L. S. Goruleva, I. I. Obabkov, and E. Yu. Prosviryakov

EXACT SOLUTIONS TO THE OBERBECK–BOUSSINESQ EQUATIONS FOR DESCRIBING MULTILAYER FLUID FLOWS IN THE STOKES APPROXIMATION

The flow of viscous incompressible fluids in engineering devices, in technological and natural processes is characterized by the stratification of hydrodynamic fields. Conventionally, the stratification of the velocity field and the pressure field is studied for isothermal flows. If fluid motion occurs in a thermal field, temperature is an important characteristic of an incompressible fluid. Convective fluid flow has a very complex topological structure of hydrodynamic fields due to the temperature dependence of density. As is known, in the description of convection in the Boussinesq approximation, the dependence of density on the spatial coordinate and time is ignored in the continuity equation, which is then transformed into the incompressibility equation. Field and experimental observations of fluid flow allow us to identify flow regions with discrete density distribution along one of the coordinates. Such fluids are referred to as stratified fluids in the scientific literature. Their mathematical description is significantly complicated since it is necessary to solve the Oberbeck–Boussinesq equations for each layer and join the analytical or numerical solutions between the layers and the boundaries. For applied studies of convective flows, the Stokes approximation for the total derivative of a vector or scalar function is often introduced. The paper considers the construction of exact Lin–Sidorov–Aristov solutions for describing slow (creeping) flows of a non-uniformly heated stratified fluid. In this case, hydrodynamic fields are described by special polynomials with functional arbitrariness. It is shown how the calculations of unknown coefficients can be automated to form hydrodynamic fields (velocities and temperatures). For steady-state Stokes-type flows, an exact solution of the Oberbeck–Boussinesq system is written out explicitly (by means of formulas).

References:

1. Ostroumov, G.A. Free convection under the condition of the internal problem. Technical Memorandum Ser., 1407, National Advisory Committee for Aeronautics, Washington, 1958.
2. Gershuni, G.Z. and Zhukhovitskii, E.M. Convective Stability of Incompressible Liquid, Wiley, Keter Press, Jerusalem, 1976.
3. Gershuni, G.Z., Zhukhovitskii, E.M., and Nepomnyashchii, A.A. Ustoychivost konvektivnykh techeniy [Stability of Convective Flows]. Nauka Publ., Moscow, 1989, 320 p. (In Russian).
4. Getling, A.V. Formation of spatial structures in Rayleigh–Bénard convection. Soviet Physics Uspekhi, 1991, 34 (9), 737–776. DOI: 10.1070/PU1991v034n09ABEH002470.
5. Andreev, V.K., Gaponenko, Ya.A., Goncharova, O.N., and Pukhnachev, V.V. Mathematical Models of Convection, De Gruyter, Berlin, Boston, 2012, 417 p. DOI: 10.1515/9783110258592.
6. Aristov, S.N. and Schwarz, K.G. Vikhrevye techeniya v tonkikh sloyakh zhidkosti [Vortex Flows in Thin Layers of Liquid]. VyatGU Publ., Kirov, 2011, 206 p. (In Russian).
7. Birikh, R.V. Thermocapillary convection in a horizontal layer of liquid. Journal of Applied Mechanics and Technical Physics, 1966, 7 (3), 43–44. DOI: 10.1007/bf00914697.
8. Shliomis, M.I. and Yakushin, V.I. The convection in a two-layer binary system with evaporation. Uchenye Zapiski Permskogo Gosuniversiteta, Seriya Gidrodinamika, 1972, 4, 129–140. (In Russian).
9. Gershuni, G.Z. On the stability of plane convective motion of a fluid. Zh. Tekh. Fiz., 1953, 23 (10), 1838–1844. (In Russian).
10. Batchelor, G.K. Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math., 1954, 12 (3), 209–233. DOI: 10.1090/qam/64563.
11. Schwarz, K.G. Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries. Fluid Dynamics, 2014, 49 (4), 438–442. DOI: 10.1134/S0015462814040036.
12. Knyazev, D.V. Two-dimensional flows of a viscous binary fluid between moving solid boundaries. Journal of Applied Mechanics and Technical Physics, 2011, 52 (2), 212–217. DOI: 10.1134/ S0021894411020088.
13. Aristov, S.N. and Prosviryakov, E.Yu. On laminar flows of planar free convection. Nelineynaya Dinamika, 2013, 9 (4), 651–657. (In Russian).
14. Aristov, S.N., Prosviryakov, E.Yu., and Spevak, L.F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows. Theoretical Foundations of Chemical Engineering, 2016, 50 (2), 132–141. DOI: 10.1134/S0040579516020019.
15. Aristov, S.N., Prosviryakov, E.Yu., and Spevak, L.F. Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid. Vychislitelnaya Mekhanika Sploshnykh Sred, 2015, 8 (4), 445–456. (In Russian). DOI: 10.7242/1999-6691/2015.8.4.38.
16. Lin, C.C. Note on a class of exact solutions in magneto-hydrodynamics. Archive for Rational Mechanics and Analysis, 1957, 1, 391–395. DOI: 10.1007/BF00298016.
17. Sidorov, A.F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory. Journal of Applied Mechanics and Technical Physics, 1989, 30 (2), 197–203. DOI: 10.1007/BF00852164.
18. Aristov, S.N. Vikhrevye techeniya v tonkikh sloyakh zhidkosti [Eddy Currents in Thin Liquid Layers: Synopsis of Doctoral Thesis]. Vladivostok, 1990, 303 p. (In Russian).
19. Aristov, S.N. and Shvarts, K.G. Convective heat transfer in a locally heated plane incompressible fluid layer. Fluid Dynamics, 2013, 48, 330–335. DOI: 10.1134/S001546281303006X.
20. Aristov, S.N. and Prosviryakov, E.Yu. A new class of exact solutions for three-dimensional thermal diffusion equations. Theoretical Foundations of Chemical Engineering, 2016, 50, 286–293. DOI: 10.1134/S0040579516030027.
21. Baranovskii, E.S., Burmasheva, N.V., and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations with couple stresses. Symmetry, 2021, 13 (8), 1355. DOI: 10.3390/sym13081355.
22. Goruleva, L.S. and Prosviryakov, E.Yu. A new class of exact solutions to the Navier–Stokes equations with allowance for internal heat release. Optics and Spectroscopy, 2022, 130 (6), 365–370. DOI: 10.1134/S0030400X22070037.
23. Burmasheva, N., Ershkov, S., Prosviryakov, E., and Leshchenko, D. Exact solutions of Navier–Stokes equations for quasi-two-dimensional flows with Rayleigh friction. Fluids, 2023, 8 (4), 123. DOI: 10.3390/fluids8040123.
24. Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations for describing the convective flows of multilayer fluids. Rus. J. Nonlin. Dyn., 2022, 18 (3), 397–410. DOI: 10.20537/nd220305.
25. Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations describing stratified fluid flows. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki, 2021, 25 (3), 491–507. DOI: 10.14498/vsgtu1860.
26. Prosviryakov, E.Yu. New class of exact solutions of Navier-Stokes equations with exponential dependence of velocity on two spatial coordinates. Theoretical Foundations of Chemical Engineering, 2019, 53 (1), 107–114. DOI: 10.1134/S0040579518060088.
27. Privalova, V.V. and Prosviryakov, E.Yu. A new class of exact solutions of the Oberbeck–Boussinesq equations describing an incompressible fluid. Theoretical Foundations of Chemical Engineering, 2022, 56 (3), 331–338. DOI: 10.1134/S0040579522030113.
28. Goruleva, L.S. and Prosviryakov, E.Yu. Nonuniform Couette–Poiseuille shear flow with a moving lower boundary of a horizontal layer. Technical Physics Letters, 2022, 48, 258–262. DOI: 10.1134/S1063785022090024.
29. Goruleva, L.S. and Prosviryakov, E.Yu. A new class of exact solutions for magnetohydrodynamics equations to describe convective flows of binary liquids. Khimicheskaya Fizika i Mezoskopiya, 2023, 25 (4), 447–462. (In Russian). DOI: 10.15350/17270529.2023.4.39.
30. Goruleva, L.S. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations for describing inhomogeneous isobaric vertical vortex fluid flows in regions with permeable boundaries. Diagnostics, Resource and Mechanics of materials and structures, 2023, 1, 41–53. DOI: 10.17804/2410-9908.2023.1.041-053. Available at: http://dream-journal.org/issues/2023-1/2023-1_393.html
31. Goruleva, L.S. and Prosviryakov, E.Yu. Unidirectional steady-state inhomogeneous Couette flow with a quadratic velocity profile along a horizontal coordinate. Diagnostics, Resource and Mechanics of materials and structures, 2022, 3, 47–60. DOI: 10.17804/2410-9908.2022.3.047-060. Available at: http://dream-journal.org/issues/2022-3/2022-3_367.html
32. Goruleva, L.S. and Prosviryakov, E.Yu. Exact solutions for the description of nonuniform unidirectional flows of magnetic fluids in the Lin–Sidorov–Aristov class. Diagnostics, Resource and Mechanics of materials and structures, 2023, 5, 39–52. DOI: 10.17804/2410-9908.2023.5.039-052. Available at: http://dream-journal.org/issues/2023-5/2023-5_415.html
33. Privalova, V.V. and Prosviryakov, E.Yu. Steady convective Coutte flow for quadratic heating of the lower boundary fluid layer. Nelineynaya Dinamika., 2018, 14 (1), 69–79. (In Russian). DOI: 10.20537/nd1801007.
34. Vlasova, S.S. and Prosviryakov, E.Yu. Parabolic convective motion of a fluid cooled from below with the heat exchange at the free boundary. Russian Aeronautics, 2016, 59 (4), 529–535. DOI: 10.3103/S1068799816040140.
35. Aristov, S.N., Privalova, V.V., and Prosviryakov, E.Yu. Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer. Nelineynaya Dinamika, 2016, 12 (2), 167–178. (In Russian). DOI: 10.20537/nd1602001.
36. Aristov, S.N., Privalova, V.V., and Prosviryakov, E.Yu. Planar linear Bénard-Rayleigh convection under quadratic heating of the upper boundary of a viscous incompressible liquid layer. Vestnik Kazanskogo Gosudarstvennogo Tekhnicheskogo Universiteta im. A.N. Tupoleva, 2015, 2, 6–13. (In Russian).
37. Aristov, S.N. and Prosviryakov, E.Yu. Exact solutions of thermocapillary convection during localized heating of a flat layer of viscous incompressible liquid. Vestnik Kazanskogo Gosudarstvennogo Tekhnicheskogo Universiteta im. A.N. Tupoleva, 2014, 3, 7–12. (In Russian).
38. Aristov, S.N. and Prosviryakov, E.Yu. On one class of analytic solutions of the stationary axisymmetric convection Bénard–Maragoni viscous incompressible fluid. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki, 2013, 3 (32), 110–118. (In Russian). DOI: 10.14498/vsgtu1205.
39. Privalova, V.V. and Prosviryakov, E.Yu. Exact solutions for a Couette–Hiemenz creeping convective flow with linear temperature distribution on the upper boundary. Diagnostics, Resource and Mechanics of materials and structures, 2018, 2, 92–109. DOI: 10.17804/2410-9908.2018.2.092-109. Available at: http://dream-journal.org/issues/2018-2/2018-2_170.html
40. Privalova, V.V. and Prosviryakov, E.Yu. Couette–Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid, with allowance made for heat recovery. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki, 2018, 22 (3), 532–548. DOI: 10.14498/vsgtu1638.
41. Andreev, V.K. and Efimova, M.V. The structure of a two-layer flow in a channel with radial heating of the lower substrate for small Marangoni numbers. Journal of Applied and Industrial Mathematics, 2024, 18 (2), 179–191. DOI: 10.1134/S1990478924020017.
42. Andreev, V.K. Thermocapillary convection of immiscible liquid in a three-dimensional layer at low Marangoni numbers. Journal of Siberian Federal University, Mathematics and Physics, 2024, 17 (2), 195–206.
43. Andreev, V.K. and Pianykh, A.A. Comparative analysis of the analytical and numerical solution of the problem of thermocapillary convection in a rectangular channel. Journal of Siberian Federal University, Mathematics and Physics, 2023, 16 (1), 48–55.
44. Andreev, V.K. and Lemeshkova, E.N. Thermal convection of two immiscible fluids in a 3D channel with a velocity field of a special type. Fluid Dynamics, 2023, 58 (7), 1246–1254. DOI: 10.1134/s0015462823602176.
45. Andreev, V.K. and Uporova, A.I. Initial boundary value problem on the motion of a viscous heat-conducting liquid in a vertical pipe. Journal of Siberian Federal University, Mathematics and Physics, 2023, 16 (1), 5–16.
46. Andreev, V.K. and Uporova, A.I. On a spectral problem for convection equations. Journal of Siberian Federal University, Mathematics and Physics, 2022, 15 (1), 88–100. DOI: 10.17516/1997-1397-2022-15-1-88-100.
47. Andreev, V.K. and Stepanova, I.V. Inverse problem for source function in parabolic equation at Neumann boundary conditions. Journal of Siberian Federal University, Mathematics and Physics, 2021, 14 (4), 445–451. DOI: 10.17516/1997-1397-2021-14-4-445-451.
48. Andreev, V.K. and Sobachkina, N.L. Two-layer stationary flow in a cylindrical capillary taking into account changes in the internal energy of the interface. Journal of Siberian Federal University, Mathematics and Physics, 2021, 14 (4), 507–518. DOI: 10.17516/1997-1397-2021-14-4-507-518.
49. Lemeshkova, E. and Andreev, V. On the asymptotic behavior of inverse problems for parabolic equation. Journal of Elliptic and Parabolic Equations, 2021, 7 (2), 905–921. DOI: 10.1007/s41808-021-00127-8.
50. Andreev, V.K. Asymptotic behavior of small perturbations for unsteady motion an ideal fluid jet. Journal of Siberian Federal University, Mathematics and Physics, 2021, 14 (2), 204–212. DOI: 10.17516/1997-1397-2021-14-2-204-212.

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