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N. V. Burmasheva, E. Yu. Prosviryakov


DOI: 10.17804/2410-9908.2019.1.006-042

The exact solution of the Oberbeck-Boussinesq equations, which describes the influence of the thermocapillary effect on the convective flows of a viscous incompressible fluid, is discussed. The class of exact solutions is chosen in such a way that it allows one to solve an overdetermined system of equations for the motion of a fluid by identically satisfying the incompressibility condition. Heat sources at both boundaries, which heat (cool) the fluid, are selected as the boundary conditions. The exact solution has been obtained and the temperature field has been studied. It is shown that the structure of the exact solution does not allow one to reduce the dimension of the boundary value problem under study. It is also shown that the obtained exact solution admits the presence of countercurrents in the fluid.

Keywords: layered flow, exact solution, counterflows, Marangoni convection, vertical swirl, vortex flow, system of Oberbeck–Boussinesq equations


  1. Gershuni G.Z., Zhukhovitskii E.M. Convective Stability of Incompressible Fluids. Israel Program for Scientific Translations. Jerusalem, Keter Publishing House, 1976, 330 p.
  2. Ekman V.W. On the Influence of the Earth’s Rotation on Ocean-Currents. Ark. Mat. Astron. Fys., 1905, vol. 2, no. 11, pp. 1–52.
  3. Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingler’s Politech. J., 1911, vol. 326, pp. 321–324.
  4. Ekman V.W. Uber Horizontazirkulation bei winder-reugten Meeresstromungen. Arkiv Mat., Astr., Phys., 1923, vol. 17, no. 26, pp. 1–74.
  5. Pedlosky J. Geophysical Fluid Dynamics, Springer-Verlag New York, 1979, 710 p.
  6. Haeusser T.M., Leibovich S. Pattern formation in the marginally unstable Ekman layer. J. Fluid Mech., 2003, vol. 479, pp. 125–144. DOI: 10.1017/S0022112002003415.
  7. Schwarz K.G. Stability of thermocapillary advective flow in a slowly rotating liquid layer under microgravity conditions. Fluid Dynamics, 2012, vol. 47, iss. 1, pp. 37–49. DOI: 10.1134/S001546281201005X.
  8. Aristov S.N., Frik P.G. Nonlinear effects of the Ekman layer on the dynamics of largescale eddies in shallow water. J. Appl. Mech. Tech. Phys., 1991, vol. 32, no. 2, pp. 189–194.
  9. Gorshkov A.V., Prosviryakov E.Y. Ekman Convective Layer Flow of a Viscous Incompressible Fluid. Izvestiya. Atmospheric and Oceanic Physics, 2018, vol. 54, no. 2, pp. 189–195. DOI: 10.1134/S0001433818020081.
  10. Kryukov N.D., Shmatkov V.A. A review of studies of wind-generated surface currents. Eurasian Union of Scientists, 2015, № 6–2 (15), pp. 109–113. (In Russian).
  11. Shvarts K.G. On an exact solution of the Navier-Stokes equations describing non-isothermal large-scale flow in a rotating fluid layer with a free upper boundary. Vestnik Permskogo Universiteta. Matematika. Mekhanika. Informatika, 2016, no. 2 (33), pp. 118–124. (In Russian).
  12. Shvarts K.G. Exact solution of the Navier–Stokes equation describing nonisothermal largescale flows in a rotating layer of liquid with free upper surface. J. Math. Sci., 2018, vol. 230, no. 5, pp. 813–817. DOI: 10.1007/s10958-018-3796-y.
  13. Brown R.A. Analytical Methods in Planetary Boundary-Layer Modelling. Hilger, 1974.
  14. Welander P. The thermocline problem. Phil. Trans. R. Soc. Lond. A., 1971, vol. 270, pp. 415–421. DOI: 10.1098/rsta.1971.0081.
  15. Chefranov S.G. Cyclone–anticyclone vortex asymmetry mechanism and linear Ekman friction. Journal of Experimental and Theoretical Physics, 2016, vol. 122, no. 4, pp. 759–768. DOI: 10.1134/S1063776116040038.
  16. Riabouchinsky D. Quelques considerations sur les mouvements plans rotationnels d’un liquid. C. R. Hebdomadaires Acad. Sci., 1924, vol. 179, pp. 1133–1136.
  17. Galaktionov V.A., Svirshchevskii S.R. Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. Boca Raton, Chapman and Hall/CRC, 2007. DOI: 10.1201/9781420011623.
  18. Pukhnachev V.V. Group Properties of the Navier-Stokes Equations in a Plane Case. Prikl. Mekh. Tekh. Fiz., no. 1, pp. 83–90. (In Russian).
  19. Atlas osesimmetrichnykh kavitatsionnykh techeniy tipa Ryabushinskogo [Atlas of Axisymmetric Cavitation Ryabushinsky-Type Flows]. Novosibirsk, Institut Gidrodinamiki SO AN SSSR, 1968. (In Russian).
  20. Guzevsky L.G. Osesimmetrichnye zadachi obtekaniya so svobodnymi granitsami. In: Issledovaniya po razvitoy kavitatsii [Advanced Cavitation Research]. Novosibirsk, Izdatelstvo SO AN SSSR, 1976. (In Russian).
  21. Aristov S. N., Knyazev D.V. Viscous fluid flow between moving parallel plates. Fluid Dynamics, 2012, vol. 47, iss. 4, pp 476–482. DOI: 10.1134/S0015462812040060.
  22. Petrov A.G. Exact solution of the Navier-Stokes equations in a fluid layer between the moving parallel plates. Journal of Applied Mechanics and Technical Physics, 2012, vol. 53, no. 5, pp. 642–646. DOI: 10.1134/S0021894412050021.
  23. Vlasova S.S., Prosviryakov E.Yu. Two-dimensional convection of an incompressible viscous fluid with the heat exchange on the free border. Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 2016, vol. 20, no. 3, pp. 567–577. DOI: https://doi.org/10.14498/vsgtu1483. (In Russian).
  24. Aristov S.N., Prosviryakov E.Yu. On one class of analytic solutions of the stationary axisymmetric convection Benard–Maragoni viscous incompreeible fluid. Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 2013, vol. 3, no. 32, pp. 110–118. DOI: 10.14498/vsgtu1205. (In Russian).
  25. Aristov S.N., Privalova V., Prosviryakov E.Y. Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer. Rus. J. Nonlin. Dyn., 2016, vol. 12, no. 2, pp. 167–178. DOI: 10.20537/nd1602001.
  26. Fomin A.A, Fomina L.N. On the solution of fluid flow and heat transfer problem in a 2D channel with backward-facing step. Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 2017, vol. 21, no. 2, pp. 362–375. DOI: 10.14498/vsgtu1545.
  27. Shtern V. Counterflows. Paradoxical Fluid Mechanics Phenomena. Cambridge, Cambridge University Press, 2012. DOI: 10.1017/CBO9781139226516.
  28. Stuart J.T. The viscous flow near a stagnation point when the external flow has uniform vorticity. Journal of the Aerospace Sciences, 1959, vol. 26, no. 2, pp. 124–125.  DOI: 10.2514/8.7963.
  29. Dorrepaal J.M. An exact solution of the Navier-Stokes equation which describes nonorthogonal stagnation-point flow in two dimensions. J. Fluid Mech., 1986, vol. 163, no. 1, pp. 141–147. DOI: 10.1017/s0022112086002240.
  30. Riesco-Chueca P., de la Mora J.F. Brownian motion far from equilibrium: a hypersonic approach. J. of Fluid Mech., 1990, vol. 214, pp. 639–663.  DOI: 10.1017/ S0022112090000301.
  31. Aristov S.N., Prosviryakov E.Yu. Nonuniform convective Couette flow. Fluid Dynamics, 2016, vol. 51, iss. 5, pp. 581–587. DOI: 10.1134/S001546281605001X.
  32. Gorshkov A.V., Prosviryakov E.Yu. Layered Benard–Marangoni convection during heat transfer according to the Newton’s law of cooling. Computer Research and Modeling, 2016, vol. 8, no. 6, pp. 927–940. (In Russian).
  33. Aristov S.N., Prosviryakov E.Y., Spevak L.F.  Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid. Computational Continuum Mechanics, 2015, vol. 8, no. 4, pp. 445–456. DOI: 10.7242/1999-6691/2015.8.4.38.
  34. Burmasheva N.V., Prosviryakov E.Yu. A large-scale layered stationary convection of an incompressible viscous fluid under the action of shear stresses at the upper boundary. Velocity field investigation. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2017, vol. 21, no. 1, pp. 180–196. DOI: 10.14498/vsgtu1527.
  35. Burmasheva N.V., Prosviryakov E.Yu. Exact solutions for layered large-scale convection induced by tangential stresses specified on the free boundary of a fluid layer. IOP Conference Series: Materials Science and Engineering, 2017, vol. 208, conf. 1. DOI: 10.1088/1757-899X/208/1/012010.
  36. Burmasheva N.V., Prosviryakov E.Yu. Exact solutions for natural convection of layered flows of a viscous incompressible fluid with specified tangential forces and the linear distribution of temperature on the layer boundaries. Diagnostics, Resource and Mechanics of materials and structures, 2017, iss. 4, pp. 16–31. DOI: 10.17804/2410-9908.2017.4.016-031. Available at: http://dream-journal.org/DREAM_Issue_4_2017_Burmasheva_N.V._et_al._016_031.pdf
  37. Burmasheva N.V., Prosviryakov E.Yu.  Exact solution for the layered convection of a viscous incompressible fluid at specified temperature gradients and tangential forces on the free boundary. AIP Conference Proceedings, 2017, vol. 1915, iss. 1.  DOI: 10.1063/1.5017353.
  38. Burmasheva N.V., Prosviryakov E.Yu. A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation. J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 2017, vol. 21, no. 4, pp. 736–751. DOI: 10.14498/vsgtu1568. (In Russian).
  39. 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. Theoretical Foundations of Chemical Engineering, 2009, vol. 43, no. 5, pp. 642–662. DOI: 10.1134/S0040579509050066.
  40. Ostroumov G.A. Free convection under the condition of the internal problem. Washington, NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, 1958.
  41. Birikh R.V. Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys., 1966, vol. 7, no. 3, p. 43.
  42. Napolitano L.G. Plane Marangoni-Poiseulle flow of two immiscible fluids. Acta Astronautica, 1980, vol. 7, no. 4, pp. 461–478.
  43. Pukhnachev V.V. Non-stationary analogues of the Birikh solution. Izvestiya AltGU, 2011, nos. 1–2, pp. 62–69. (In Russian).
  44. Goncharova O.N., Rezanova E.V. Example of an exact solution of the stationary problem of two-layer flows with evaporation at the interface. Journal of Applied Mechanics and Technical Physics, 2014, vol. 55, no. 2, pp. 247–257. DOI: 10.1134/s0021894414020072.
  45. Aristov S.N., Prosviryakov E.Y.  On laminar flows of planar free convection. Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 4, pp.  651–657. DOI: 10.20537/nd1304004.
  46. Schwarz K.G. Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries. Fluid Dynamics, 2014, vol. 49, iss. 4, pp. 438–442. DOI: 10.1134/S0015462814040036.
  47. Bratsun D.A., Vyatkin V.A., Mukhamatullin A.R. On exact nonstationary solutions of equations of vibrational convection. Computational Continuum Mechanics, 2017, vol. 10, no. 4, pp. 433–444. DOI: 10.7242/1999-6691/2017.10.4.35.
  48. Schwarz K.G. Plane parallel advective flow in a horizontal layer of incompressible fluid with an internal linear heat source. Prikladnaya Matematika i Mekhanika, 2018, vol. 82, no. 1, pp. 25–30. (In Russian).
  49. Aristov S.N., Shvarts K.G. Convective heat transfer in a locally heated plane incompressible fluid layer. Fluid Dynamics, 2013, vol. 48, iss. 3, pp 330–335. DOI: 10.1134/S001546281303006X.
  50. Gorshkov A.V., Prosviryakov E.Yu. Analytic solutions of stationary complex convection describing a shear stress field of different signs. In: Trudy instituta matematiki i mekhaniki UrO RAN [Proccedings of Institute of Mechanics and Mathematics of RAS, Ural Branch, 2017, vol. 23, no. 2, pp. 32–41]. (In Russian).
  51. Rafiq Sh., Nawaz M., Mustahsan M. Casson Fluid Flow due to Non-Coaxial Rotation of a Porous Disk and the Fluid at Infinity Through a Porous Medium. Journal of Applied Mechanics and Technical Physics, 2018, vol. 59, iss. 4, pp. 601–607. DOI: 10.1134/S0021894418040053.
  52. Andreev V.K. Resheniya Birikha uravneniy konvektsii i nekotorye ego obobshcheniya: preprint [Birikh Solutions of Convection Equations and Some of its Generalizations: Preprint]. Krasnoyarsk, 2010, № 1–10. (In Russian).
  53. Birikh R.V. Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys., 1966, vol. 7, no. 3, pp. 43–44.
  54. Pukhnachev V.V. Group-theoretical methods in the convection problems. In: Application of Mathematics in Technical and Natural Sciences, M.D. Todorov and C.I. Christov, eds., American Institute of Physics, CP 1404, Melwille, NY, 2011, pp. 31–42.
  55. 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, vol. 30, iss. 2, pp. 197–203. DOI: 10.1007/BF00852164.
  56. Aristov S.N., Shvarts K.G. Advective flow in a rotating liquid film. J. Appl. Mech. Tech. Phys., 2016, vol. 57, no. 1, pp. 188–194. DOI: 10.1134/S0021894416010211.
  57. Aristov S.N., Prosviryakov E.Yu. A new class of exact solutions for three-dimensional thermal diffusion equations. Theor. Found. Chem. Eng., 2016, vol. 50, no. 3, pp. 286–293. DOI: 10.1134/S0040579516030027.
  58. Aristov S.N., Frik P.G. Nonlinear effects of the Ekman layer on the dynamics of largescale eddies in shallow water. J. Appl. Mech. Tech. Phys., 1991, vol. 32, no. 2, pp. 189–194. DOI: 10.1007/BF00858033.
  59. Aristov S.N., Shvarts K.G. Vikhrevye Techeniya Advektivnoy Prirody vo Vrashchayushchemsya Sloe Zhidkosti [Vortex Flows of Advective Nature in the Rotary Layer of Fluid]. Perm, Perm Univ. Publ., 2006. (In Russian).
  60. Aristov S.N., Shvarts K.G. Vikhrevye Techeniya v Tonkikh Sloyakh Zhidkosti [Vortical Flows in Thin Fluid Layers]. Kirov, VyatGU, 2011, 207 pp. (In Russian).
  61. Ingel L.Kh., Aristov S.N. The class of exact solutions of nonlinear problems on thermal circulation associated with volumetric heat release in the atmosphere. In: Tr. In-ta Eksperim. Meteorol., 1996, no. 27 (162), pp. 142–157. (In Russian).
  62. Aristov S.N., Prosviryakov E.Yu. On laminar flows of planar free convection. Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 651–657. DOI: 10.20537/nd1304004.
  63. Andreev V.K., Gaponenko Ya.A., Goncharova O.N., Pukhnachev V.V. Mathematical Models of Convection. Berlin, Boston, De Gryuter Publ., 2012, 417 p.
  64. Bekezhanova V.B., Goncharova O.N. Evaporative convection problems (review). Prikladnaya Matematika i Mekhanika, 2018, vol. 82, no. 2, pp. 219–260. (In Russian).
  65. Galkin V.A., Dubovik A.O. About Layered Flow Modeling of Magnetic Viscous Incompressible Fluid in the Rotating Coaxial Pipe. Vestnik Kibernetiki, 2017, no. 4 (28), pp. 56–61. (In Russian).
  66. Kraiko A.N., Vatazhin A.B., Lyubimov G.A. Mekhanika zhidkosti i gaza. Izbrannoe [[Fluid Mechanics. Selected, ed. by A.N. Kraiko]. Moscow, Fizmatlit Publ., 2013, 752 p. (In Russian).
  67. Vatazhin A.B., Klimenko A.Yu., Lebedev A.B., Sorokin A.A. Homogeneous condensation in turbulent submerged isobaric jets. Fluid Dynamics, 1988, vol. 23, iss. 2, pp. 194–203. DOI: 10.1007/BF01051887.
  68. Vasil’kov A.P. Calculation of a turbulent two-phase isobaric stream. Fluid Dynamics, 1976, vol. 11, iss. 5, pp. 699–704. DOI: 10.1007/BF01012960.
  69. Shmiglevskiy Yu.D. On isobaric plane flows of a viscous incompressible fluid. USSR Computational Mathematics and Mathematical Physics, 1985, vol. 25, iss. 6, pp. 191–193. DOI: 10.1016/0041-5553(85)90030-8.
  70. Shmyglevsky Yu. D. Analiticheskie issledovanija dinamiki gaza i zhidkosti. [Analytical Studies of Gas and Liquid Dynamics]. Moscow, 1999, 232 p. (In Russian).
  71. Koterov V.N., Shmiglevskiy Yu.D., Scheprov A.V. A survey of analytical studies of steady viscous incompressible flows (2000–2004). Computational Mathematics and Mathematical Physics, 2005, vol. 45, no. 5, pp. 867–888.
  72. Aristov S.N., Prosviryakov E.Y. Inhomogeneous Couette flow. Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 2, pp. 177–182. DOI: 10.20537/nd1402004.
  73. Aristov S.N., Prosviryakov E.Y. Stokes waves in vortical fluid. Rus. J. Nonlin. Dyn., 2014, vol. 10, no. 3, pp. 309–318. DOI: 10.20537/nd1403005.
  74. Aristov S.N., Prosviryakov E.Y. Large-scale flows of viscous incompressible vortical fluid. Russian Aeronautics (IzVUZ), 2015, vol. 58, no. 4, pp. 413–418. DOI: 10.3103/S1068799815040091.
  75. Aristov S.N., Prosviryakov E.Y. Unsteady layered vortical fluid flows. Fluid Dynamics, 2016, vol. 51, no. 2, pp. 148–154. DOI: 10.1134/S0015462816020034.
  76. 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. Eng., 2009, vol. 43, no. 5, pp. 642–662. DOI: 10.1134/S0040579509050066.
  77. Gershuni G.Z., Zhukhovitskii E.M. Instability of a system of horizontal layers of immiscible fluids heated from above. Fluid Dynamics, 1980, vol. 15, iss. 6, pp. 816–822. DOI: 10.1007/BF01096629.
  78. Fikhtengolts G.M. Kurs differentsial’nogo i integral’nogo ischisleniya [A Course in Differential and Integral Calculus]. Moscow, Fizmatlit, 2001. (In Russian).
  79. Kostrikin A. I. Vvedenie v algebru. Chast I. Osnovy algebry: Uchebnik dlya vuzov [Introduction to algebra. Part 1. Bases of algebra: Textbook for institutions of higher learning]. Moscow, Fizmatlit, 2004, 272. (In Russian).


Article reference

Burmasheva N. V., Prosviryakov E. Yu. Temperature Field Investigation in Layered Flows of a Vertically  swirling Viscous Incompressible Fluid under Two Thermocapillar Forces at a Free Boundary // Diagnostics, Resource and Mechanics of materials and structures. - 2019. - Iss. 1. - P. 6-42. -
DOI: 10.17804/2410-9908.2019.1.006-042. -
URL: http://eng.dream-journal.org/issues/2019-1/2019-1_236.html
(accessed: 06/22/2024).


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