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Thermal Cnvection of Two Immiscible Liquids in a 3D Channel with a Velocity Field of a Special Type
- Authors: Andreev V.K.1, Lemeshkova E.N.1
-
Affiliations:
- Institute of Computational Modelling SB RAS
- Issue: Vol 87, No 2 (2023)
- Pages: 200-210
- Section: Articles
- URL: https://journals.rcsi.science/0032-8235/article/view/138852
- DOI: https://doi.org/10.31857/S0032823523020029
- EDN: https://elibrary.ru/TYSYHA
- ID: 138852
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Abstract
The three-dimensional stationary flow of two immiscible liquids in a layer bounded by solid parallel walls is investigated. The upper wall is thermally insulated, and the lower one has a temperature field quadratic in horizontal coordinates. Velocity fields in liquids have a special form: their horizontal components are linear in the coordinates of the same name. The resulting conjugate boundary value problem in the framework of the Oberbeck–Boussinesq model is inverse and is reduced to a system of ten integro-differential equations. For small Marangoni numbers (creeping current), the problem is solved analytically. The nonlinear problem is solved by the tau method. It is shown that the solution of the nonlinear problem with a decrease in the Marangoni number is approximated by the solution of the creeping flow problem. The analysis of the influence of physical and geometric parameters, as well as the behavior of temperature on the substrate, on the structure of convection in layers is carried out.
About the authors
V. K. Andreev
Institute of Computational Modelling SB RAS
Author for correspondence.
Email: andr@icm.krasn.ru
Russia, Krasnoyarsk
E. N. Lemeshkova
Institute of Computational Modelling SB RAS
Author for correspondence.
Email: elena_cher@icm.krasn.ru
Russia, Krasnoyarsk
References
- Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder // Dinglers Poliytech. J., 1911, vol. 326, pp. 321–440.
- Howann F. Der Einfluss grosser Zahigkeit bei der Stromung um den Zylinder und um die Kugel // Zeitschrift für Angewandte Mathematik und Mechanik, 1936, vol. 16, pp. 153–164.
- Howarth L. The boundary layer in three-dimensional flow. Pt. II. The flow near a stagnation point // Lond. Edinb. Dubl. Phil. Mag.: Ser. 7, 1951, vol. 42, no. 335, pp. 1433–1440.
- Davey A. Boundary-layer flow at a saddle point of attachment // J. Fluid Mech., 1961, vol. 10, no. 4, pp. 593–610.
- Wang C.Y. Axisymmetric stagnation flow on a cylinder // Q. Appl. Math., 1974, vol. 32, no. 2, pp. 207–213.
- Gorla R.S.R. Unsteady laminar axisymmetric stagnation flow over a circular cylinder // Develop. Mech., 1977, vol. 9, pp. 286–288.
- Bekezhanova V.B., Andreev V.K., Shefer I.A. Influence of heat defect on the characteristics of a two-layer flow with the Hiemenz-type velocity // Interfac. Phenom.&Heat Transfer., 2019, vol. 7, iss. 4, pp. 345–364.
- Lin C.C. Note on a class of exact solutions in magnetohydrodynamics // Arch. Rat. Mech. Anal., 1958, vol. 1, pp. 391–395.
- Sidorov A.F. On two classes of solutions of equations of fluid and gas mechanics and their connection with the theory of traveling waves // J. Appl. Mech.&Tech. Phys., 1989, no. 2, pp. 34–40.
- Andreev V.K. On a creeping 3D convective motion of fluids with an isothermal interface // J. Sib. Fed. Univ. Math.&Phys., 2020, vol. 13, no. 6, pp. 661–669.
- Azanov A.A., Andreev V.K. 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 Math.&Math. Educ. Herzen readings – 2021. Mater. Sci. Conf. St. Petersburg: A.I. Herzen SPU Pub., 2021. pp. 42–54.
- Andreev V.K., Lemeshkova E.N. Two-layer steady creeping thermocapillary flow in a three-dimensional channel // J. Appl. Mech.&Tech. Phys., 2022, vol. 63, no. 1, pp. 82–88.
- Andreev V.K. On a creeping 3D Convective Motion of Fluids with an Isothermal Interface // J. Sib. Fed. Univ. Math.&Phys., 2020, vol. 13 (6), pp. 661–669.
- Andreev V.K., Gaponenko Yu.A., Goncharova O.N., Pukhnachev V.V. Mathematical Models of Convection. Berlin; Boston: De Gruyter, 2020.
- Aristov S.N., Knyazev D.V., Polyanin A.D. Exact solutions of the Navier–Stokes equations with the linear dependence of velocity components on two space variables // Theor. Found. Chem. Engng., 2009, vol. 43, no. 5, pp. 547–566.
- Joseph D. Stability of Fluid Movements. Moscow: Mir, 1981. 638 p.
- Pukhnachev V.V. Model of viscous layer deformation by thermocapillary forces // Eur. J. Appl. Math., 2002, vol. 13, no. 2, pp. 205–224.
- Rezanova E.V. Numerical modelling of heat transfer in the layer of viscous incompressible liquid with free boundaries // EPJ Web Conf., 2017, no. 159, pp. 00047.
- Zeytounian R. Kh. The Benard–Marangoni thermocapillary instability problem // UFN, 1998, vol. 168, no. 3, pp. 259–286.
- Bogdanov S.N., Burtsev S.I., Ivanov O.P., Kupriyanova A.V. Refrigeration Equipment. Air Conditioning. Properties of Substances. / Ed. by Bogdanov S.N. St. Petersburg: SPbGAHPT, 1999. 320 p.
- Fletcher C.A.J. Computational Galerkin Method. Springer, 1984.
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