Abstract
Unsteady-state Bénard–Marangoni convection in large-scale liquid flows with a linear temperature distribution at the layer boundaries has been investigated by the boundary element method. Two variants of boundary conditions are considered. In the case of temperature gradient components distributed at both boundaries, the boundary problem cannot be reduced to a one-dimensional one. The structure of layered convective flows has been studied. It has been demonstrated that the initial and boundary value problems considered here describe convective liquid counterflows and the formation of extremum (local and global) values of temperature fields. The existence of stagnant points (in which the liquid velocity is zero) inside the layer of the moving nonisothermal liquid has been discovered.
Similar content being viewed by others
References
Kutepov, A.M., Polyanin, A.D., and Zapryanov, Z., Khimicheskaya gidrodinamika (Chemical Fluid Dynamics), Moscow: Byuro Kvantum, 1996
Polyanin, A.D., Kutepov, A.M., Vyazmin, A.V., and Kazenin, D.A., Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Boca Raton, Fla.: Taylor and Francis, 2002
Landau, L.D. and Lifshits, E.M., Teoreticheskaya fizika: Gidrodinamika (Course of Theoretical Physics: Fluid Mechanics), Moscow: Fizmatlit, 2006, vol. 6, 5th ed.
Gershuni, G.Z. and Zhukhovitskii, E.M., Konvektivnaya ustoichivost’ neszhimaemoi zhidkosti (Convective Stability of Incompressible Liquids), Moscow: Nauka, 1972
Aristov, S.N., Knyazev, D.V., and Polyanin, A.D., Exact solutions of the Navier–Stokes equations with the linear dependence of velocity components on two space variables, Theor. Found. Chem. Eng., 2009, vol. 43, no. 5, pp. 642–662.
Polyanin, A.D. and Aristov, S.N., A new method for constructing exact solutions to three-dimensional Navier-Stokes and Euler equations, Theor. Found. Chem. Eng., 2011, vol. 45, no. 6, pp. 885–890.
Napolitano, L.G., Plane Marangoni–Poiseuille flow of two immiscible fluids, Acta Astronaut., 1980, vol. 7, no. 4, pp. 461–478.
Goncharova, O. and Kabov, O., Gas flow and thermocapillary effects of fluid flow dynamics in a horizontal layer, Micrograv. Sci. Technol., 2009, vol. 21, suppl. 1, pp. 129–137.
Andreev, V.K., Birikh solutions of convection equations and some of their generalizations, Preprint of Inst. Computational Mathematics, Siberian Branch, Ross. Acad. Sci., Krasnoyarsk, 2010, no. 1–10.
Aristov, S.N. and Shvarts, K.G., Vikhrevye techeniya advektivnoi prirody vo vrashchayushchemsya sloe zhidkosti (Advective Eddy Flows in a Rotating Liquid Layer), Perm: Perm. Gos. Univ., 2006
Aristov, S.N. and Shvarts, K.G., Vikhrevye techeniya v tonkikh sloyakh zhidkosti (Eddy Flows in Thin Liquid Layers), Kirov: Vyat. Gos. Univ., 2011
Ingel’, L.Kh. and Kalashnik, M.V., Nontrivial features in the hydrodynamics of seawater and other stratified solutions, Phys.-Usp., 2012, vol. 55, no. 4, pp. 356–381.
Andreev, V.K. and Bekezhanova, V.B., Stability of nonisothermal fluids (review), J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 2, pp. 171–184.
Ostroumov, G.A., Svobodnaya konvektsiya v usloviyakh vnutrennei zadachi (Free Convection under the Condition of the Internal Problem), Moscow: Gostekhteorizdat, 1952
Birikh, R.V., Thermocapillary convection in a horizontal layer of liquid, J. Appl. Mech. Tech. Phys., 1966, no. 7, pp. 43–49.
Aristov, S.N. and Prosviryakov, E.Yu., On laminar flows of planar free convection, Nelineinaya Dinam., 2013, vol. 9, no. 4, pp. 651–657.
Schwarz, E.G., Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries, Fluid Dyn., 2014, vol. 49, no. 4, pp. 438–442.
Pukhnachev, V.V., Unsteady-state analogues of the Birikh solution, Izv. Altaisk. Gos. Univ., 2011, nos. 1–2, p. 62.
Brebbia, C.A., Telles, J.C.F., and Wrobel, L.C., Boundary Element Techniques, Berlin: Springer, 1984
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.N. Aristov, E.Yu. Prosviryakov, L.F. Spevak, 2016, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2016, Vol. 50, No. 2, pp. 137–146.
Rights and permissions
About this article
Cite this article
Aristov, S.N., Prosviryakov, E.Y. & Spevak, L.F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows. Theor Found Chem Eng 50, 132–141 (2016). https://doi.org/10.1134/S0040579516020019
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040579516020019