×

Solutions of the buoyancy-drag equation with a time-dependent acceleration. (English) Zbl 1420.37114

Summary: We perform the analytic study of the buoyancy-drag equation with a time-dependent acceleration \(\gamma(t)\) by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of \(\gamma(t)\) define a time-dependent Hamiltonian from which the buoyancy-drag equation can be derived. We then determine the Lie point symmetries of the buoyancy-drag equation, which only exist for values of \(\gamma(t)\) including the previous ones, plus additional classes of accelerations for which the equation is reducible to an Abel equation. This allows us to exhibit two régimes for the asymptotic (large time \(t\)) solution of the buoyancy-drag equation. It is shown that they describe a mixing zone driven by the Rayleigh-Taylor instability and the Richtmyer-Meshkov instability, respectively.

MSC:

37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
22E99 Lie groups
34Mxx Ordinary differential equations in the complex domain
30Dxx Entire and meromorphic functions of one complex variable, and related topics
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
76Fxx Turbulence
37-XX Dynamical systems and ergodic theory
60Gxx Stochastic processes
60Jxx Markov processes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alon, U.; Hecht, J.; Hofer, D.; Shvarts, D., Power laws and similarity of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts at all density ratios, Physical review letters, 74, 4, 534-537 (1995) · doi:10.1103/PhysRevLett.74.534
[2] Appell, P., Sur les invariants de quelques eéquations différentielles, Journal de mathématiques pures et appliquées, 5, 360-424 (1889)
[3] Babich, M. V.; Bordag, L. A., Projective differential geometrical structure of the Painlevé equations, J. differential equations, 157, 452-485 (1999) · Zbl 0938.34081 · doi:10.1006/jdeq.1999.3628
[4] Bouquet, S.; Gandeboeuf, P.; Pailhoriès, P., Analytic study of the buoyancy-drag equation, Math. meth. appl. sci., 30, 2027-2035 (2007) · Zbl 1149.76023 · doi:10.1002/mma.944
[5] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability (1961), Oxford university press: Oxford university press, Oxford · Zbl 0142.44103
[6] Chazy, J., Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, Acta Math., 34, 317-385 (1911) · JFM 42.0340.03 · doi:10.1007/BF02393131
[7] Cheb-Terrab, E. S.; Roche, A. D., Abel ODEs: equivalence and integrable classes, Computer physics communications, 130, 204-231 (2000) · Zbl 0961.65067 · doi:10.1016/S0010-4655(00)00042-4
[8] Cheng, B.; Glimm, J.; Sharp, D. H., Dynamical evolution of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts, Physical review E, 66, 3 (2002)
[9] Conte, R.; Musette, M., The Painlevé handbook (2008), Springer: Springer, Berlin · Zbl 1153.34002
[10] Davies, R. M.; Sir Geoffroy Taylor, F. R.S., The mechanics of large bubbles rising through extended liquids and through liquids in tubes, Proceedings of the royal society of London, Ser. A, 200, 375-390 (1950) · doi:10.1098/rspa.1950.0023
[11] Dimonte, G., Spanwise homogeneous buoyancy-drag model for Rayleigh-Taylor mixing and experimental evaluation, Physics of plasmas, 7, 6, 2255-2269 (2000) · doi:10.1063/1.874060
[12] Dimonte, G.; Schneider, M., Density ratio dependence of Rayleigh-Taylor mixing for sustained and impulsive acceleration histories, Physics of fluids, 12, 2, 304-321 (2000) · Zbl 1149.76361 · doi:10.1063/1.870309
[13] Dimonte, G.; Youngs, D. L.; Dimits, A.; Weber, S.; Marinak, M.; Wunsch, S.; Garasi, C.; Robinson, A.; Andrews, M. J.; Ramaprabhu, P.; Calder, A. C.; Fryxell, B.; Biello, J.; Dursi, L.; Macneice, P.; Olson, K.; Ricker, P.; Rosner, R.; Timmes, F.; Tufo, H.; Young, Y.-N.; Zingale, M., A comparative study of the turbulent Rayleigh-Taylor instability using high-resolution three-dimensional numerical simulations: the Alpha-group collaboration, Physics of fluids, 16, 5, 1668-1693 (2004) · Zbl 1186.76143 · doi:10.1063/1.1688328
[14] Gradshteyn, I. S.; Ryzhik, I. M., Tables of integrals, series, and products (1980), Academic press: Academic press, New York · Zbl 0521.33001
[15] Gréa, B. J., The rapid acceleration model and the growth rate of a turbulent mixing zone induced by Rayleigh-Taylor instability, Physics of fluids, 25, 1-20 (2013) · doi:10.1063/1.4775379
[16] Hecht, J.; Alon, U.; Shvarts, D., Potential flow models of Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts, Physics of fluids, 6, 12, 4019-4030 (1994) · Zbl 0922.76169 · doi:10.1063/1.868391
[17] Kamke, E., Differentialgleichungen: Lösungsmethoden und Lösungen
[18] Kelsch, V.; Bouquet, S.; Conte, R.
[19] Layzer, D., On the instability of superposed fluids in a gravitational field, The astrophysical journal, 122, 1, 1-12 (1955) · doi:10.1086/146048
[20] Liouville, R., Sur les invariants de certaines équations différentielles et sur leurs applications, Journal de l’École polytechnique, 59, 7-76 (1889) · JFM 21.0317.01
[21] Liouville, R., Sur une équation différentielle du premier ordre, Acta mathematica, 26, 55-78 (1902) · JFM 34.0364.03
[22] Llor, A., Bulk turbulent transport and structure in Rayleigh-Taylor, Richtmyer-Meshkov, and variable acceleration instabilities, Laser and particle beams, 21, 305-310 (2003)
[23] Llor, A., Analytical “0D” evaluation criteria, and comparison of single-and two-phase flow approaches, Statistical hydrodynamic models for developed mixing instability flows (2005), Springer: Springer, Berlin
[24] Louvet, F.; Bouquet, S.
[25] Murphy, G. M., Ordinary differential equations and their solutions (1960), Van Nostrand: Van Nostrand, Princeton · Zbl 0095.06405
[26] Neuvazhaev, V. E., Theory of turbulent mixing, Soviet physics doklady, 20, 6, 398-400 (1975)
[27] Neuvazhaev, V. E., Properties of a model for the turbulent mixing of the boundary between accelerated liquids differing in density, Journal applied mechanics technical physics, 24, 5, 680-687 (1983) · doi:10.1007/BF00905883
[28] Olver, P. J., Applications of Lie groups to differential equations (1986), Springer: Springer, Berlin · Zbl 0588.22001
[29] Ovsiannikov, L. V., Group properties of differential equations, (1962), Siberian section of the Academy of Sciences of the USSR: Siberian section of the Academy of Sciences of the USSR, Novosibirsk
[30] Panayotounakos, D. E.; Zarmpoutis, T. I., Construction of exact parametric or closed form solutions of some unsolvable classes of nonlinear ODEs (Abel’s nonlinear ODEs of the first kind and relative degenerate equations), International journal of mathematics and mathematical sciences, 2011 (2011) · Zbl 1238.34002 · doi:10.1155/2011/387429
[31] Pandian, A.; Swisher, N. C.; Abarzhi, S. I., Deterministic and stochastic dynamics of Rayleigh-Taylor mixing with a power-law time-dependent acceleration, Physica scripta, 92 (2017)
[32] Polyanin, A. D.; Zaitsev, V. F., Handbook of exact solutions for ordinary differential equations (1995), CRC Press: CRC Press, Boca Raton · Zbl 0855.34001
[33] Ramshaw, J. D., Simple model for linear and nonlinear mixing at unstable fluid surfaces with variable acceleration, Physical review E, 58, 5, 5834-5840 (1998) · doi:10.1103/PhysRevE.58.5834
[34] Srebro, Y.; Elbaz, Y.; Sadot, O.; Arazi, L.; Shvarts, D., A general buoyancy-drag model for the evolution of the Rayleigh-Taylor and Richtmyer-Meshkov instability, Laser particle beams, 21, 347-353 (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.