Thermocapillary convection of immiscible liquid in a three-dimensional layer at low Marangoni numbers

Viktor K. Andreev

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 2, pp. 195-206.

Résumé

The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by solid flat walls is studied. The upper wall is thermally insulated, and a non-stationary temperature field is set on the lower wall. Liquids are assumed to be immiscible and complex conjugation conditions are set at the flat interface between them. The evolution of this system is described by the Oberbeck-Boussinesq equations in each fluid. The solution of this problem is sought in the class of velocity fields linear in two coordinates, and temperature fields are quadratic functions of the same coordinates. In this case, the problem is reduced to a system of 10 nonlinear integro-differential equations. It is conjugate and inverse with respect to 4 functions of time. To find them, integral redefinition conditions are set. The physical meaning of these conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection in a two-layer system that occurs near the temperature extremum point on the lower solid wall. For small Marangoni numbers, the problem is approximated by a linear one (the Marangoni number plays the role of the Reynolds number for the Navier-Stokes equations). A stationary solution to this problem has been found. The linear nonstationary problem is solved by the Laplace transform method, and the temperature can have discontinuities of the 1st kind (change by a jump). In Laplace images, the solution is obtained in quadratures. It is proved that with increasing time, it tends to stationary mode if the temperature on the lower wall stabilizes over time. The evolution of the behavior of the velocity field in the transformer oil-water system has been studied using the numerical inversion of the Laplace transform.

Détail

Keywords: Oberbeck-Boussinesq equations, Marangoni number, thermocapillarity, inverse problem
Mots-clés : interface, Laplace transform.

Bibliographie

[1] C.C.Lin, “Note on a class of exact solutions in magnetohydrodynamics”, Arch. Rational Mech. Anal., 1 (1958), 391–395 | DOI | MR | Zbl

[2] A.F.Sidorov, “On two classes of solutions of equations of fluid and gas mechanics and their connection with the theory of traveling waves”, Applied Mechanics and Technical Physics, 1989, no. 2, 34–40 | MR

[3] V.V.Pukhnachev, “Model of viscous layer deformation by thermocapillary forces”, Eur. Journal Appl. Math., 13:2 (2002), 205–224 | DOI | MR | Zbl

[4] V.K.Andreev, Yu.A.Gaponenko, O.N.Goncharova, V.V.Pukhnachev, Mathematical Models of Convection, De Gruyter, Berlin–Boston, 2020 | MR | Zbl

[5] E.V.Rezanova, “Numerical medelling of heat transfer in the layer of viscous incompressible liquid with free boundaries”, EPJ Web of Conference, 159 (2017), 00047 | DOI

[6] S.N.Aristov, D.V.Knyazev, A.D.Polyanin, “Exact solutions of the Navier-Stokes equations with linear dependence of the velocity components on two spatial variables”, Theoretical Foundations of Chemical Engineering, 43:5 (2009), 547–566 | DOI

[7] A.A.Azanov, V.K.Andreev, “Solving the problem of the creeping motion of a fluid with a free boundary with a special velocity field in a three-dimensional strip”, Some actual problems of modern mathematics and mathematical education, Herzen Readings — 2021, Materials of the scientific conference, Publishing House of the A.I. Herzen State Pedagogical University. Publishing House of the VVM, St. Petersburg, 2021, 42–54

[8] V.K.Andreev, E.N.Lemeshkova, “Two-layer steady creeping thermocapillary flow in a three-dimensional channel”, Journal of Applied Mechanics and Technical Physics, 63:1 (2022), 82–88 | DOI | MR | Zbl

[9] V.K.Andreev, “On a creeping 3D Convective Motion of Fluids with an Isothermal Interface”, Journal of Siberian Federal University. Mathematics $\$ Physics, 13:6 (2020), 661–669 | DOI | MR

[10] V.K.Andreev, “The solution of three-dimensional equations of thermal convection and its interpretation”, Some actual problems of modern mathematics and mathematical education, Herzen Readings — 2020, Materials of the scientific conference, Publishing House of the A. I. Herzen State Pedagogical University, VVM, St. Petersburg, 2021, 4–8

[11] V.K.Andreev, E.N.Lemeshkova, “Thermal convection of two immiscible liquids in a three-dimensional channel with a velocity field of a special type”, PMM, 87:2 (2023), 200–210 (in Russian) | DOI | MR

[12] V.K.Andreev, E.N.Lemeshkova, Linear problems of convective motions with interface surfaces, Sib. Fed. Univ., Krasnoyarsk, 2018

[13] R.H.Zeitunyan, “The problem of thermocapillary instability of Benara-Marangoni”, UFN, 168 (1998), 259–286 (in Russian) | DOI

[14] M.A.Lavrentiev, B.V.Shabat, Methods of the theory of functions of a complex variable, Nauka, M., 1973 | MR

[15] N.B.Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Nauka, 1972