Models of low-speed flow for near-critical fluids with gravitational and capillary effects
Authors:
D. L. Denny and R. L. Pego
Journal:
Quart. Appl. Math. 58 (2000), 103-125
MSC:
Primary 76N10; Secondary 35Q35, 76D45
DOI:
https://doi.org/10.1090/qam/1738560
MathSciNet review:
MR1738560
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Abstract: We study low-speed flows of a highly compressible, single-phase fluid in the presence of gravity, for example, in a regime appropriate for modeling recent space-shuttle experiments on fluids near the liquid-vapor critical point. In the equations of motion, we include forces due to capillary stresses that arise from a contribution made by strong density gradients to the free energy. We derive formally simplified sets of equations in a low-speed limit analogous to the zero Mach number limit in combustion theory.
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H. Boukari, R. Pego, and R. W. Gammon, Calculation of the dynamics of gravity-induced density profiles near a liquid-vapor critical point, Phys. Rev. E 52, 1614–1625 (1995)
H. Boukari, J. N. Shaumeyer, M. E. Briggs, and R. W. Gammon, Critical speeding up in pure fluids, Phys. Rev. A 41, 2260–2263 (1990)
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B. Zappoli, S. Amiroudine, P. Carlès, and J. Ouazzani, Thermoacoustic and buoyancy-driven transport in a square side heated cavity filled with a near critical fluid, J. Fluid Mech. 316, 53–72 (1996)
F. Zhong and H. Meyer, Density equilibration near the liquid-vapor critical point of a pure fluid: Single phase, Phys. Rev. E 51 3223–3241 (1995)
D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech. 30, 139–165 (1998)
G. K. Batchelor, The conditions for dynamical similarity of motions of a frictionless perfect-gas atmosphere, Quart. J. Roy. Meteo. Soc. 79, 224–235 (1953)
R. F. Berg, Thermal equilibration near the critical point: Effects due to three dimensions and gravity, Phys. Rev. E 48, 1799–1805 (1993)
H. Boukari, M. E. Briggs, J. N. Shaumeyer, and R. W. Gammon, Critical speeding up observed, Phys. Rev. Lett. 65, 2654–2657 (1990)
H. Boukari, R. Pego, and R. W. Gammon, Calculation of the dynamics of gravity-induced density profiles near a liquid-vapor critical point, Phys. Rev. E 52, 1614–1625 (1995)
H. Boukari, J. N. Shaumeyer, M. E. Briggs, and R. W. Gammon, Critical speeding up in pure fluids, Phys. Rev. A 41, 2260–2263 (1990)
A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd. ed., Springer-Verlag, New York, 1993
D. L. Denny and R. L. Pego, Solutions for a model of low-speed flow for highly compressible fluids with capillary effects, in preparation
J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal. 88, 95–133 (1985)
D. Durran, Improving the anelastic approximation, J. Atmos. Sci. 46, 1453–1461 (1989)
J. A. Dutton and G. H. Fichtl, Approximate equations of motion for gases and liquids, J. Atmos. Sci. 26, 241–254 (1969)
P. Embid, Well-posedness of the Nonlinear Equations for Zero Mach Number Combustion, Ph. D. Thesis, University of California, Berkeley, 1984
R. Gammon, personal communication. Also see the ZENO home page at the URL http://roissy.umd.edu/. The experimental design is described at the URL http://roissy. umd.edu/usmp3/reminder.html.
C. Ikier, H. Klein, and D. Woermann, Optical observation of the gas/liquid, phase transition in near-critical $SF_{6}$ under reduced gravity, J. Coll. Internat. Sci. 178, 368–370 (1996)
C. Ikier, H. Klein, and D. Woermann, Density equilibration in a near-critical fluid under reduced gravity, Ber. Bunsenges. Phys. Chem. 100 (8), 1308–1311 (1996)
S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34, 481–524 (1981)
A. B. Kogan, F. Zhong, and H. Meyer, Dynamics of density equilibration near the liquid-vapor critical point of He—3, Czechoslovak J. Phys. 46, Suppl. 1, 71–72 (1996)
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed., Pergamon, Oxford, 1987
F. B. Lipps and R. S. Hemler, A scale analysis of deep moist convection and some related numerical calculations, J. Atmos. Sci. 39, 2192–2210 (1982)
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53, Springer, New York, 1984
A. Majda and J. Sethian, The derivation and numerical solution of the equations for zero Mach number combustion, Combust. Sci. Tech. 42, 185–205 (1985)
M. R. Moldover, J. V. Sengers, R. W. Gammon, and J. R. Hocken, Gravity effects in fluids near the gas-liquid critical point, Rev. Mod. Phys. 51, 79–99 (1979)
Y. Ogura and N. A. Phillips, Scale analysis of deep and shallow convection in the atmosphere, J. Atmos. Sci. 19, 173–179 (1962)
A. Onuki and R. A. Ferrell, Adiabatic heating effect near the gas-liquid critical point, Physica A 164, 245–264 (1990)
A. Onuki, H. Hao, and R. A. Ferrell, Fast adiabatic equilibration in a single-component fluid near the liquid-vapor critical point, Phys. Review A 41, 2256–2259 (1990)
V. A. Rabinovich, Thermophysical Properties of Neon, Argon, Krypton, and Xenon, Hemisphere Publishing Corp., New York, 1988
R. G. Rehm and H. R. Baum, The equations of motion for thermally driven, buoyant flows, J. Res. Natl. Bur. Stand. 83, 297–308 (1973)
J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Clarendon Press, Oxford, 1982
J. V. Sengers, Transport properties of fluids near critical points, Internat. J. Thermophysics 6, 203–232 (1985)
J. V. Sengers, R. S. Basu, and J. M. H. Levelt Sengers, Representative equations for the thermodynamic and transport properties of fluids near the gas-liquid critical point, NASA Contractor Report 3424, 1981
H. L. Swinney and D. L. Henry, Dynamics of fluids near the critical point: Decay rate of order-parameter fluctuations, Phys. Rev. A 8, 2586–2617 (1973), and references therein
R. Wilhelmson and Y. Ogura, The pressure perturbation and the numerical modeling of a cloud, J. Atmos. Sci. 29, 1295–1307 (1972)
C.-S. Yih, Stratified Flows, Academic Press, New York, 1980
B. Zappoli and P. Carles, The thermo-acoustic nature of the critical speeding up, Eur. J. Mech. B/Fluids 14, 41–65 (1995)
B. Zappoli, S. Amiroudine, P. Carlès, and J. Ouazzani, Thermoacoustic and buoyancy-driven transport in a square side heated cavity filled with a near critical fluid, J. Fluid Mech. 316, 53–72 (1996)
F. Zhong and H. Meyer, Density equilibration near the liquid-vapor critical point of a pure fluid: Single phase, Phys. Rev. E 51 3223–3241 (1995)
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