N. V. Burmasheva, E. Yu. Prosviryakov

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

A new exact solution of the Oberbeck-Boussinesq equations for the convective flow of a viscous incompressible fluid is considered. Layered flows of a viscous incompressible fluid are investigated within the class of the Sidorov-Lin exact solutions, which generalizes the family of the Ostroumov-Birikh solutions. The use of an exact solution allows an overdetermined system of fluid motion equations to be solved. The fluid is heated by setting a heat source at both boundaries. The dimension of the studied boundary value problem cannot be lowered by the transformation of the rotation. The obtained exact solution describes the counterflow in the fluid. Thermocline and a boundary layer occur near one of the boundary layers in the fluid flow.

Acknowledgments: This work was supported by the Foundation for Assistance to Small Innovative Enterprises in Science and Technology (the UMNIK program); the agreement no. 12281 GU/2017.

References:

1. Joseph D.D. Stability of fluid motions. Berlin, Heidelberg, New York, Springer–Verlag, 1976.
2. Gershuni G.Z., Zhukhovitskii E.M. Convective Stability of Incompressible Fluids. Israel Program for Scientific Translations. Jerusalem, Keter Publishing House, 1976, 330 p.
3. Shtern V. Counterflows. Paradoxical Fluid Mechanics Phenomena. Cambridge University Press, 2012, 469 p. DOI: 10.1017/CBO9781139226516
4. Dorrepaal J.M. An exact solution of the Navier-Stokes equation which describes nonorthogonal stagnation-point flow in two dimensions. Journal of Fluid Mechanics, 1986, vol. 163, no. 1, pp. 141–147. DOI: 10.1017/s0022112086002240
5. 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
6. Riesco-Chueca P., De la Mora J.F. Brownian motion far from equilibrium: a hypersonic approach. Journal of Fluid Mechanics, 1990, vol. 214, pp. 639–663. DOI: 10.1017/S0022112090000301
7. Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingler’s Politech. J., 1911, vol. 326, pp. 321–324.
8. 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.
9. 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, iss. 5, pp. 642–661. DOI: 10.1134/S0040579509050066
10. Aristov S.N., Prosviryakov E.Yu. Nonuniform convective Couette flow. Fluid Dynamics, 2016, vol. 51, no. 5, pp. 581–587. DOI: 10.1134/S001546281605001X.
11. Gorshkov A.V., Prosviryakov E.Yu. Layered B´enard-Marangoni convection during heat transfer according to the Newton's law of cooling. Kompyuternye Issledovaniya i Modelirovanie, 2016, vol. 8, no. 6, pp. 927–940. (In Russian).
12. Gorshkov A.V., Prosviryakov E.Yu. Complex stationary convection with third-kind boundary conditions at the boundaries of a fluid layer. Diagnostics, Resource and Mechanics of Materials and Structures, 2016, iss. 2, pp. 34–47. DOI: 10.17804/2410-9908.2016.2.034-047 Available at: http://dream-journal.org/issues/2016-2/2016-2_81.html (accessed: 01.10.2017).
13. Gorshkov A.V., Prosviryakov E.Yu. Analytic solutions of stationary complex convection describing a shear stress field of different signs. Trudy Inst. Mat. i Mekh. UrO RAN, 2017, vol. 23, no. 2, pp. 32–41. (In Russian). DOI: 10.21538/0134-4889-2017-23-2-32-4114
14. Aristov S.N., Prosviryakov E.Yu., Spevak L.F. Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid. Vychislitelnaya Mekhanika Sploshnykh Sred, 2015, vol. 8, no. 4, pp. 445–456. (In Russian).
15. 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. Vestn. Samar. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2017, vol. 21, no. 1, pp. 180–196. (In Russian). DOI: 10.14498/vsgtu152716
16. 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
17. Ostroumov G.A. Free convection under the condition of the internal problem. Washington, NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, 1958.
18. Birikh R. V. Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys., 1966, vol. 7, no. 3, pp. 43–44.
19. Sidorof A.F. On two classes of solutions of the equations of fluid and gas mechanics and their relation to the theory of traveling waves. Prikl. Mech. i Tekhnich. Fizika, 1989, no. 2, pp. 34–40. (In Russian).
20. 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
21. 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.
22. 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. Tr. In-ta Eksperim. Meteorol., 1996, no. 27 (162), pp. 142–157. (In Russian).
23. Aristov S.N., Shvarts K.G. Vikhrevye Techeniya Advektivnoy Prirody vo Vrashchayushchemsya Sloe Zhidkosti [Vortical Flows of the Advective Nature in a Rotating Fluid Layer]. Perm, Perm State Univ. Publ., 2006, 155 pp. (In Russian).
24. Aristov S.N., Shvarts K.G. Vikhrevye Techeniya v Tonkikh Sloyakh Zhidkosti [Vortical Flows in Thin Fluid Layers]. Kirov, Vyatka State Univ. Publ., 2011, 207 pp. (In Russian).
25. 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
26. Aristov S.N., Prosviryakov E.Yu. On laminar flows of planar free convection. Nelin. Dinam., 2013, vol. 9, no. 4, pp. 651–657. (In Russian). DOI: 10.20537/nd1304004
27. Andreev V.K. Resheniya Birikha uravneniy konvektsii i nekotorye ego obobshcheniya [Birikh Solutions to Convection Equations and Some of its Extensions]. Krasnoyarsk, IBM SO RAN Publ., 2010, 68 p. (In Russian).
28. Aristov S.N., Prosviryakov E.Yu., Spevak L.F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows. Theor. Found. Chem. Eng., 2016, vol. 50, no. 2, pp. 132–141. DOI: 10.1134/S0040579516020019.
29. Andreev V.K., Gaponenko Ya.A., Goncharova O.N., Pukhnachev V.V. Mathematical Models of Convection. Berlin, Boston, De Gryuter Publ., 2012, 417 p.
30. 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.
31. Pukhnachev V.V. Non-stationary analogues of the Birikh solution. Izvestiya AltGU, 2011, no. 1–2, pp. 62–69. (In Russian).
32. Andreev V.K., Bekezhanova V.B. Stability of non-isothermal fluids (Review). J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 2, pp. 171–184. DOI: 10.1134/S0021894413020016
33. Andreev V.K., Stepanova I.V. Unidirectional flows of binary mixtures within the framework of the Oberbeck–Boussinesq model. Fluid Dyn., 2016, vol. 51, no. 2, pp. 136–147. DOI: 10.1134/S0015462816020022
34. Goncharova O.N., Kabov O.A. Gravitational-thermocapillary convection of fluid in the horizontal layer in co-current gas flow. Dokl. Phys., 2009, vol. 54, no. 5, pp. 242–247. DOI: 10.1134/S1028335809050061
35. Goncharova O.N., Rezanova E.V. Example of an exact solution of the stationary problem of two-layer flows with evaporation at the interface. J. Appl. Mech. Tech. Phys., 2014, vol. 55, no. 2, pp. 247–257. DOI: 10.1134/S0021894414020072
36. Birikh R.V., Pukhnachev V.V. An axial convective flow in a rotating tube with a longitudinal temperature gradient. Dokl. Phys., 2011, vol. 56, no. 1, pp. 47–52. DOI: 10.1134/S1028335811010095
37. Birikh R.V., Pukhnachev V.V., Frolovskaya O.A. Convective flow in a horizontal channel with non-Newtonian surface rheology under time-dependent longitudinal temperature gradient. Fluid Dyn., 2015, vol. 50, no. 1, pp. 173–179. DOI: 10.1134/S0015462815010172
38. Ryzhkov I.I. Termodiffuziya v Smesyakh: Uravneniya, Simmetrii, Resheniya i ikh Ustoychivost [Thermodiffusion in Mixtures: Equations, Symmetries, Solutions and their Stability]. Novosibirsk, SB RAS Publ., 2013, 200 p. (In Russian).

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