A. V. Gorshkov, E. Y. Prosviryakov

COMPLEX STATIONARY CONVECTION WITH THIRD-KIND BOUNDARY CONDITIONS AT THE BOUNDARIES OF A FLUID LAYER

In this paper presents analytical solutions of the Oberbeck-Boussinesq system describing layered Benard-Marangoni convection flows in an incompressible viscous fluid. On the free surface a third-kind boundary condition is used, namely Newton’s heat transfer law.

Bibliography:

1. Thomson J. On certain curious motions observable on the surfaces of wine and other alcoholic liquours. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. X, fourth series. London, Taylor and Francis, Red Lion Court, Fleet Street, 1855, pp. 330–333.
2. Bénard H. Les Tourbillons cellulaires dans une nappe liquide propageant de la chaleur par convection, en régime permanent. Paris, Gauthier-Villars, 1901.
3. B'enard H. Etude exp'erimentale des courants de convection dans une nappe liquide. R'egime permanent: tourbillons cellulaires. J. Phys. Theor. Appl., 1900, vol. 9, no. 1, pp. 513–524.
4. Getling A.V. Formation of the spatial structures of Rayleigh-Benard convection. Uspekhi Fizicheskikh Nauk, 1991, vol. 161, no. 9, pp. 1–80. (In Russian).
5. Landa P.S. Avtokolebaniya v raspredelennykh sistemakh [Self-Excited Vibrations in Distribution Systems]. M., Knizhnyi Dom Publ., 2010, pp. 320. (In Russian).
6. Marangoni С. Sull espansione delle goccie di un liquido galleggiante sulla superficie di altro liquid. Pavia: Tipografia dei fratelli Fusi, 1865.
7. Birikh R.V. Thermocappillary convection on a horizontal fluid layer. J. Appl. Mech. Technol. Phys., 1966, vol. 7, iss. 3, pp. 43–44. DOI: 10.1007/BF00914697.7.
8. Ostroumov G.A. Svobodnaya konvektsiya v usloviyakh vnutrenney zadachi [Free Convection under Interior Problem Conditions]. M.; L., Gostekhteoretizdat Publ., 1952, 256 p. (In Russian).
9. Gershuni G.Z., Zhukhovitsky E.M. Konvektivnaya ustoichivost neszhimaemoy zhidkosti [Convective Stability of Incompressible Fluid]. M., Nauka Publ., 1972. (In Russian).
10. Napolitano L.G. Plane Marangoni-Poiseuille flow of two immissible fluids. Acta Astronautica, 1980, vol. 7, iss. 4–5, pp. 461–478. DOI: 10.1016/0094-5765(80)90036-3.
11. Sidorov A.F. On one class of solutions to equations of gas dynamics and natural convection. In: Chislennye i analiticheskie metody resheniya zadach sploshnoi sredy [Numerical and Analytical Methods of Solving Continuum Problems]. Sverdlovsk, UNTs AN SSSR Publ., 1981, pp. 101–117. (In Russian).
12. Sidorov A.F. Two classes of solution of the fluid and gas mechanics equatins and their con[1]nection to traveling wave theory. Journal of Applied Mechanics and Technical Physics, 1989, vol. 30, iss. 2, pp. 197–203. DOI: 10.1007/BF00852164.
13. Kirdyashkin A.G. Teplovye gravitatsionnye techeniya i teploobmen v astenosfere [Heat and Gravitational Flows and Heat Exchange in the Asthenosphere]. Novosibirsk, Nauka, SO RAN Publ., 1989, 81 p. (In Russian).
14. Goncharova O., Kabov O. Gas flow and thermocapillary effects of fluid flow dynamics in a horizontal layer. Microgravity Sci. Technol., 2009, vol. 21, iss. 1 suppl., pp. 129–137. DOI: 10.1007/s12217-009-9108-x.
15. Andreev V.K. Resheniya Birikha uravneniy konvektsii i nekotorye ego obobshcheniya [Birikh Solutions to Convection Equations and Some of its Extensions]. Krasnoyarsk, IBM SO RAN Publ., 2010, 68 p. (In Russian).
16. Ingel L.Kh., Kalashnik Maksim V. Nontrivial features in the hydrodynamics of seawater and other stratified solutions. Physics-Uspekhi, vol. 55, no. 4, pp. 356–381. DOI: 10.3367/UFNe.0182.201204b.0379.
17. Aristov S.N. Shvarts K.G. Vikhrevye techeniya v tonkikh sloyakh zhidkosti [Vortex Flows in Thin Fluid Layers]. Kirov, VyatGU Publ., 2011, 207 p. (In Russian).
18. Aristov S.N., Prosviryakov E.Yu. On layered flows of plane free convection. Nelineynaya dinamika, 2013, vol. 9, no. 4, pp. 651–657. (In Russian).
19. Shvarts K.G. Plane-parallel advective flow in a horizontal layer of incompressible fluid with solid boundaries. Izvestiya RAN. MZhG, 2014, no. 4, pp. 26–30. (In Russian).
20. Yudovich V.I. On the challenges and prospects of advanced mathematical hydrodynamics. Uspekhi mekhaniki, 2002, vol. 1, no. 1, pp. 61–102. (In Russian).
21. Aristov S.N., Shvarts K.G. Vikhrevye techeniya advektivnoy prirody vo vrashchayushchemsya sloe zhidkosti [Vortex Flows of Advective Nature in a Rotating Fluid Layer]. Perm, Izd-vo PGU Publ., 2006, 154 p. (In Russian).
22. Aristov S.N., Shvarts K.G. Advective flow in a rotating liquid film. Journal of Applied Mechanics and Technical Physics, 2016, vol. 57, no. 1, pp. 188–194. DOI: 10.1134/S0021894416010211.
23. Andreev V.K., Bekezhanova V.B. Stability of non-isothermal fluids (Review). Journal of Applied Mechanics and Technical Physics, 2013, vol. 54, iss. 2, pp. 171–184. DOI: 10.1134/S0021894413020016.
24. Pukhnachev V.V. Non-stationary analogs of the Birikha solution. Izvestiya Altaiskogo gos. Universiteta, 2011, no. 1–2, pp. 62–69. (In Russian).
25. Nikitin N.V., Nikitin S.A., Polezhaev V.I. Convective instabilities in the hydrodynamic model of crystal growth by the Chokhralsky method. Uspekhi Mekhaniki, 2003, vol. 2, no. 4, pp. 63–105. (In Russian).
26. Aristov S.N., Zimin V.D. Advektivnye volny vo vrashchayushchemsya sharovom sloe [Adequate Waves in a Rotating Spherical Layer]. IMMS, UNTs, AN SSSR Publ., Sverdlovsk, 1986, 50 p. (In Russian).
27. Aristov S.N., Frik P.G. Dinamika krupnomasshtabnykh techeniy v tonkikh sloyakh zhidkosti [Dynamics of Large-Scale Flows in Thin Fluid Layers]. IMMS, UNTs, AN SSSR Publ., Sverdlovsk, 1987, 48 p. (In Russian).
28. Aristov S.N., Shvarts K.G. Convective heat exchange in a plane layer of incompressible fluid under localized heating. Izvestiya RAN. MZhG, 2013, no. 3, pp. 53–58. (In Russian).
29. Aristov S.N., Prosviryakov E.Yu. On one class of analytical solutions to stationary axisymmetric Benard-Marangoni convection of viscous incompressible fluid. Vestnik Sam. gos. Un-ta. Ser. fiz.-mat. nauki, 2013, no. 3 (32), pp. 110–118. (In Russian).
30. Aristov S.N., Knyazev D.V. Localized convective flows in a layer of nonuniformly heated fluid. Izvestiya RAN. MZhG, 2014, no. 5, pp. 5. (In Russian).
31. Aristov S.N., Frik P.G. Large-scale turbulency in a thin layer of nonisothermal rotating fluid. Izvestiya RAN SSSR. MZhG, 1988, no. 4, pp. 48–55. (In Russian).
32. Landau L.D. Teoreticheskaya fizika. T. 6. Gidrodinamika [Theoretical Physics. Vol. 6. Hydrodynamics]. М., Nauka Publ., 2006, 736 p. (In Russian).
33. Lin C.C. Note on a class of exact solutions in magneto-hydrodynamics. Arch. Rational Mech. Anal., 1958, vol. 1, pp. 391–395.
34. Aristov S.N., Knyazev D.V., Polyanin A.D. Exact solutions to Navier-Stokes equations with the linear dependence of the velocity components on two spatial variables. Teoreticheskie osnovy khim. tekhnologii, 2009, vol. 43, no. 5, pp. 547–566. (In Russian).
35. Gantmakher F.R. Teoriya matrits [Theory of Matrices]. M., Nauka Publ., 1967, 576 p. (In Russian).