Geodesic
Steady thermo-diffusive shear Couette flow of~incompressible fluid. Velocity field analysis
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 25 (2021) no. 4, pp. 763-775.

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An exact solution that describes steady flow of viscous incompressible fluid with coupled convective and diffusion effects (coupled dissipative Soret and Dufour effects) has been found. To analyze shear fluid flow an over-determined boundary value problem has been solved. The over-determination of the boundary value problem is caused by the advantage of number of equations in non-linear Oberbeck–Boussinesq system against number of unknown functions (two components of velocity vector, pressure, temperature and concentration of dissolved substance). Non-trivial exact solution of system consisting of Oberbeck–Boussinesq equations, incompressibility equation, heat conductivity equation and concentration equation has been built as Birich–Ostroumov class exact solution. Since the exact solution a priori satisfies the incompressibility equation the over-determined system is solvable. Existence of stagnation points is shown both in general flow and in secondary fluid motion without vorticity. Conditions of countercurrent appearance are found.
Keywords: Navier–Stokes equations, stratified fluid, mass force field, overdetermined reduced system.
Mots-clés : exact solution 
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Vyach. V. Bashurov; E. Yu. Prosviryakov. Steady thermo-diffusive shear Couette flow of~incompressible fluid. Velocity field analysis. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 25 (2021) no. 4, pp. 763-775. http://geodesic.mathdoc.fr/item/VSGTU_2021_25_4_a8/

[1] Gershuni G. Z., Zhukhovitskii E. M., Convective Stability of Incompressible Fluids, Israel Program for Scientific Translations, Keter Publishing House, Jerusalem, 1976, 330 pp.

[2] Kutepov A. M., Polyanin A. D., Zapryanov Z. D., Vyazmin A. V., Kazenin D. A., Khimicheskaya gidrodinamika [Chemical Hydrodynamics], Quantum, Moscow, 1996, 336 pp. (In Russian)

[3] Aristov S. N., Shvarts K. G., Vihrevye techenija advektivnoj prirody vo vrashhajushhemsja sloe zhidkosti [Vortex Flows of Advective Nature in a Rotating Fluid Layer], Perm. State Univ., Perm, 2006, 155 pp. (In Russian)

[4] Aristov S. N., Shvarts K. G., Vikhrevye techenija v tonkikh slojakh zhidkosti [Vortical Flows in Thin Fluid Layers], Vyatka State Univ., Kirov, 2011, 207 pp. (In Russian)

[5] Andreev V. K., Gaponenko Yu. A., Goncharova O. N., Pukhnachev V. V., Mathematical Models of Convection, De Gruyter Studies in Mathematical Physics, 5, De Gryuter, Berlin, Boston, 2012, xv+417 pp. | DOI | Zbl

[6] Ryzhkov I. I., Termodiffuziia v smesiakh: uravneniia, simmetrii, resheniia i ikh ustoichivost' [Thermodiffusion in Mixtures: Equations, Symmetries, Solutions and its Stability], Novosibirsk, 2013, 200 pp. (In Russian)

[7] Ostroumov G. A., Free Convection under the Condition of the Internal Problem, NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, Washington, 1958

[8] Birikh R. V., “Thermocapillary convection in a horizontal layer of liquid”, J. Appl. Mech. Tech. Phys., 7:3 (1966), 43–44 | DOI

[9] 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), 142–157 (In Russian)

[10] Andreev V. K., Birikh Solutions to Convection Equations and Some of its Extensions, ICM SB RAS Preprint no. 1–10, Krasnoyarsk, 2010, 68 pp. (In Russian)

[11] Pukhnachev V. V., “Non-stationary analogues of the Birikh solution”, Izv. Alt. Gos. Univ., 2011, no. 1–2, 62–69 (In Russian)

[12] Birikh R. V., Pukhnachev V. V., “An axial convective flow in a rotating tube with a longitudinal temperature gradient”, Dokl. Phys., 56:1 (2011), 47–52 | DOI

[13] Andreev V. K., Bekezhanova V. B., “Stability of non-isothermal fluids (Review)”, J. Appl. Mech. Tech. Phys., 54:2 (2013), 171–184 | DOI | Zbl

[14] Aristov S. N., Prosviryakov E. Yu., Spevak L. F., “Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid”, Computational Continuum Mechanics, 8:4 (2015), 445–456 (In Russian) | DOI

[15] 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. Velocity field investigation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:1 (2017), 180–196 (In Russian) | DOI

[16] 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”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:4 (2017), 736–751 (In Russian) | DOI | Zbl

[17] 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., 30:2 (1989), 197–203 | DOI

[18] Aristov S. N., Prosviryakov E. Yu., “A new class of exact solutions for three-dimensional thermal diffusion equations”, Theor. Found. Chem. Technol., 50:3 (2016), 286–293 | DOI

[19] Aristov S. N., Prosviryakov E. Yu., “Nonuniform convective Couette flow”, Fluid Dyn., 51:5 (2016), 581–587 | DOI | Zbl

[20] Burmasheva N. V., Prosviryakov E. Yu., “Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:2 (2019), 341–360 | DOI | Zbl

[21] Burmasheva N. V., Prosviryakov E. Yu., “On Marangoni shear convective flows of inhomogeneous viscous incompressible fluids in view of the Soret effect”, J. King Saud Univ. – Science, 32:8 (2020), 3364–3371 | DOI

[22] Burmasheva N. V., Prosviryakov E. Yu., “Convective layered flows of a vertically whirling viscous incompressible fluid. Temperature field investigation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 24:3 (2020), 528–541 | DOI | Zbl

[23] Burmasheva N. V., Privalova V. V., Prosviryakov E. Yu., “Layered Marangoni convection with the Navier slip condition”, Sādhanā, 46:1 (2021), 55 | DOI

[24] Burmasheva N. V., Prosviryakov E. Yu., “Exact solutions for steady convective layered flows with a spatial acceleration”, Russ. Math., 65:7 (2021), 8–16 | DOI | Zbl

[25] Burmasheva N. V., Prosviryakov E. Yu., “Exact solutions to the Oberbeck–Boussinesq equations for shear flows of a viscous binary fluid with allowance made for the Soret effect”, The Bulletin of Irkutsk State University. Series Mathematics, 37 (2021), 17–30 | DOI | Zbl

[26] 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., 43:5 (2009), 642 | DOI

[27] Ekman V. W., “On the influence of the Earth's rotation on ocean-currents”, Ark. Mat. Astron. Fys., 2:11 (1905), 1–52 http://jhir.library.jhu.edu/handle/1774.2/33989 | Zbl

[28] Couette M., “Études sur le frottement des liquids”, Ann. de Chim. et Phys. (6), 21 (1890), 433–510 (In French)