Maxwell and Cattaneo’s Time-Delay Ideas Applied to Shockwaves and the Rayleigh-Bénard Problem
Uribe F. J. 1*, Hoover Wm.G. , Hoover C.G. 2
1Department of Physics, Universidad Autónoma Metropolitana
México City, México 09340
E-mail: paco@xanum.uam.mx
2Ruby Valley Research Institute, Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
Received:
(Received: 24 November 2012; revised: 7 January 2013; accepted: 14 January 2013; published online: 21 January 2013)
DOI: 10.12921/cmst.2013.19.01.5-12
OAI: oai:lib.psnc.pl:447
Abstract:
We apply Maxwell and Cattaneo’s relaxation approaches to the analysis of strong shockwaves in a two-dimensional viscous heat-conducting fluid. Good agreement results for reasonable values of Maxwell’s relaxation times. Instability results if the viscous relaxation time is too large. These relaxation terms have negligible effects on slower-paced subsonic problems, as is shown here for two-roll and four-roll Rayleigh-Bénard flow.
Key words:
heat-conducting fluids, non-equilibrium computer simulations, Rayleigh-Bénard flow, shockwaves, twodimensional systems, viscous fluids
References:
[1] J. C. Maxwell, “On the Dynamical Theory of Gases”, Philosophical Transactions of the Royal Society of London 157, 49-88 (1867).
[2] M. C. Cattaneo, “Sur une Forme de l’Équation de la Chaleur Éliminant le Paradoxe d’une Propagation Instantaneé”, Comptes Rendus De L’Académie des Sciences-Series I-Mathematics 247, 431-433 (1958).
[3] D. D. Joseph and L. Preziosi, “Heat Waves”, Reviews of Modern Physics 61, 41-73 (1989).
[4] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, Second Edition (Springer-Verlag, Berlin, 1993).
[5] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, New York, 1959). Chapter IX is devoted to shockwaves.
[6] R. E. Duff, W. H. Gust, E. B. Royce, M. Ross, A. C. Mitchell, R. N. Keeler, and W. G. Hoover, “Shockwave Studies in Condensed Media”, in Behavior of Dense Media under High Dynamic Pressures”, Proceedings of the 1967 Paris Conference, pages 397-406 (Gordon and Breach, New York, 1968).
[7] V. Y. Klimenko and A. N. Dremin, “Structure of Shockwave Front in a Liquid”, pages 79-83 in Detonatsiya, Chernogolovka (Akademia Nauk, Moscow, 1978).
[8] B. L. Holian, W. G. Hoover, B. Moran, and G. K. Straub, “Shockwave Structure via Nonequilibrium Molecular Dynamics and Navier-Stokes Continuum Mechanics”, Physical Review A 22, 2798-2808 (1980).
[9] O. Kum, Wm. G. Hoover, and C. G. Hoover, “Temperature Maxima in Stable Two-Dimensional Shockwaves”, Physical Review E 56, 462 (1997).
[10] Wm. G. Hoover and C. G. Hoover, “Tensor Temperature and Shockwave Stability in a Strong Two-Dimensional Shockwave”, Physical Review E 80, 011128 (2009).
[11] Wm. G. Hoover and C. G. Hoover, “Well-Posed Two- Temperature Constitutive Equations for Stable Dense Fluid Shockwaves using Molecular Dynamics and Generalized Navier-Stokes-Fourier Continuum Mechanics”, Physical Review E 81, 046302 (2010).
[12] Wm. G. Hoover and C. G. Hoover, “Shockwaves and Local Hydrodynamics; Failure of the Navier-Stokes Equations”, in New Trends in Statistical Physics – Festschrift in Honor of Leopoldo García-Colín’s 80th Birthday, A. Macias and L. Dagdug, Editors, pages 15-26 (World Scientific, Singapore, 2010). See arXiv:0909.2882 [physics.flu-dyn].
[13] Wm. G. Hoover, C. G. Hoover, and F. J. Uribe, “Flexible Macroscopic Models for Dense-Fluid Shockwaves: Partitioning Heat and Work; Delaying Stress and Heat Flux; Two-Temperature Thermal Relaxation”, Proceedings of the XXXVIII International Summer School-Conference: Advanced Problems in Mechanics (Saint Petersburg, Russia, July 2010), pages 261-273. See arXiv:1005.1525 [cond-mat.statmech].
[14] Wm. G. Hoover and C. G. Hoover, “Three Lectures: NEMD, SPAM, and Shockwaves”, 11th Granada Seminar at La Herradura, 13-17 September 2010. See arXiv:1008.4947 [condmat. stat-mech].
[15] B. L. Holian and M. Mareschal, “Heat-Flow Equation Motivated by the Ideal-Gas Shockwave”, Physical Review E 82, 026707 (2010). B. L. Holian, M. Mareschal, and R. Ravelo, “Burnett-Cattaneo Continuum Theory for Shockwaves”, Physical Review E 83, 026703 (2011).
[16] W. G. Hoover, “Structure of a Shockwave Front in a Liquid”, Physical Review Letters 42, 1531-1534 (1979).
[17] L. B. Lucy, “A Numerical Approach to the Testing of the Fission Hypothesis”, The Astronomical Journal 82, 1013-1024 (1977).
[18] Wm. G. Hoover, Smooth Particle Applied Mechanics – The State of the Art (World Scientific Publishers, Singapore, 2006)
[19] Wm. G. Hoover, Molecular Dynamics (Springer- Verlag, Berlin, 1986, available at the homepage http://williamhoover.info/MD.pdf).
[20] Wm. G. Hoover, Computational Statistical Mechanics (Elsevier, Amsterdam, 1991, available at the homepage http://williamhoover.info).
[21] O. Kum, Wm. G. Hoover, and H. A. Posch, “Viscous Conducting Flows with Smooth-Particle Applied Mechanics”, Physical Review E 52, 4899-4908 (1995).
[22] V. M. Castillo, Wm. G. Hoover, and C. G. Hoover, “Coexisting Attractors in Compressible Rayleigh-Bénard Flow”, Physical Review E 55 5546-5550 (1997).
[23] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley & Sons, New York, 1954).
We apply Maxwell and Cattaneo’s relaxation approaches to the analysis of strong shockwaves in a two-dimensional viscous heat-conducting fluid. Good agreement results for reasonable values of Maxwell’s relaxation times. Instability results if the viscous relaxation time is too large. These relaxation terms have negligible effects on slower-paced subsonic problems, as is shown here for two-roll and four-roll Rayleigh-Bénard flow.
Key words:
heat-conducting fluids, non-equilibrium computer simulations, Rayleigh-Bénard flow, shockwaves, twodimensional systems, viscous fluids
References:
[1] J. C. Maxwell, “On the Dynamical Theory of Gases”, Philosophical Transactions of the Royal Society of London 157, 49-88 (1867).
[2] M. C. Cattaneo, “Sur une Forme de l’Équation de la Chaleur Éliminant le Paradoxe d’une Propagation Instantaneé”, Comptes Rendus De L’Académie des Sciences-Series I-Mathematics 247, 431-433 (1958).
[3] D. D. Joseph and L. Preziosi, “Heat Waves”, Reviews of Modern Physics 61, 41-73 (1989).
[4] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, Second Edition (Springer-Verlag, Berlin, 1993).
[5] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, New York, 1959). Chapter IX is devoted to shockwaves.
[6] R. E. Duff, W. H. Gust, E. B. Royce, M. Ross, A. C. Mitchell, R. N. Keeler, and W. G. Hoover, “Shockwave Studies in Condensed Media”, in Behavior of Dense Media under High Dynamic Pressures”, Proceedings of the 1967 Paris Conference, pages 397-406 (Gordon and Breach, New York, 1968).
[7] V. Y. Klimenko and A. N. Dremin, “Structure of Shockwave Front in a Liquid”, pages 79-83 in Detonatsiya, Chernogolovka (Akademia Nauk, Moscow, 1978).
[8] B. L. Holian, W. G. Hoover, B. Moran, and G. K. Straub, “Shockwave Structure via Nonequilibrium Molecular Dynamics and Navier-Stokes Continuum Mechanics”, Physical Review A 22, 2798-2808 (1980).
[9] O. Kum, Wm. G. Hoover, and C. G. Hoover, “Temperature Maxima in Stable Two-Dimensional Shockwaves”, Physical Review E 56, 462 (1997).
[10] Wm. G. Hoover and C. G. Hoover, “Tensor Temperature and Shockwave Stability in a Strong Two-Dimensional Shockwave”, Physical Review E 80, 011128 (2009).
[11] Wm. G. Hoover and C. G. Hoover, “Well-Posed Two- Temperature Constitutive Equations for Stable Dense Fluid Shockwaves using Molecular Dynamics and Generalized Navier-Stokes-Fourier Continuum Mechanics”, Physical Review E 81, 046302 (2010).
[12] Wm. G. Hoover and C. G. Hoover, “Shockwaves and Local Hydrodynamics; Failure of the Navier-Stokes Equations”, in New Trends in Statistical Physics – Festschrift in Honor of Leopoldo García-Colín’s 80th Birthday, A. Macias and L. Dagdug, Editors, pages 15-26 (World Scientific, Singapore, 2010). See arXiv:0909.2882 [physics.flu-dyn].
[13] Wm. G. Hoover, C. G. Hoover, and F. J. Uribe, “Flexible Macroscopic Models for Dense-Fluid Shockwaves: Partitioning Heat and Work; Delaying Stress and Heat Flux; Two-Temperature Thermal Relaxation”, Proceedings of the XXXVIII International Summer School-Conference: Advanced Problems in Mechanics (Saint Petersburg, Russia, July 2010), pages 261-273. See arXiv:1005.1525 [cond-mat.statmech].
[14] Wm. G. Hoover and C. G. Hoover, “Three Lectures: NEMD, SPAM, and Shockwaves”, 11th Granada Seminar at La Herradura, 13-17 September 2010. See arXiv:1008.4947 [condmat. stat-mech].
[15] B. L. Holian and M. Mareschal, “Heat-Flow Equation Motivated by the Ideal-Gas Shockwave”, Physical Review E 82, 026707 (2010). B. L. Holian, M. Mareschal, and R. Ravelo, “Burnett-Cattaneo Continuum Theory for Shockwaves”, Physical Review E 83, 026703 (2011).
[16] W. G. Hoover, “Structure of a Shockwave Front in a Liquid”, Physical Review Letters 42, 1531-1534 (1979).
[17] L. B. Lucy, “A Numerical Approach to the Testing of the Fission Hypothesis”, The Astronomical Journal 82, 1013-1024 (1977).
[18] Wm. G. Hoover, Smooth Particle Applied Mechanics – The State of the Art (World Scientific Publishers, Singapore, 2006)
[19] Wm. G. Hoover, Molecular Dynamics (Springer- Verlag, Berlin, 1986, available at the homepage http://williamhoover.info/MD.pdf).
[20] Wm. G. Hoover, Computational Statistical Mechanics (Elsevier, Amsterdam, 1991, available at the homepage http://williamhoover.info).
[21] O. Kum, Wm. G. Hoover, and H. A. Posch, “Viscous Conducting Flows with Smooth-Particle Applied Mechanics”, Physical Review E 52, 4899-4908 (1995).
[22] V. M. Castillo, Wm. G. Hoover, and C. G. Hoover, “Coexisting Attractors in Compressible Rayleigh-Bénard Flow”, Physical Review E 55 5546-5550 (1997).
[23] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley & Sons, New York, 1954).
