Introduction to the generalized theory of non-equilibrium Cahn-Hilliard phase transitions

E. A. Lukashev ; E. V. Radkevich ; N. N. Yakovlev ; O. A. Vasil'eva

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 3, pp. 437-472.

Résumé

The occurrence of convective currents and their development from regular forms with the subsequent transition to irregular turbulent currents draw attention to the fact that they are responsible for the efficiency of many technological processes of heat and mass transfer. Such technological processes are basic in the chemical, petrochemical, power, metallurgical and other industries. Convective flows arise in liquids and gases in the gravitational field in the presence of spatial inhomogeneity of the density created by the inhomogeneity of the temperature and the concentration of components arising during, for example, chemical reactions or other causes. With increasing temperature difference, the resting liquid loses its stability, which then leads to the appearance of a convective flow (Rayleigh–Bénard instability). A further increase in the temperature difference leads to an instability of the primary convective flow, and the hydrodynamic crisis leads to a heat transfer crisis. The paper reconstructs the early stage of the Rayleigh–Bénard convective instability considered as a nonequilibrium phase transition with the spinodal decomposition (diffusion separation) mechanism.

Détail

Keywords: critical processes, Rayleigh–Bénard instability, nonequilibrium phase transition, Ginzburg–Landau potential, pumping of internal energy, free Gibbs energy, models of continuum mechanics.
Mots-clés : diffusion separation

Bibliographie

[1] Gershuni G. Z., Zhukhovitskii E. M, Konvektivnaia ustoichivost' neszhimaemoi zhidkosti [Convective Stability of Incompressible Fluids], Nauka, Moscow, 1972, 392 pp. (In Russian)

[2] Gershuni G.Z., Zhukhovitskii E. M., Nepomnyashchii A. A., Ustoichivost' konvektivnykh techenii [Stability of convective flows], Nauka, Moscow, 1989, 320 pp. (In Russian) | MR | Zbl

[3] Bratsun D. A., Dynamics of multiphase multicomponent fluids with elements of external control, Dr. Phys. and Math. Sci. Thesis, Perm, 2010, 375 pp. (In Russian)

[4] Ziuzgin A. V., Experimental study of thermal convection in varying force fields, Dr. Phys. and Math. Sci. Thesis, Perm, 2011, 180 pp. (In Russian)

[5] Prokudina L. A., Instability of physical and chemical systems at phase transitions and violation of spatial symmetry, Dr. Phys. and Math. Sci. Thesis, Chelyabinsk, 1999, 259 pp. (In Russian)

[6] Landau L. D., Lifshits E. M., Gidrodinamika [Fluid dynamics], Teoreticheskaia fizika [Theoretical physics], 6, Nauka, Moscow, 1986, 736 pp. (In Russian) | MR | Zbl

[7] Criminale W. O., Jackson T. L., Joslin R. D., Theory and Computation in Hydrodynamic Stability, Cambridge University Press, Cambridge, 2003, xxii+441 pp | DOI | MR

[8] Gol'dshtik M. A., Shtern V. N., Gidrodinamicheskaia ustoichivost' i turbulentnost' [Hydrodynamic stability and turbulence], Nauka, Novosibirsk, 1977, 367 pp. (In Russian)

[9] Joseph D. D., Stability of Fluid Motions I, Springer Tracts in Natural Philosophy, 27, Springer, Berlin, 1976, xiii+282 pp. | DOI | MR | Zbl

[10] Schlichting H., “Entstehung der Turbulenz”, Fluid Dynamics I / Strömungsmechanik I, Encyclopedia of Physics / Handbuch der Physik, 3/8/1, ed. C. Truesdell, Springer, Berlin, 1959, 351–450 | DOI | MR

[11] Shkadov V. Ya., “Some methods and problems of the theory of hydrodynamic stability”, Scientific Proceedings of the Institute of Mechanics of Moscow State University, 25, Moscow, 1973, 192 pp. (In Russian)

[12] Kachanov Yu. S., Kozlov V. V., Levchenko V. Ya., Vozniknovenie turbulentnosti v pogranichnom sloe [Onset of Turbulence in Boundary Layers], Nauka, Novosibirsk, 1982, 151 pp. (In Russian)

[13] Getling A. V., Rayleigh-Bńard Convection, Advanced Series in Nonlinear Dynamics, 11, World Scientific, Singapore, ix+245 pp. | DOI | MR

[14] Getling A. V., “Formation of spatial structures in Rayleigh—Bńard convection”, Sov. Phys. Usp., 34:9 (1991), 737–776 | DOI | DOI | MR

[15] Samoilova A. E., Convective stability of horizontal layers of fluid with deformable boundary, Cand. Phys. and Math. Sci. Thesis, Perm, 2015, 120 pp. (In Russian)

[16] Andreev V. K., Bekezhanova V. B., “Stability of non-isothermal fluids (Review)”, J. Appl. Mech. Tech. Phys., 54:2 (2013), 171–184 | DOI | MR | Zbl

[17] Myshkis A. D., Babskii V. G., Kopachevskii N. D., Slobozhanin L. A., Tyuptsov A. D., Low-Gravity Fluid Mechanics. Mathematical Theory of Capillary Phenomena, Springer-Verlag, Berlin, 1987, xix+583 pp. | MR

[18] Pukhnachev V. V., “Model of convective motion with reduced gravity”, Modelirovanie v mekhanike [Modelling in Mechanics], 6:4 (1992), 47–56 (In Russian) | MR

[19] Zeytounian R. Kh., “The Benard-Marangoni thermocapillary-instability problem”, Phys. Usp., 41:12 (1998), 241–267 | DOI | DOI

[20] Radkevich E.V., “Some problems of hydrodynamics”, Sobolevskie chteniia. [Sobolev Readings. International School-Conference] (Novosibirsk, 18–22 December 2016.), eds. V. L. Vaskevich, G. V. Demidenko, Novosibirsk State Univ., Novosibirsk, 2016, 22–34 (In Russian)

[21] Andreev V. K., Zakhvataev V. E., Ryabitskii E. A., Termokapilliarnaia neustoichivost' [Thermocapillary Instability], Nauka, Novosibirsk, 2000, 280 pp. (In Russian) | MR

[22] Napolitano L. G., “Plane Marangoni–Poiseuille flow of two immiscible fluids”, Acta Astronaut., 7:4–5 (1980), 461–478 | DOI | Zbl

[23] Bekezhanova V. B., “Convective Instability of Marangoni–Poiseuille Flow Under a Longitudinal Temperature Gradient”, J. Appl. Mech. Tech. Phys., 52:1 (2011), 74–81 | DOI | MR | Zbl

[24] Ermolenko A. N., “Rayleigh–Benard problem for an anomalous fluid”, J. Appl. Mech. Tech. Phys., 48:2 (2007), 166–175 | DOI | MR | Zbl

[25] Krishnamoorthy S., Ramaswamy B., Joo S. W., “Spontaneous rupture of thin liquid films due to thermocapillarity: A full-scale direct numerical simulation”, Phys. Fluids, 7:9 (1995), 2291–2293 | DOI | Zbl

[26] VanHook S. J., Schatz M., Swift J., McCormick W., Swinney H., “Long-wavelength surface-tension-driven Bénard convection: experiment and theory”, J. Fluid Mech., 345 (1997), 45–78 | DOI | MR | Zbl

[27] Oron A., “Three-dimensional nonlinear dynamics of thin liquid films”, Phys. Rev. Lett., 85:10 (2000), 2108–2111 | DOI

[28] Oron A., Davis S. H., Bankoff S. G., “Long-scale evolution of thin liquid films”, Rev. Mod. Phys., 69:3 (1997), 931–980 | DOI | MR

[29] Barmakova T. V., Uvarova L. A., Barmakova N. M., “Dynamics Thermocapillary instability in the non-isothermal evaporation of multicomponent liquid mixtures”, Skladni sistemi i protsesi [Complex Systems and Processes], 2012, no. 2, 33–39 (In Russian)

[30] Bograchev D. A., Preobrazhenskii A. A., Davydov A. D., “Rayleigh-Benard instability in a flat electrolyte solution layer between two horizontal ion-selective membranes”, Russ. J. Phys. Chem., 82:11 (2008), 1938–1942 | DOI

[31] Haken H., Synergetics, Springer Series in Synergetics, 1, Berlin, Springer, 1983, xiv+390 pp. | DOI | Zbl

[32] Yakovlev N. N., Lukashev E. A., Radkevich E. V., “Problems of reconstruction of the process of directional solidification”, Doklady Physics, 53:8 (2008), 442–446 | DOI | Zbl

[33] Yakovlev N. N., Lukashev E. A., Radkevich E. V., “Investigation of directed crystallization by the method of mathematical reconstruction”, Doklady Physics, 57:8 (2012), 297–300 | DOI | Zbl

[34] Lukashev E. A., Yakovlev N. N., Radkevich E. V., Vasil'yeva O. A., “On problems of the laminar–turbulent transition”, Doklady Mathematics, 94:3 (2016), 649–653 | DOI | DOI | MR | Zbl

[35] Glansdorff P., Prigogine I., Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, New York, 1971, xxvi+305 pp. | MR | Zbl

[36] Cahn J. W., Hillard J. E., “Free Energy of a Nonuniform System. I. Interfacial Free Energy”, J. Chem. Phys., 28:2 (1958), 258–267 ; | DOI | DOI

[37] Cahn J. W., Hillard J. E., “Free energy of a nonuniform system. II. Thermodynamic”, J. Chem. Phys., 30:5 (1958), 1121–1134 | DOI

[38] Cahn J. W., Hillard J. E., “Free Energy of a Nonuniform System. III. Nucleation in a two-component incommpressible fluid”, J. Chem. Phys., 31:3 (1959), 688–699 ; | DOI | DOI

[39] Cahn J. W., “On spinodal decomposition in cubic crystals”, Acta Met., 10:3 (1962), 179–183 | DOI

[40] Cahn J. W., “Coherent fluctuation and nucleation in isotropic solids”, Acta Met., 10:10 (1962), 907–913 ; | DOI | MR | DOI

[41] Cahn J. W., “Magnetic Aging of Spinodal Alloys”, J. Appl. Phys., 34:12 (1963), 3581–3586 ; | DOI | DOI

[42] Cahn J. W., “Phase Separation by Spinodal Decomposition in Isotropic Systems”, J. Chem. Phys., 42:1 (1965), 93–99 | DOI

[43] Cahn J. W., “On spinodal decomposition”, Acta Met., 9:9 (1961), 795–801 ; | DOI | DOI

[44] Hoffman D. W., Cahn J. W., “A vector thermodynamics for anisotropic surface”, Surface Sciences, 31 (1972), 368–388 ; | DOI | DOI

[45] Danilov V. G., Omel'yanov G. A., Radkevich E. V., “Asymptotic solution of the conserved phase field system in the fast relaxation case”, Eur. J. Appl. Math., 9:1 (1998), 1–21 | DOI | MR | Zbl

[46] Danilov V. G., Omel'yanov G. A., Radkevich E. V., “Hugoniot-type conditions and weak solutions to the phase-field system”, Eur. J. Appl. Math., 10:1 (1999), 55–77 | DOI | MR | Zbl

[47] Monin A. S., Iaglom A. M., Staticheskaia gidromekhanika [Statistical Fluid Mechanics], Part 1, Nauka, Moscow, 1965, 640 pp. (In Russian)

[48] Tikhonov A. I., Samarskii A. A., Uravnenie matematicheskoi fiziki [Equation of Mathematical Physics], Nauka, Moscow, 1972, 763 pp. (In Russian) | MR | Zbl

[49] Lukashev E. A., Yakovlev N. N., Radkevich E. V., Palin V. V., “On the possibility of the Cahn-Hilliard approach extension to the solution of gas dynamics problems (inner turbulence)”, AIP Conference Proceedings, 1631 (2014), 197–208 | DOI

[50] Lukashev E. A., Radkevich E. V., Yakovlev N. N., “On visualization for initial stage of crystallization of binary alloys”, Nanostruktury. Matematicheskaia fizika i modelirovanie [Nanostructures, Mathematical Physics and Simulation], 11:2 (2014), 5–36 (In Russian)

[51] Lukashev E. A., Yakovlev N. N., Radkevich E. V., Vasil'eva O. A., “On the theory of nonequilibrium phase transitions on the laminar-turbulent transition”, Nanostruktury. Matematicheskaia fizika i modelirovanie [Nanostructures, Mathematical Physics and Simulation], 14:1 (2016), 5–40 (In Russian)

[52] Münster A., Classical Thermodynamics, Wiley-Interscience, New York, 1970

[53] Prigogine I. Stengers I., The End of Certainty. Time, Chaos and the New Laws of Nature, The Free Press, New York, 1997, ix+228 pp. | MR

[54] Loitsianskii L. G., Mekhanika zhidkosti i gaza [Fluid Mechanics], Nauka, Moscow, 1973 (In Russian) | MR

[55] Zhukov V. T., Zaitsev N. A., Lysov V. G., Rykov Yu. G., Feodoritova O. B., “Numerical analysis of a model of metal solidification, 2D case”, Math. Models Comput. Simul., 4:4 (2012), 440–453 | DOI | MR | Zbl

[56] Kozlov V. V., “The generalized Vlasov kinetic equation”, Russian Math. Surveys, 63:4 (2008), 691–726 | DOI | DOI | MR | Zbl

[57] Palin V. V., Radkevich E. V., “Navier–Stokes approximation and problems of the Chapman-Enskog projection for kinetic equations”, J. Math. Sci. (N. Y.), 135:1 (2006), 2721–2748 | DOI | MR | Zbl