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Continuum perspective of bulk viscosity in compressible fluids

  • Xin-Dong Li (a1), Zong-Min Hu (a1) and Zong-Lin Jiang (a1)


Kinetic theory and acoustic measurements have proven that the bulk viscosity associated with the expansion or compression effect cannot be ignored in compressible fluids except for monatomic gases. A new theoretical formula for the bulk viscosity coefficient (BVC) $\unicode[STIX]{x1D701}$ is derived by the continuum medium methodology, which provides a further understanding of the bulk viscosity, i.e. $\unicode[STIX]{x1D701}$ is equal to the product of the bulk modulus $K$ and the relaxation time $\unicode[STIX]{x1D70F}$ ( $\unicode[STIX]{x1D701}=K\unicode[STIX]{x1D70F}$ ). The continuum and kinetic theories present consistent results from macro- and microperspectives respectively, only differing in terms of a coefficient. The theoretical predictions of the BVC in diatomic molecules, such as $\text{N}_{2}$ , $\text{O}_{2}$ and CO, show good agreement with the experimental data over a wide range of temperature. In addition, the vibrational contributions to $\unicode[STIX]{x1D701}$ are controlled by a rapid exponential decrease at high temperatures, while at low temperatures a slow linear increase proceeds for the rotational cases. The relaxation time $\unicode[STIX]{x1D70F}$ , collision number $Z$ , BVC $\unicode[STIX]{x1D701}$ and ratio of bulk-to-shear viscosities $\unicode[STIX]{x1D701}/\unicode[STIX]{x1D707}$ in the vibrational mode are found to be several orders of magnitude larger than those in the rotational mode.


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Anderson, J. D. 2006 Hypersonic and High-Temperature Gas Dynamics, 2nd edn. AIAA.
Bahmani, F. & Cramer, M. S. 2014 Suppression of shock-induced separation in fluids having large bulk viscosities. J. Fluid Mech. 756, 110.
Bauer, H. J. & Kosche, H. 1966 Rotational relaxation in carbon monoxide compared with other diatomic gases. Acustica 17 (2), 9697.
Billet, G., Giovangigli, V. & Gassowski, G. D. 2008 Impact of volume viscosity on a shock/hydrogen bubble interaction. Combust. Theor. Model. 12 (2), 221248.
Blackman, V. 1956 Vibrational relaxation in oxygen and nitrogen. J. Fluid Mech. 1 (1), 6185.
Borgnakke, C. & Sonntag, R. E. 2009 Fundamentals of Thermodynamics. Wiley.
Carnevale, E. H., Carey, C. & Larson, G. 1967 Ultrasonic determination of rotational collision numbers and vibrational relaxation times of polyatomic gases at high temperatures. J. Chem. Phys. 27 (8), 28292835.
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge University Press.
Chikitkin, A. V., Rogov, B. V., Tirsky, G. A. & Utyuzhnikov, S. V. 2015 Effect of bulk viscosity in supersonic flow past spacecraft. Appl. Numer. Maths 93, 4760.
Cramer, M. S. 2012 Numerical estimates for the bulk viscosity of ideal gases. Phys. Fluids 24, 066102.
Cramer, M. S. & Bahmani, F. 2014 Effect of large bulk viscosity on large-Reynolds-number flows. J. Fluid Mech. 751, 142163.
Connor, J. V. 1958 Ultrasonic dispersion in oxygen. J. Acoust. Soc. Am. 30 (4), 297300.
Elizarova, T. G., Khokhlov, A. A. & Montero, S. 2007 Numerical simulation of shock wave structure in nitrogen. Phys. Fluids 19, 068102.
Emanuel, G. 1990 Bulk viscosity of a dilute polyatomic gas. Phys. Fluids 2, 22522254.
Emanuel, G. 1992 Effect of bulk viscosity on a hypersonic boundary layer. Phys. Fluids 4, 491495.
Emanuel, G. 1994 Linear dependence of the bulk viscosity on shock wave thickness. Phys. Fluids 6, 32023205.
Emanuel, G. 1998 Bulk viscosity in the Navier–Stokes equations. Intl J. Engng Sci. 36 (11), 13131323.
Emanuel, G. 2016 Analytical Fluid Dynamics, 3rd edn. CRC Press.
Eringen, A. C. 1980 Mechanics of Continua, 2nd edn. Krieger.
Fru, G., Janiga, G. & Thevenin, D. 2011 Direct numerical simulations of the impact of high turbulence intensities and volume viscosity on premixed methane flames. J. Combust. 2011, 746719.
Fru, G., Janiga, G. & Thevenin, D. 2012 Impact of volume viscosity on the structure of turbulent premixed flames in the thin reaction zone regime. Flow Turbul. Combust. 88, 451478.
Fujii, Y., Lindsay, R. B. & Urushihara, K. 1963 Ultrasonic absorption and relaxation times in nitrogen, oxygen, and water vapor. J. Acoust. Soc. Am. 35 (7), 961966.
Gaydon, A. G. & Hurle, I. R.1961 Measurement of times of vibrational relaxation and dissociation behind shock waves in $\text{N}_{2}$ , $\text{O}_{2}$ , air, CO, $\text{CO}_{2}$ and $\text{H}_{2}$ . Tech. Rep. 309-318; 8th Symp. Combust.
Gonzalez, H. & Emanuel, G. 1993 Effect of bulk viscosity on Couette flow. Phys. Fluids 5, 12671268.
Graves, R. E. & Argrow, B. M. 1999 Bulk viscosity: past to present. J. Thermophys. Heat Transfer 13 (3), 337342.
Greenspan, M. 1959 Rotational relaxation in nitrogen, oxygen, and air. J. Acoust. Soc. Am. 31 (2), 155160.
Henderson, M. C. 1962 Vibrational relaxation in nitrogen and other gases. J. Acoust. Soc. Am. 34, 349350.
Herzfeld, K. F. & Litovitz, T. A. 1959 Absorption and Dispersion of Ultrasonic Waves. Academic.
Holmes, R., Jones, G. R., Pusat, N. & Tempest, W. 1962 Rotational relaxation in helium + oxygen and helium + nitrogen mixtures. Trans. Farad. Soc. 58, 23422347.
Holmes, R., Simth, F. A. & Tempest, W. 1963 Vibrational relaxation in oxygen. Proc. Phys. Soc. 81 (2), 311319.
Hooker, W. J. & Millikan, R. C. 1963 Shock-tube study of vibrational relaxation in carbon monoxide for the fundamental and first overtone. J. Chem. Phys. 38 (1), 214220.
Hurle, L. R. 1964 Line-reversal studies of the sodium excitation process behind shock waves in N2 . J. Chem. Phys. 41 (12), 39113920.
Kistemaker, P. G., Tom, A. & Vries, A. E. D. 1970 Rotational relaxation numbers for the isotopic molecules of N2 and CO. Physica 48, 414424.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon.
Lukasik, S. J. & Young, J. E. 1957 Vibrational relaxation times in nitrogen. J. Chem. Phys. 27 (5), 11491155.
Matthews, D. L. 1961 Vibrational relaxation of carbon monoxide in the shock tube. J. Chem. Phys. 34 (2), 639642.
Michel, A. A. 1985 Compressible Fluid Flow. Prentice-Hall.
Millikan, R. C. & White, D. R. 1963a Vibrational energy exchange between N2 and CO. The vibrational relaxation of nitrogen. J. Chem. Phys. 39 (1), 98101.
Millikan, R. C. & White, D. R. 1963b Systematic of vibrational relaxation. J. Chem. Phys. 39 (12), 32093213.
Monchick, L., Yun, K. S. & Mason, E. A. 1963 Formal kinetic theory of transport phenomena in polyatomic gas mixtures. J. Chem. Phys. 39 (3), 654669.
Parker, J. G. 1959 Rotational and vibrational relaxation in diatomic gases. Phys. Fluids 2 (4), 449462.
Parker, J. G. 1961 Effect of several light molecules on the vibrational relaxation time of oxygen. J. Chem. Phys. 34 (5), 17631772.
Parker, J. G. 1964 Comparison of experimental and theoretical vibrational relaxation times for diatomic gases. J. Chem. Phys. 41 (6), 16001609.
Parker, J. G., Adams, C. E. & Stavseth, R. M. 1953 Absorption of sound in argon, nitrogen, and oxygen at low pressure. J. Acoust. Soc. Am. 25 (2), 263269.
Prangsma, G. J., Alberga, A. H. & Beenakker, J. J. M. 1973 Ultrasonic determination of the volume viscosity of N2 , CO, CH4 and CD4 between 77 and 300 K. Physica 64 (2), 278288.
Rajagopal, K. R. 2013 A new development and interpretation of the Navier–Stokes fluid which reveals why the ‘Stokes assumption’ is inapt. Intl J. Non-Linear Mech. 50, 141151.
Sherman, F. S.1955 A low-density wind-tunnel study of shock-wave structure and relaxation phenomena in gases. NACA Tech. Rep. TN-3298.
Shileds, F. D. & Lee, K. P. 1963 Sound absorption and velocity measurements in oxygen. J. Acoust. Soc. Am. 35, 251252.
Sivian, L. J. 1947 High frequency absorption in air and other gases. J. Acoust. Soc. Am. 19 (5), 914916.
Smith, F. A. & Tempest, W. 1961 Low-frequency sound propagation in gases. J. Acoust. Soc. Am. 33, 16261627.
Stokes, G. G. 1845 On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Camb. Phil. Soc. 8 (22), 287342.
Tempest, W. T. & Parbrook, H. D. 1957 The absorption of sound in argon, nitrogen and oxygen. Acustica 7 (6), 354362.
Thompson, P. A. 1972 Compressible-Fluid Dynamics. McGraw-Hill.
Tip, A., Los, J. & Vries, A. E. D. 1967 Rotational relaxation numbers from thermal transpiration measurements. Physica 35, 489498.
Tisza, L. 1942 Supersonic absorption and Stokes’ viscosity relation. Phys. Rev. 61, 531536.
Vincenti, W. G. & Kruger, G. H. 1965 Introduction to Physical Gas Dynamics. Krieger.
Wang, C. & Uhlenbeck, G. E.1951 Transport phenomena in polyatomic gases. Tech. Rep. CM-681. US Navy Department.
White, D. R. & Millikan, R. C. 1963 Vibrational relaxation of oxygen. J. Chem. Phys. 39 (1), 18031806.
Windsor, M. W., Davidson, N. & Taylor, R. 1957 Measurement of the vibrational relaxation time of CO behind a shock wave by infrared emission. J. Chem. Phys. 27, 315316.
Winter, T. G. & Hill, G. 1967 High-temperature ultrasonic measurements of rotational relaxation in hydrogen, deuterium, nitrogen, and oxygen. J. Acoust. Soc. Am. 42 (4), 848858.
Zartman, I. F. 1949 Ultrasonic velocities and absorption in gases at low pressures. J. Acoust. Soc. Am. 21 (3), 171174.
Zel’dovich, Y. B. & Raizer, Y. P. 1966 Physics of Shock Waves and High-Temperature Hydrodynamics Phenomena, vol. 2. Academic.
Zmuda, A. J. 1951 Dispersion of velocity and anomalous absorption of ultrasonics in nitrogen. J. Acoust. Soc. Am. 23 (4), 472477.
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