Pattern Formation in Convective Media
Abstract
The several models of convection in a thin layer of liquid (gas) with poorly heat conducting boundaries are considered. These models demonstrate a rich dynamics of pattern formation and structural phase transitions. The primary analysis of pattern formation in such a system is performed with using of the well-studied Swift-Hohenberg model. The more advanced Proctor-Sivashinsky model is examined in order to study the second-order structural phase transitions both between patterns with translational invariance and between structures with broken translational invariance but keeping a long-range order. The spatial spectrum of arising structures and visual estimation of the number of defects are analyzed. The relation between the density of defects and the spectral characteristics of the structure is found. We also discuss the effect of noise on the formation of structural defects. It is shown that within the framework of the Proctor-Sivashinsky model with additional term, taking into account the inertial effects, the large-scale vortex structures arise as a result of the secondary modulation instability.
Downloads
References
Busse F.H., Riahi N. Nonlinear convection in a layer with nearly insulating boundaries // J.Fluid Mech. – 1980. - Vol.96. - P.243.
Chandrasekhar S. Hydrodenamic and hydromagnetic stability. Third printing of the Clarendon Press. Oxford University Press edition, Dover Publication Inc.: New York, 1970. - 704 p.
Getling A.V. Structures in heat convection // Usp. Fiz. Nauk. – 1991. - Vol.11. – P.1-80.
Karpman V.I. Nonlinear waves in dispersive media. - М.: Nauka, 1973. – 175с.
Autowave processes in systems with diffusion / edited by M.T. Grekhov. - Gorky: IPF SA USSR, 1981. – 246s.
Engelbrecht J.K. Evolution equations and self-wave processes in the active medium // On nonlinear continuum mechanics. - Tallinn: Valgus, 1985. - P.119-131.
Kosevich A.M., Ivanov B.A., Kovalev A.S. Nonlinear waves of magnetization. – Kiev: Naukova Dumka, 1983. - 240s.
Schwartz L.W., Fenton J.D. Strongly nonlinear waves // Ann. Rev. Fluid. Mech. – 1982. - Vol.14. - P.39-60. Translation: Schwartz L., Fenton J. Strongly nonlinear waves. In Sat Nonlinear wave processes. Collection of articles. Series «Mechanics», № 42, Mir: 1987. - S.296.
Davydov A.S. Solitons in molecular systems. - Kyiv: Naukova Dumka, 1984. – 288s.
Yanovsky V.V. Lectures on nonlinear phenomena. Vol.2. - Kharkov: Institute for Single Crystals, 2007. - 448s.
Zaslavsky G.M., Sagdeev R.Z., Usikov D.A., Chernikov A.A. Weak chaos and quasi-regular structure. - Moscow: Nauka, 1991. – 320s.
Kuklin V.M. The role of absorption and dissipation of energy in the formation of spatial structures in nonlinear nonequilibrium media // Ukrainian Journal of Physics, Reviews. – 2004. – Vol.1, № 1. - P.49-81.
Dorodnitsyn V.A., Elenin G.G. Symmetry in solutions of equations of mathematical physics. - Knowledge, 1984. - 175s.
Ovsyannikov L.V. Group analysis of differential equations. – Moscow: Nauka, 1978. - 240s.
Ibragimov N.H. Transformation groups in mathematical physics. – Moscow: Nauka, 1983. - 240s.
Nikolis G., Prigogine I. Self-organization in nonequilibrium systems. - Academic Press, 1979. - 240s.
Haken H. Synergetics. - World: 1980. – 404p.
Tsytovich V.N. Nonlinear effects in plasma. - Moscow: Nauka, 1967. - 288s.
Zakharov V.E. Wave collapse in physics continuum / Problems of physical kinetics and solid state physics. Ed. Sitenko AG , Academy of Sciences of the USSR. ITP. - Kiev: Naukova Dumka, 1990. - 488s.
Petviashvili V.I., Pokhotelov O.A. Solitary waves in the plasma and the atmosphere. – Moscow: Energoatomizdat, 1989. - 200s.
Klimontovich L., Vilhelmson H., Yakimenko I.P. Statisticheskaya theory of plasma-molecular systems. - Moscow: Mosk. University Press, 1990. – 224s.
L'vov V.S. Wave Turbulence under Parametric Excitations. Applications to Magnetics. - Springer-Verlag, 1994; L'vov V.S. Nonlinear Spin Waves. – Moscow: Nauka, 1987. - 270s.
Kuklin V.M., Panchenko I.P., Khakimov F.H. Multiwave processes in plasma physics. – Dushanbe: Donish, 1989. - 175s.
Buts V.A., Lebedev A.N. Coherent emission of intense electron beams. - M. Ed. LPI RAS, 2006. - 333p.
Moiseev S.S., Hovhannisyan K., Rutkevich P.B., Tur A.V., Khomenko G.A., Yanovsky V.V. Vortex dynamos in helical turbulence / Integrability and kinetic equations for solitons. Ed. Bar'yakhtar V.G., Zakharov V.E, Chernousenko V.M. - Kiev: Naukova Dumka, 1990. – 472s.
Horsthemke B., Lefebvre R. Noise-Induced Transitions. Translated from English. - Academic Press, 1987. – 400p.
Kirichok A.V., Kuklin V.M. Allocated Imperfections of Developed Convective Structures // Physics and Chemistry of the Earth Part A. - 1999, № 6. - P.533-538.
Belkin E.V. Gushchin I.V., Kirichok A.V. Kuklin V.M. Struсture transitions in the model Proctor-Sivashinsky // VANT, Ser. Plasma electronics and new methods of acceleration. - 2010, № 4(68). - S.296-298.
Kamyshanchenko N.V., Krasilnikov V.V., Neklyudov I.M., Parkhomenko A.A. Formation of spatial inhomogeneities in deformable irradiated materials // Condensed Matter and interphase boundaries. - 2001. - Vol.2, № 4. - P.339-341.
Krasil'nikov. V.V., Parkhomenko A.A., Savotchenko S.E. Internal-Stress distribution in deformed irradiated materials // Russian Metallurgy, Metally. – 2003. - №6. - P.559-565.
Bryk V.V., Krasilnikov V.V., Parkhomenko A., Savotchenko S. Self-organization mechanisms of radiation-induced defects in complex alloys of zirconium // Izvestiya. Metals. – 2005. - № 4. - P.81-87.
Krasilnikov V.V, Klepikov V.F., Parkhomenko A.A. Savotcheoko S.E. Features self-dislocation and vacancy ensemble in irradiated deformable materials // VANT. Ser. FRP and PM. - 2005. - № 5. - P.26-32.
Landau L.D., Lifshitz E.M. Theoretical Physics. Vol.5. Statistical physics. Part 1. - Moscow: Francis, London, 2002.
Yuhnovsky I.R. Phase transitions of the second order: Collective variables method. – Singapore: World Scientific, 1987. - 327p.
Patashinskii A.Z., Pokrovskii V.L. Fluctuation theory of phase transitions. - 2nd ed., Rev. Moscow: Science - Home Edition physical and mathematical literature, 1982.
Hartri D. Calculations of atomic structures. - New York: Oxford, 1960. – 256s.
Sleter J. Methods of self-consistent field for molecules and solids. - Academic Press, 1978. - 664p.
Suhl H. Effective Nuclear Spin Interactions in Ferromagnets // Phys Rev. - 1958. - Vol.109, №2. - P.606.
Zaharov V.E., Lvov V.S. Starobinets S.S. Spin-wave turbulence beyond the threshold of parametric excitation // UFN. - 1974. – T.114, №4. - S.609-654.
Bogolyubov N.N., Shirkov D.V. Introduction to the theory of quantized fields. 4th ed. - Moscow: Nauka, 1984.
Abrikosov A.A., Gorkov L.P., Dzyaloshinskii I.E. Methods of quantum field theory in statistical physics. – M: Fizmatgiz, 1963.
Vasilev A.N. Quantum field renormalization group in the theory of critical behavior and stochastic dynamics. - St. Petersburg: Petersburg Nuclear Physics Institute, 1998.
Guld H., Tobochnik Ya. Computer Simulation in Physics. - Mir, 1990.
Lazarev N.P. Molecular dynamics simulation of phase transitions in liquids and solids // The Journal of Kharkiv National University, physical series “Nuclei, Particles, Fields”. – 2007- №763. – Iss.1(33). - S.3-31.
Ziman J.M. Models of Disorder. - Cambridge: University Press, 1979.
White R.M., Geballe T.N .Long range order in solid. - New York: Academic Press, 1979, translation: Bumm P., Geball T. Long-range order in solids. - Academic Press, 1982.
Bonch-Bruevich V.D. et al. Electronic theory of disordered semiconductors. - Moscow: Nauka, 1981.
Efros A.L. Physics and geometry of the disorder. - Moscow: Nauka, 1982.
Rabinovich M.I., Fabrikant A.L., Tsimring L.S. A finite-dimensional disorder // Usp. Fiz. Nauk.. – 1992. - T.162, № 8. - S.1-42.
Senechal M., Quasicrystals and Geometry. - Cambridge: Cambridge Univ. Press, 1996.
Vekilov Y.F., Chernikov M.A. Quasicrystals // Usp. Fiz. Nauk. – 2010. - T.180, №6. - P.561-586.
Kuklin V.M. About interference nature of the formation of the fine structure of laser pulses and bursts of abnormal oscillation amplitude in the model Lighthill // VIII Khariton scientific reading. March 21-24, 2006, Sarov, Russia, Sat reports. - Sarov, 2006. - S.450-456; Effect of induced interference and the formation of spatial perturbation fine structure in nonequlibrium open-ended system // Problems of Atomic Science and Technology, Ser. Plasma electronics and new methods of acceleration. - 2006. - № 5(5). - S.63-68.
de Gennes P.G. The Physics of Uquid Crystals. - Oxford: Clarendon, 1974.
Eckmann J.-P., Procaccia I. The generation of spatio-temporal chaos in large aspect ratio hydrodynamics // Nonlinearity. – 1991. - Vol. 4, №2. - P.567-582.
Zaslavsky G.M., Sagdeev R.Z. Vvedenie in nonlinear physics: from the pendulum to turbulence and chaos, 1988. – 368s.
Landau, L.D. On the problem of turbulence // Doklady Akademii Nauk SSSR. – 1944. – T.44. - S.339-342.
Landau L.D., Lifshitz E.M. Theoretical physics. T.6. Hydrodynamics. - Moscow: Nauka, 1986. - 736s.
Hopf E. A mathematical example displaying features of turbulence // Comm. Pure and Appl. Math. - 1948. - Vol.1. - P. 303-322.
Swift J.V., Hohenberg P.C. Hydrodynamic fluctuations at the convective instability // Phys. Rev. A. – 1977. - Vol.15. - P.319.
Rabinovich M.I., Sushchik M.M. Regular and chaotic dynamics of structures in fluid flows // UFN. – 1990. - T.160, №1. - P.3-64.
Dimensions and Entropies in Chaotic Systems. / Ed. A Mayer-Kress. - Berlin: Springer-Verlag, 1986.
Chapman J., Proctor M.R.E. Nonlinear Rayleigh-Benard convection between poorly conducting boundaries // J. Fluid Mech. - 1980, №101. - P.759-765.
Gertsberg V., Sivashinsky G.E. Large cells in nonlinear Rayleigh-Benard convection // Prog. Theor. Phys. - 1981, №66. - P.1219-1229.
Malomed B.A., Nepomniachtchi A.A., Tribel'skii M.P. Two-dimensional quasi-periodic structures in nonequilibrium systems // Lett. – 1989. - Vol.96. - P.684-699.
Pismen, L., Inertial effects in long-scale thermal convection // Phys. Lett. A. – 1986. – Vol.116. – P.241-243.
Kirichok A.V. Kuklin V.M., Panchenko I.P., Moiseev S.S., Pismen L. Dynamics of large-scale vortices in the mode of convective instability / Inter. Conf. "Physics in Ukraine", Kiev, 22-27 June 1993. Proc. Contr. Pap. ITP, 1993. - P. 76-80.
Kirichok A.V. Kuklin V.M., Panchenko I.P. On the possibility of dynamo mechanism in non-equilibrium convective environment // Reports of the National Academy of Sciences. - 1997, № 4. - P.87-92.
Belkin E.V., Gushchin I.V. A mathematical model of convection of the liquid layer with a temperature gradient / Nauchno-tehnicheskaja konferencija s mezhdunarodnym uchastiem «Komp'juternoe modelirovanie v naukoemkih tehnologijah», Kharkov, 2010. - Vol.1. - P.39-40.
Busse F.G. Hydrodynamic instability and transition to turbulence / Ed. X. Sweeney, J. Gollaba. – M: Mir, 1984. – 124s.
Gershuni G.Z. Zhukovitsky E.M. Convective stability of incompressible fluid. - Moscow: Nauka, 1972. – 392s.
Whitehead J.A. Dislocations in convection and the onset of chaos // Phys. Fluids. – 1983. - Vol.26, №10. - P.2899.
Siggia E.D., Zippelius A. Dynamics of defects in Rayleigh Benard convection // Phys. Ref. A. – 1981. - Vol.24(2). - P.1036-1049.
Bodenschatz E., de Bruyn J.R., Ahlers G., Connell D. Experiments on three systems with non-variatinal aspects. - Preprint. - Santa Barbara, 1991.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
