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Rayleigh–Benard Instability: a Study by the Methods of Cahn–Hillard Theory of Nonequilibrium Phase Transitions

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Abstract

This article is an attempt to study the process of Rayleigh–Benard convective instability by the methods used for mathematical modeling of critical phenomena as nonequilibrium phase transitions in their initial stages of spinodal decomposition. We show that it is possible to extend the formalism adopted in the Cahn–Hillard theory of nonequilibrium phase transitions and perfected on problems of highgradient crystallization to other types of problems, in particular, those pertaining to the Rayleigh–Benard convective instability. For the initial stage of instability, a model is constructed that represents it as a nonequilibrium phase transition due to diffusive stratification. It is shown that the Gibbs free energy of deviation from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginsburg–Landau potential. Numerical experiments, by means of boundary temperature control, have been conducted with regard to self-excitation of the homogeneous state. Numerical analysis shows that convective flows may appear and proceed from regular forms (the so-called regular structures) to nonregular flows through a chaotization of the process. External factors, such as temperature growth, may lead to chaos via period doubling bifurcations.

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References

  1. E. A. Lukashev, N. N. Yakovlev, E. V. Radkevich, and V. V. Palin, “On the possibility of the Cahn–Hillard approach extension to the solution of gas dynamics problems (inner turbulence),” in: 40th Intern. Conf. Applications of Mathematics in Engineering and Economics (AMEE-14), 2014, AIP Conf. Proc., 16 31, 197–207 (2014).

  2. E. A. Lukashev, E. V. Radkevich, and N. N. Yakovlev, “On the reconstruction of the initial stage of internal turbulence,” Nanostrukt. Mat. Fiz. Model., 11, No. 1, 73–99 (2014).

    Google Scholar 

  3. N. N. Yakovlev, E. A. Lukashev, E. V. Radkevich, and V. V. Palin, “On the internal turbulence paradigm,” Vestn. Samarsk. Gos. Univ. Ser. Fiz. Mat., 19, No. 1, 155–185 (2015).

    MATH  Google Scholar 

  4. E. A. Lukashev, N. N. Yakovlev, E. V. Radkevich, and O. A. Vasil’yeva, “On the theory of nonequilibrium phase transitions on the laminar-turbulent transition,” Nanostruct. Math. Phys. Model., 14, No. 1, 5–40 (2016).

    MATH  Google Scholar 

  5. E. A. Lukashev, N. N. Yakovlev, E. V. Radkevich, and O. A. Vasil’yeva, “On laminar-to-turbulent transition,” Dokl. RAN, 471, No. 3, 1–5 (2016).

    MathSciNet  MATH  Google Scholar 

  6. E. V. Radkevich, E. A. Lukashev, M. I. Sidorov, and O. A. Vasil’eva, “Methods of nonlinear dynamics of nonequilibrium processes in fracture mechanics,” Euras. J. Math. Comput. Appl., 6, No. 2, 43–80 (2018).

    Google Scholar 

  7. E. V. Radkevich, E. A. Lukashev, N. N. Yakovlev, and O. A. Vasil’yeva, “Study of the Rayleigh–Benard instability by methods of the theory of nonequilibrium phase transitions in the Cahn–Hillard form,” Euras. J. Math. Comput. Appl., 5, No. 2, 36–65 (2017).

    Google Scholar 

  8. E. V. Radkevich, E. A. Lukashev, N. N. Yakovlev, and O. A. Vasil’yeva, “On the nature of the Rayleigh–Benard convective instability,” Dokl. RAN, 475, No. 6, 1–6 (2017).

    Google Scholar 

  9. N. N. Yakovlev, E. A. Lukashev, and E. V. Radkevich, “Problems of reconstruction of the oriented crystallization process,” Dokl. RAN, 421, No. 5, 625–629 (2008).

    MATH  Google Scholar 

  10. E. A. Lukashev and E. V. Radkevich, “Solidification and structurization of instability zones,” Appl. Math., 1, 159–178 (2010).

    Google Scholar 

  11. E. A. Lukashev, E. V. Radkevich, and N. N. Yakovlev, “Structurization of the instability zone and crystallization,” Tr. Semin. Petrovskogo, 28, 229–265 (2011).

    MathSciNet  MATH  Google Scholar 

  12. N. N. Yakovlev, E. A. Lukashev, and E. V. Radkevich, “A study of the directed crystallization by the nethod of mathematical reconstruction,” Dokl. RAN, 445, No. 4, 398–401 (2012).

    MATH  Google Scholar 

  13. E. A. Lukashev, E. V. Radkevich, and N. N. Yakovlev, “On visualization of the initial stage of binary alloy crystallization,” Nanostrukt. Mat. Fiz. Model., 11, No. 2, 5–36 (2014).

    Google Scholar 

  14. E. A. Lukashev, E. V. Radkevich, M. I. Sidorov, and O. A. Vasil’yeva, “Fracture of a structural material as a nonequilibrium phase transition,” Dokl. RAN, 480, No. 2, 1–6 (2018).

    Google Scholar 

  15. J. W. Cahn and J. E. Hillard, “Free energy of a nonuniform system. I. Interfacial free energy,” J. Chem. Phys., 28, No. 2, 258–271 (1958).

    Google Scholar 

  16. J. W. Cahn and J. E. Hillard, “Free energy of a nonuniform system. II. Thermodynamic,” J. Chem. Phys., 30, No. 5, 1121–1134 (1958).

    Google Scholar 

  17. J. W. Cahn and J. E. Hillard, “Free energy of a nonuniform system. III. Nucleation in a two-component incommpressible fluid,” J. Chem. Phys., 31, No. 3, 688–699 (1959).

    Google Scholar 

  18. J. W. Cahn, “Spinodal decomposition,” Acta Met., 9, No. 8, 795–811 (1961).

    Google Scholar 

  19. J. W. Cahn, “Spinodal decomposition in cube crystals,” Acta Met., 10, No. 3, 179–183 (1962).

    Google Scholar 

  20. J. W. Cahn, “Coherent fluctuation and nucleation in isotropic solids,” Acta Met., 10, No. 10, 907–913 (1962).

    Google Scholar 

  21. J. W. Cahn, “Magnetic aging of spinodal alloys,” J. Appl. Phys., 34, No. 12, 3581–3586 (1963).

    Google Scholar 

  22. J. W. Cahn, “Phase separation by spinodal decomposition in isotropic solids,” J. Chem. Phys., 42, No. 1, 93–99 (1965).

    Google Scholar 

  23. J. W. Cahn, “Spinodal decomposition,” Trans. AIME, 242, No. 2, 166–180 (1968).

    Google Scholar 

  24. D. W. Hoffman and J. W. Cahn, “A vector thermodynamics for anisotropic surfaces. I. Fundamentals and applications to plane surface junctions,” Surface Sci., 31, 368–388 (1972).

    Google Scholar 

  25. G. Z. Gershuni, E. M. Zhukhovitskii, and A. A. Nepomnyashchii, Stability of Convective Currents [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  26. P. Debye, Selected Works [Russian translation], Nauka, Leningrad (1987).

    Google Scholar 

  27. P. G. de Gennes, “Dynamics of fluctuation and spinodal decomposition in polimer blends,” J. Chem. Phys., 72, No. 9, 4756–4763 (1980).

    MathSciNet  MATH  Google Scholar 

  28. P. G. de Gennes, “Dynamics of fluctuation and spinodal decomposition in polimer blends,” J. Chem. Phys., 74, No. 5, 3086 (1981).

    MathSciNet  Google Scholar 

  29. Proc. of the 8-th Pacific Rim Int. Congress on Advanced Materials and Processing. USA, Hawaii, 4–9 August, 2013.

  30. P. de Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford (1974).

    MATH  Google Scholar 

  31. P. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press (1980).

    Google Scholar 

  32. S. Gorsse, P. B. Pereira, R. Decourt, and E. Sellier, “Microstructure engineering design for thermoelectric materials: An approach to minimize thermal diffusivity,” Chem. Mater., No. 22, 988–993 (2010).

    Google Scholar 

  33. A. A. Alekseev, E. N. Kablov, N. V. Petrushin, E. V. Filonova, A. Ya. Kochubei, E. A. Lukina, D. V. Zaitsev, and I. A. Treninkov, “The mechanism of plastic flow stability loss in highrhenium high-temperature nickel alloys,” in: Proc. of the Int. Sci. and Tech. Conf. Dedicated to the 100th Anniversary of the Birth of Academician S. T. Kishkin “Scientific Ideas of S. T. Kishkin and Modern Materials science,” VIAM, Moscow (2006), pp. 168–171.

  34. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, Springer (1977).

  35. H. Ziegler, Some Extremum Principles in Irreversible Thermodynamics, with Application to Continuum Mechanics, North-Holland, Amsterdam (1962).

    Google Scholar 

  36. L. M. Matyushev and V. D. Seleznev, The Principle of Maximal Entropy Production in Physics and Related Sciences [in Russian], Uralsk. Gos. Tekhn. Univ., Ekaterinburg, 2006.

    Google Scholar 

  37. L. A. Prokudina, Instability of Physical and Chemical Systems at Phase Transitions and Violation of Spatial Symmetry [in Russian], Diss. Doct. Sci., South Ural State Univ., Chelyabinsk (1999).

    Google Scholar 

  38. L. D. Landau and E. M. Lifshitz, Theoretical Physics, Vol. 6, Hydrodynamics [in Russian], Nauka, Moscow (1988).

  39. D. A. Bratsun, Dynamics of Multiphase Multicomponent Fluids with Elements of External Control [in Russian], Diss. Doct. Sci., Perm. State Univ., Perm (2010).

  40. A. V. Zyuzgin, Experimental Study of Thermal Convection in Varying Force Fields [in Russian], Diss. Doct. Sci., Perm. State Univ., Perm (2011).

    Google Scholar 

  41. G. Z. Gershuni and E. M. Zhukhovitskii, Convective Stability of Incompressible Fluid [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  42. G. Z. Gershuni, E. M. Zhukhovitskii, and A. A. Nepomnyashchii, Stability of Convective Currents [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  43. R. Betchov and V. Criminale, Questions of Hydrodynamic Stability [Russian translation], Mir, Moscow (1977).

    Google Scholar 

  44. M. A. Gol’dshtik and V. N. Stern, Hydrodynamic Stability and Turbulence [in Russian], Nauka, Novosibirsk (1977).

    Google Scholar 

  45. D. Joseph, Stability of Fluid Motion, Mir, Moscow (1981).

    Google Scholar 

  46. G. Schlichting, Turbulence Emergence, Inostr. Lit., Moscow (1962).

    Google Scholar 

  47. V. J. Shkadov, Several Methods and Problems of Hydrodynamic Stability Theory [in Russian], Proc. of the Inst. of Mechanics, Moscow State Univ., No. 25, Moscow State Univ., Moscow (1973).

  48. Y. S. Kachanov, V. V. Kozlov, and V. J. Levchenko, The Emergence of Turbulence in the Boundary Layer [in Russian], Nauka, Novosibirsk (1982).

    Google Scholar 

  49. A. V. Getling, Rayleigh–Benard Convection. The Structure and Dynamics, Editorial URSS, Moscow (1999).

    MATH  Google Scholar 

  50. A. V. Getling, “Spatial patterns formed by Rayleigh–Benard convection,” Usp. Fiz. Nauk, 161, No. 9, 1–80 (1991).

    Google Scholar 

  51. A. E. Samoilova, Convective Stability of Horizontal Layers of Fluid with Deformable Boundary [in Russian], Diss. Cand. Sci., Perm. State Univ., Perm (2015).

    Google Scholar 

  52. V. K. Andreev and V. B. Bekezhanova, Stability Nonisothermal Fluids [in Russian], Sib. Fed. Univ., Krasnoyarsk (2010).

    Google Scholar 

  53. V. G. Babskii, N. D. Kopachevskii, and A. D. Myshkis, Fluid Weightlessness [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  54. V. V. Pukhnachov, “Model convective movement at low gravity,” Model. Mech., 6, No. 4, 47–56 (1992).

    Google Scholar 

  55. A. D. Zeytounian, “The problem thermocapillary Benard–Marangoni instability,” UFN, 168, No. 3, 259–286 (1998).

    Google Scholar 

  56. V. K. Andreev, V. S. Zakhvataev, and E. A. Ryabitsky, Thermocapillary Instability, Nauka, Novosibirsk (2000).

    Google Scholar 

  57. L. G. Napolitano, “Plane Marangoni–Poiseulle flow of two immiscible fluids,” Acta Astronaut., 7, No. 4, 461–478 (1980).

    MATH  Google Scholar 

  58. S. J. VanHook, M. Schatz, J. Swift, W. McCormick, and H. Swinney, “Long-wavelength surface-tension-driven Benard convection: Experiment and theory,” J. Fluid Mech., 345, 45–78 (1997).

    MathSciNet  MATH  Google Scholar 

  59. A. Oron, “Three-dimensional nonlinear dynamics of thin liquid films,” Phys. Rev. Lett., 85, No. 10, 2108 (2000).

    MATH  Google Scholar 

  60. A. Oron, S. H. Davis, and S. G. Bankoff, “Long-scale evolution of thin liquid films,” Rev. Mod. Phys., 69, No. 3, 931 (1997).

    Google Scholar 

  61. T. V. Barmakova, L. A. Uvarova, and N. M. Barmakova, “Dynamics thermocapillary instability in the non-isothermal evaporation of multicomponent liquid mixtures,” Complex Systems Processes, No. 2, 33–39 (2012).

  62. D. A. Bograchev, A. A. Preobrazenskii, and A. D. Davydov, “Rayleigh–Benard convection in a plane layer of electrolyte solution between the two horizontal ion-selective membranes,” JPhCh, 82, No. 11, 2154–2159 (2008).

    Google Scholar 

  63. H. Haken, Synergetics. An Introduction, Springer, Berlin (1977).

    MATH  Google Scholar 

  64. E. A. Lukashev, N. N. Yakovlev, E. V. Radkevich, and O. A. Vasil’yeva, “On the problems of the laminar-turbulent transition,” Rep. Russ. Acad. Sci., 471, No. 3, 1–5 (2016).

    MathSciNet  MATH  Google Scholar 

  65. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, Mir, Moscow (1973).

    MATH  Google Scholar 

  66. V. G. Danilov, G. A. Omel’yanov, and E. V. Radkevich, “Asymptotic solution of the conserved phase field system in the fast relaxation case,” Euras. J. Appl. Math., 9, 1–21 (1998).

    MathSciNet  MATH  Google Scholar 

  67. V. G. Danilov, G. A. Omel’yanov, and E. V. Radkevich, “Hugoniot-type conditions and weak solutions to the phase field system,” Euras. J. Appl. Math., 10, 55–77 (1999).

    MathSciNet  MATH  Google Scholar 

  68. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics [in Russian], Vol. 1 Nauka, Moscow (1965).

    Google Scholar 

  69. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  70. L. G. Loitsianskii, Mechanics of Liquids and Gases [in Russian], Drofa, Moscow (2003).

    Google Scholar 

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    E. V. Radkevich, E. A. Lukashev & O. A. Vasil’yeva

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  1. E. V. Radkevich
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  2. E. A. Lukashev
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  3. O. A. Vasil’yeva
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Correspondence to E. V. Radkevich.

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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 32, pp. 283–324, 2019.

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Radkevich, E.V., Lukashev, E.A. & Vasil’yeva, O.A. Rayleigh–Benard Instability: a Study by the Methods of Cahn–Hillard Theory of Nonequilibrium Phase Transitions. J Math Sci 244, 294–319 (2020). https://doi.org/10.1007/s10958-019-04620-3

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