A system of equations of unsteady spatial free convection of an incompressible viscous fluid in the Boussinesq approximation is considered. The analysis is based on the methods of reduction of linear and nonlinear partial differential equations (PDEs) and systems of PDEs to ordinary differential equations (ODEs) and systems of ODEs. These methods were proposed by the authors earlier, and their general principles are given in the paper. The methods are based on the construction of a system of equations of characteristics for a first-order PDE (the basic equation). This equation is constructed in a certain way by analyzing the original system of equations. The reductions lead to ODEs or systems of ODEs in which an independent variable $\psi$ is such that the equation $\psi(x,y,z,t)=\mathrm{const}$ defines a level surface for all unknown functions of the original system of PDEs. The methods are applicable to PDEs and systems of PDEs regardless of their type. The Oberbeck–Boussinesq equations are reduced to a system of ODEs with a functional arbitrariness, and an exact solution with a constant arbitrariness is found for the original system. The functional arbitrariness in the constructed reduction also yielded a system of ODEs in which the temperature $T$ is an independent variable. For this system exact solutions are found. A possible (vortex or vortex-free) motion of an incompressible fluid with free convection is analyzed. The cases of vortex and vortex-free motion of the fluid are identified. An exact solution defining a vortex-free motion of the fluid is written as a result of reductions for the original system of PDEs.
Keywords: free convection of viscous fluid, Oberbeck–Boussinesq equations, partial differential equations, reductions, exact solutions
Received March 7, 2023
Revised April 24, 2023
Accepted May 15, 2023
Oleg Nikolaevich Ulyanov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: uon@imm.uran.ru
Lyudmila Ilyinichna Rubina, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: rli@imm.uran.ru
REFERENCES
1. Sidorov A.F. On a class of solutions of the gas dynamics and natural convection equations. In: Numerical and analytical methods of solving problems of a continuous mechanics, Collect. Article, Sverdlovsk, 1981, pp. 101–117 (in Russian).
2. Sidorov A.F. Analytical methods for constructing solutions in nonlinear problems of spatial natural convection (survey report). In: Sixth All-Soviet School–Seminar on models of continuum mechanics, Collect. Article, Alma-Ata, Novosibirsk, 1981, pp. 236–250 (in Russian).
3. Sidorov A.F., Khairullina O.B. Application of Bernstein polynomials for approximate solution of the problem of natural convection in a horizontal layer. In: Approximate methods for solving boundary value problems of continuum mechanics, Collect. Article, Sverdlovsk, 1985, pp. 52–63 (in Russian).
4. Sidorov A.F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory. J. Appl. Mech. Tech. Phys., 1989, vol. 30, no. 2, pp. 197–203. doi: 10.1007/BF00852164
5. Sidorov A.F., Khairullina O.B. Calculation of hexagonal convection in Benard cells using special trigonometric series. In: Approximate methods of investigation of nonlinear problems of continuum mechanics, Collection of scientific papers, Sverdlovsk, 1992, pp. 35–50 (in Russian).
6. Boussinesq J. Theorie analitique de la chaleur. Vol. 2. Paris, GauthierVillars, 1903, 625 p.
7. Oberbeck A. Ueber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen in Folge von Temperaturdifferenzen. Annalen der Physik, 1879, vol. 243, iss. 6, pp. 271–292. doi: 10.1002/andp.18792430606
8. Andreev V.K., Gaponenko Yu.A., Goncharova O.N., Pukhnachev V.V. Mathematical models of convection, Berlin, Boston, De Gruyter, 2012, 417 p. doi: 10.1515/9783110258592 . Original Russian text was published in Andreev V.K., Gaponenko Yu.A., Goncharova O.N., Pukhnachev V.V. Sovremennye matematicheskie modeli konvektsii, Moscow, Phys. Math. Liter Publ., 2008, 368 p. ISBN 978-5-9221-0905-5 .
9. Mayeli P., Sheard G.J. Buoyancy-driven flows beyond the Boussinesq approximation: a brief review. Int. Commun. Heat and Mass Transfer, 2021, vol. 125, article no. 105316. doi: 10.1016/j.icheatmasstransfer.2021.105316
10. Lappa M. Incompressible flows and the Boussinesq approximation: 50 years of CFD. Comptes Rendus. Mecanique, 2022, vol. 350, pp. 1–22. doi: 10.5802/crmeca.134
11. Ovsiannikov L.V. Group analysis of differential equations. NY, Acad. Press, 1982, 416 p. doi: 10.1016/C2013-0-07470-1 . Original Russian text was published in Ovsiannikov L.V., Gruppovoi analiz differentsial’nykh uravnenii, Moscow, Nauka Publ., 1978, 399 p.
12. Pukhnachev V.V. Group-theoretical methods in convection theory. AIP Conf. Proc., 2011, vol. 1404, pp. 27–38. doi: 10.1063/1.3659901
13. Sidorov A.F., Shapeev V.P., Yanenko N.N. Metod differentsial’nykh svyazei i ego prilozheniya k gazovoi dinamike [The method of differential constraints and its applications to gas dynamics], Novosibirsk, Nauka Publ., 1984, 272 p.
14. Ostroumov G.A. Free convection under the condition of the internal problem. Washington, NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, 1958.
15. Birikh R.V. Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys., 1966, vol. 7, no. 3, pp. 43–44. doi: 10.1007/BF00914697
16. Andreev V. K., Stepanova I.V. Ostroumov–Birikh solution of convection equations with nonlinear buoyancy force. Appl. Math. Comput., 2014, vol. 228, pp. 59–67. doi: 10.1016/j.amc.2013.11.002
17. Barna I.F., Matyas L. Analytic self-similar solutions of the Oberbeck–Boussinesq equations. Chaos, Solitons and Fractals, 2015, vol. 78, pp. 249–255. doi: 10.1016/j.chaos.2015.08.002
18. Burmasheva N.V., Prosviryakov E.Y. Exact solutions to the Oberbeck–Boussinesq equations for shear flows of a viscous binary fluid with allowance made for the Soret effect. Izvestiya Irkutskogo Gos. Univ. Ser. Mathematics, 2021, vol. 37, pp. 17–30. doi: 10.26516/1997-7670.2021.37.17
19. Rubina L.I., Ul’yanov O.N. One method for solving systems of nonlinear partial differential equations. Proc. Steklov Inst. Math., 2015, vol. 288, suppl. 1, pp. 180–188. doi: 10.1134/S0081543815020182
20. Ulyanov O.N., Rubina L.I. On the reduction of one system of magnetic gas dynamics system of equations to systems of ordinary differential equations. Vestnik Natsional’nogo Issledovatel’skogo Yadernogo Univ. “MIFI”, 2022, vol. 11, no. 2, pp. 122–132 (in Russian). doi: 10.56304/S2304487X22020122
21. Clarkson P.A., Ludlow D.K., Priestley T.J. The classical, direct, and nonclassical methods for symmetry reductions of nonlinear partial differential equations. Methods and Appl. of Anal., 1997, vol. 4, no. 2, pp. 173–195. doi: 10.4310/MAA.1997.v4.n2.a7
22. Polyanin A.D. Reductions and new exact solutions of the convective heat and mass transfer equations with a nonlinear source. Vestnik natsional’nogo issledovatel’skogo yadernogo universiteta “MIFI”, 2018, vol. 7, no. 6, pp. 458–469 (in Russian). doi: 10.1134/S2304487X18060093
23. Courant R., Hilbert D. Methods of mathematical physics. Vol. 2: Partial differential equations, NY, Interscience, 1962, 830 p. Translated to Russian under the title Metody matematicheskoi fiziki: Uravneniya v chastnykh proizvodnykh, Moscow, Mir Publ., 1964, 831 p.
24. Kochin N.E., Kibel I.A., Roze N.V. Teoreticheskaya gidromekhanika [Theoretical Hydromechanics], Moscow, Fizmatgiz Publ., 1963, 584 p.
Cite this article as: O.N. Ul’yanov, L.I. Rubina. On some classes of free convection motions. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 2, pp. 189–206; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S239–S256.
