, A. 1974 Shock-tube for generating weak shock-waves. AIAA J. 12 (8), 1132–1134.Google Scholar
Gilbarg, D. 1951 The existence and limit behavior of the one-dimensional shock layer. Am. J. Maths 73 (2), 256–274.Google Scholar
Gilbarg, D. & Paolucci, D. 1953 The structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2 (4), 617–642.Google Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2 (4), 331–407.Google Scholar
Grad, H. 1952 The profile of a steady plane shock wave. Commun. Pure Appl. Maths 5 (3), 257–300.Google Scholar
Greene, E. F. & Hornig, D. F. 1953 The shape and thickness of shock fronts in argon, hydrogen, nitrogen, and oxygen. J. Chem. Phys. 21 (4), 617–624.Google Scholar
Greenshields, C. J. & Reese, J. M. 2007 The structure of shock waves as a test of Brenner’s modifications to the Navier–Stokes equations. J. Fluid Mech. 580, 407–429.Google Scholar
Jin, S & Slemrod, M 2001 Regularization of the Burnett equations via relaxation. J. Stat. Phys. 103 (5–6), 1009–1033.Google Scholar
Khonkin, A. D. & Orlov, A. V. 1993 Weak shock structure on the basis of modified hydrodynamical equations. Phys. Fluids A 5 (7), 1810–1813.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics, Course of Theoretical Physics, vol. 6. Pergamon Press.Google Scholar
Liepmann, H. W., Narasimha, R. & Chahine, M. T. 1962 Structure of a plane shock layer. Phys. Fluids 5 (11), 1313–1324.CrossRefGoogle Scholar
Linzer, M. & Hornig, D. F. 1963 Structure of shock fronts in argon and nitrogen. Phys. Fluids 6 (12), 1661–1668.Google Scholar
Liu, I.-S. 2002 Continuum Mechanics. Springer.Google Scholar
Liu, I.-S. & Salvador, J. A. 1990 Hyperbolic system for viscous fluids and simulation of shock tube flows. Contin. Mech. Thermodyn. 2 (3), 179–197.Google Scholar
Ludford, G. S. S. 1951 The classification of one-dimensional flows and the general shock problem of a compressible, viscous, heat-conducting fluid. J. Aero. Sci. 18 (12), 830–834.Google Scholar
Lumpkin, F. E. & Chapman, D. R. 1992 Accuracy of the Burnett equations for hypersonic real gas flows. J. Thermophys. Heat Transfer 6 (3), 419–425.Google Scholar
von Mises, R. 1950 On the thickness of a steady shock wave. J. Aero. Sci. 17 (9), 551–554; 594.Google Scholar
Morduchow, M. & Libby, P. A. 1949 On a complete solution of the one-dimensional flow equations of a viscous, heat-conducting, compressible gas. J. Aero. Sci. 16 (11), 674–684; 704.CrossRefGoogle Scholar
Paolucci, S. 2016 Continuum Mechanics and Thermodynamics of Matter. Cambridge University Press.CrossRefGoogle Scholar
Reese, J. M., Woods, L. C., Thivet, F. J. P. & Candel, S. M. 1995 A second-order description of shock structure. J. Comput. Phys. 117 (2), 240–250.Google Scholar
Rieutord, E.1970 Contribution à l’étude de l’onde de choc primaire en tube à choc. PhD thesis, Institut National des Sciences Appliquées de Lyon.Google Scholar
Robben, F. & Talbot, L. 1966 Measurement of shock wave thickness by electron beam fluorescence method. Phys. Fluids 9 (4), 633–643.Google Scholar
Russell, D.1965 In Proceedings of the Fourth International Symposium on Rarefied Gas Dynamics (ed. J. H. de Leeuw), vol. 1, p. 265. Academic Press.Google Scholar
Salomons, E. & Mareschal, M. 1992 Usefulness of the Burnett description of strong shock waves. Phys. Rev. Lett. 69 (2), 269–272.Google Scholar
Schmidt, B. 1969 Electron beam density measurements in shock waves in argon. J. Fluid Mech. 39 (2), 361–373.Google Scholar
Schultz-Grunow, F. & Frohn, A. 1965 In Proceedings of the Fourth International Symposium on Rarefied Gas Dynamics (ed. de Leeuw, J. H.), vol. 1, p. 250. Academic Press.Google Scholar
She, R. S. & Sather, N. F. 1967 Kinetic theory of polyatomic gases. J. Chem. Phys. 47, 4978–4993.Google Scholar
Sherman, F. S.1955 A low-density wind-tunnel study of shock-wave structure and relaxation phenomena in gases. NACA Technical Note 3298. National Advisory Committee for Aeronautics.Google Scholar
Struchtrup, H. & Torrilhon, M. 2003 Regularization of Grad’s 13 moment equations: derivation and linear analysis. Phys. Fluids 15 (9), 2668–2680.Google Scholar
Taniguchi, S., Arima, T., Ruggeri, T. & Sujiyama, M. 2014a Shock wave structure in a rarefied polyatomic gas based on extended thermodynamics. Acta Appl. Maths 132 (1), 583–593.CrossRefGoogle Scholar
Taniguchi, S., Arima, T., Ruggeri, T. & Sujiyama, M. 2014b Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: beyond the Bethe–Teller theory. Phys. Rev. E 89, 013025.Google Scholar
Thomas, L. H. 1944 Note on Becker’s theory of the shock front. J. Chem. Phys. 12 (11), 449–453.Google Scholar
Truesdell, C. & Noll, W. 2004 Non-linear Field Theories of Mechanics, 3rd edn. Springer.CrossRefGoogle Scholar
Weiss, W. 1995 Continuous shock structure in extended thermodynamics. Phys. Rev. E 52 (6), R5760–R5763.Google ScholarPubMed
Woods, L. C. 1983 Frame-indifferent kinetic theory. J. Fluid Mech. 136, 423–433.CrossRefGoogle Scholar
Woods, L. C. 1993 An Introduction to the Kinetic Theory of Gases and Magnetoplasmas. Oxford University Press.Google Scholar
Zheng, Q.-S. 1993 On the representations for isotropic vector-valued, symmetric tensor-valued and skew-symmetric tensor-valued functions. Intl J. Engng Sci. 31 (7), 1013–1024.Google Scholar
Zheng, Q.-S. 1994 Theory of representations for tensor functions: a unified invariant approach to constitutive equations. Appl. Mech. Rev. 47 (11), 545–587.Google Scholar