N. V. Burmasheva, E. Yu. Prosviryakov
EXACT SOLUTIONS OF THE NAVIER–STOKES EQUATIONS FOR DESCRIBING AN ISOBARIC ONEDIRECTIONAL VERTICAL VORTEX FLOW OF A FLUID
DOI: 10.17804/24109908.2021.2.030051 The article proposes a family of exact solutions to the Navier–Stokes equations for describing isobaric inhomogeneous unidirectional fluid motions. Due to the incompressibility equation, the velocity of the inhomogeneous Couette flow depends on two coordinates and time. The expression for the velocity field has a wide functional arbitrariness. This exact solution is obtained by the method of separation of variables, and both algebraic operations (additivity and multiplicativity) are used to substantiate the importance of modifying the classical Couette flow. The article contains extensive bibliographic information that makes it possible to trace a change in the exact Couette solution for various areas of the hydrodynamics of a Newtonian incompressible fluid. The fluid flow is described by a polynomial depending on one variable (horizontal coordinate). The coefficients of the polynomial functionally depend on the second (vertical) coordinate and time; they are determined by a chain of the simplest homogeneous and inhomogeneous partial differential parabolictype equations. The chain of equations is obtained by the method of undetermined coefficients after substituting the exact solution into the Navier–Stokes equation. An algorithm for integrating a system of ordinary differential equations for studying the steady motion of a viscous fluid is presented. In this case, all the functions defining velocity are polynomials. It is shown that the topology of the vorticity vector and shear stresses has a complex structure even without convective mixing (creeping flow).
Keywords: exact solution, Couette flow, Navier–Stokes equation, inhomogeneous unidirectional flow, method of separation of variables, shear stress References:
 Aristov S.N., Knyazev D.V., Polyanin A.D. Exact solutions of the NavierStokes 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.
 Drazin P.G., Riley N. The Navier–Stokes Equations: A classification of flows and exact solutions, Cambridge, Cambridge Univ. Press, 2006, 196 p.
 Polyanin A.D., Zaitsev V.F. Handbook of nonlinear partial differential equations, Boca Raton, Chapman & Hall / CRC Press, 2004, 840 p.
 Yerin G.B. The NavierStokes equations of motion, Oxford, Clarendon, Laminar Boundary Layers, 1963, ed. L. Rosenhead, pp. 114–162.
 Dryden H.L., Murnaghan F.D., Bateman H. Report of the Committee on hydrodynamics. Bull. Natl. Res. Counc. (US), 1932, vol. 84, pp. 155–332.
 Berker R. Sur quelques cas d'lntegration des equations du mouvement d'un fuide visquex incomprcssible, Paris–Lille, Taffin–Lefort, 1936.
 Berker R. Integration des equations du mouvement d'un fluide visqueux incompressible, Berlin, Springer–Verlag. Handbuch der Physik, ed. S. Flugge, 1963, VIII/2, 384 p.
 Wang C.Y. Exact solution of the NavierStokes equationsthe generalized Beltrami flows, review and extension. Acta Mech., 1990, vol. 81, pp. 69–74. DOI: 10.1007/BF01174556.
 Wang C.Y. Exact solutions of the steadystate NavierStokes equations. Annu. Rev. Fluid Mech., 1991, vol. 23, pp. 159–177. DOI: 10.1146/annurev.fl.23.010191.001111.
 Wang C.Y. Exact solutions of the unsteady NavierStokes equations. Appl. Mech. Rev. 1989, vol. 42 (11S), pp. 269–282. DOI: 10.1115/1.3152400.
 Pukhnachev V.V. Symmetries in the NavierStokes equations. Uspekhi Mekhaniki, 2006, No. 1, pp. 6–76. (In Russian).
 Couette M. Etudes sur le frottement des liquids. Ann. Chim. Phys., 1890, vol. 21, pp. 433–510.
 Stokes G.G. On the effect of the internal friction of fluid on the motion of pendulums. Transactions of the Cambridge Philosophical Society, 1851, vol. 9, pp. 8–106.
 Taylor G.I. Stability of a viscous fluid contained between two rotating cylinders. J. Phil. Trans. Royal Society A., 1923, vol. 223, No. 605–615, pp. 289–343. DOI: 10.1098/rsta.1923.0008.
 Holodniok M., Kubíček M., Hlaváček V. Computation of the flow between two rotating coaxial disk: multiplicity of steadystate solutions. J. Fluid Mech., 1981, vol. 108, pp. 227–240. DOI: 10.1017/S0022112081002097.
 Aristov S.N., Gitman I.M. Viscous flow between two moving parallel disks: exact solutions and stability analysis. J. Fluid Mech., 2002, vol. 464, pp. 209–215. DOI: 10.1017/S0022112002001003.
 Zhilenko D.Y., Krivonosova O.E. Transitions to chaos in the spherical Couette flow due to periodic variations in the rotation velocity of one of the boundaries. Fluid Dynamics, 2013, vol. 48, No. 4, pp. 452–460. DOI: 10.1134/S0015462813040042.
 Zhilenko D., Krivonosova O., Gritsevich M., Read P. Wave number selection in the presence of noise: Experimental results. Chaos, 2018, vol. 28 (5), pp. 053110. DOI: 10.1063/1.5011349.
 Zhilenko D.Y., Krivonosova O.E. Origination and evolution of turbulent flows in a rotating spherical layer. Technical Physics, 2010, vol. 55, No. 4, pp. 449456. DOI: 10.1134/S1063784210040031.
 Belyaev Yu.N., Monakhov A.A., Yavorskaya I.M. Stability of a spherical Couette flow in thick layers with rotation of the inner sphere. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, 1978, No. 2, pp. 9–15. (In Russian).
 Pukhnachev V.V., Pukhnacheva T.P. The Couette problem for a Kelvin–Voigt medium. J. Math. Sci., 2012, vol. 186, pp. 495–510. DOI: 10.1007/s1095801210030.
 Zhuk V.I., Protsenko I.G. Asymptotic model for the evolution of perturbations in the plane CouettePoiseuille flow. Doklady Mathematics, 2006, vol. 74, No. 3, pp. 896–900. DOI: 10.1134/S1064562406060287.
 Gavrilenko S.L., Shil'ko S.V., Vasin R.A. Characteristics of a viscoplastic material in the Couette flow. Journal of Applied Mechanics and Technical Physics, 2002, vol. 43, No. 3, pp. 439–444. DOI: 10.1023/A:1015378622918.
 Troshkin O.V. Nonlinear stability of Couette, Poiseuille, and Kolmogorov plane channel flows. Dokl. Math., 2012, vol. 85, pp. 181–185. DOI: 10.1134/S1064562412020068.
 Rudyak V.Y., Isakov E.B. & Bord E.G. Instability of plane Couette flow of twophase liquids. Tech. Phys. Lett., 1998, vol. 24, pp. 199–200. DOI: 10.1134/1.1262051.
 Shalybkov D.A. Hydrodynamic and hydromagnetic stability of the Couette flow. PhysicsUspekhi, 2009, vol. 52, No. 9, pp. 915–935. DOI: 10.3367/UFNe.0179.200909d.0971.
 Boronin S.A. Stability of the plane Couette flow of a disperse medium with a finite volume fraction of the particles. Fluid Dynamics, 2011, vol. 46, pp. 64–71. DOI: 10.1134/S0015462811010078.
 Kudinov V.A. and Kudinov I.V. Calculation of Exact Analytic Solutions of Hyperbolic Equations of Motion in the Accelerated Couette Flow. Izv. Ross. Akad. Nauk. Energetika, 2012, No. 1, pp. 119–133. (In Russian).
 Babkin V.A. Plane Turbulent Couette Flow. Journal of Engineering Physics and Thermophysics, 2003, vol. 76, pp. 1251–1254. DOI: 10.1023/B:JOEP.0000012026.19646.c6.
 Abramyan A.K., Mirantsev L.V., Kuchmin A.Yu. Modeling of processes at Couette simple fluid flow in flat nanoscopic canal. Matem. Mod., 2012, vol. 24, Nos. 4, pp. 3–21. (In Russian).
 Malyshev V. and Manita A. Stochastic Micromodel of the Couette Flow. Theor. Prob. Appl., 2009, vol. 53, no. 4. pp. 716727. DOI: 10.1137/S0040585X97983924.
 Georgievskii D.V. Generalized Joseph estimates of stability of plane shear flows with scalar nonlinearity. Bull. Russ. Acad. Sci. Phys., 2011, vol. 75, pp. 140–143. DOI: 10.3103/S1062873810121044.
 Belyaeva N.A., Kuznetsov K.P. Analysis of a nonlinear dynamic model of the Couette flow for structured liquid in a flat gap. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, vol. 2 (27), pp. 85–92. (In Russian).
 Neto C., Evans D., Bonaccurso E., Butt H.J., Craig V.S.J. Boundary slip in Newtonian liquids: a review of experimental studies. Rep. Prog. Phys., 2005, vol. 68 (12), pp. 2859–2897. DOI: 10.1088/00344885/68/12/R05.
 Beirão Da Veiga H. Regularity for Stokes and generalized Stokes systems under nonhomogeneous sliptype boundary conditions. Adv. Differential Equations, 2004, 9 (9–10), pp. 1079–1114.
 Bocquet L., Charlaix E. Nanofluidics, from bulk to interfaces. Chem. Soc. Rev., 2010, vol. 39, pp. 1073–1095. DOI: 10.1039/b909366b.
 Bouzigues C. I., Tabeling P., Bocquet L. Nanofluidics in the debye layer at hydrophilic and hydrophobic surfaces. Phys. Rev. Lett., 2008, vol. 101, pp. 114503.
 Ng C.O., Wang C.Y. Apparent slip arising from Stokes shear flow over a bidimensional patterned surface. Microfluid Nanofluid, 2010, vol. 8, pp. 361–371. DOI: 10.1007/s104040090466x.
 Wang Y., Bhushan B. Boundary slip and nanobubble study in micro/nanofluidics using atomic force microscopy. Soft Matter., 2010, vol. 6, pp. 29–66. DOI: 10.1039/B917017K.
 Schwarz K.G., Schwarz Y.A. Stability of Advective Flow in a Horizontal Incompressible Fluid Layer in the Presence of the Navier Slip Condition. Fluid Dyn., 2020, iss. 55, pp. 31–42. DOI: 10.1134/S0015462820010115.
 Burmasheva N.V., Privalova V.V., Prosviryakov E.Y. Layered Marangoni convection with the Navier slip condition. Sādhanā, 2021, vol. 46, pp. 55. DOI: 10.1007/s12046021015855.
 Privalova V.V., Prosviryakov E.Yu. Nonlinear isobaric flow of a viscous incompressible fluid in a thin layer with permeable boundaries. Computational Continuum Mechanics, 2019, vol. 12, No. 2, pp. 230–242. DOI: 10.7242/19996691/2019.12.2.20. (In Russian).
 Birikh R.V. Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys., 1966, No. 7, pp. 43–44. DOI: 10.1007/BF00914697.
 Ostroumov G.A. Free convection under the condition of the internal problem. Washington, NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, 1958.
 Smith M.K., Davis S.H. Instabilities of dynamic thermocapillary liquid layers. Part. 1. Convective instabilities. J. Fluid Mech, 1983, vol. 132, pp. 119–144. DOI: 10.1017/S0022112083001512.
 OrtizPérez A.S., DávalosOrozco L.A. Convection in a horizontal ﬂuid layer under an inclined temperature gradient. Phys. Fluid, 2011, vol. 23, iss. 8, pp. 084107. DOI: 10.1063/1.3626009.
 Burmasheva N.V., Prosviryakov E.Yu. Exact solution for stable convective concentration flows of a Couette type. Computational Continuum Mechanics, 2020, vol. 13, No. 3, pp. 337–349. DOI: 10.7242/19996691/2020.13.3.27. (In Russian).
 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 p. (In Russian).
 Aristov S.N., Prosviryakov E.Y. A new class of exact solutions for threedimensional thermal diffusion equations. Theoretical Foundations of Chemical Engineering, 2016, vol. 50, No. 3, pp. 286–293. DOI: 10.1134/S0040579516030027.
 Burmasheva N.V., Prosviryakov E.Y. Thermocapillary convection of a vertical swirling liquid. Theoretical Foundations of Chemical Engineering, 2020, vol. 54, No. 1, pp. 230–239. DOI: 10.1134/S0040579519060034.
 Birikh R.V., Pukhnachev V.V., Frolovskaya O.A. Convective flow in a horizontal channel with nonnewtonian surface rheology under timedependent longitudinal temperature gradient. Fluid Dynamics, 2015, vol. 50, No. 1, pp. 173–179. DOI: 10.1134/S0015462815010172.
 Burmasheva N.V., Prosviryakov E.Yu. A largescale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Velocity field investigation. J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 2017, vol. 21, No.1, pp. 180–196. DOI: 10.14498/vsgtu1527. (In Russian).
 Burmasheva N.V., Prosviryakov E.Yu. A largescale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and pressure 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).
 Burmasheva N.V., Prosviryakov E.Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation. J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 2019, vol. 23, No. 2, pp. 341–360. DOI: 10.14498/vsgtu1670.
 Aristov S.N., Privalova V.V., Prosviryakov E.Yu. Stationary nonisothermal Couette flow. quadratic heating of the upper boundary of the fluid layer. Nelineynaya Dinamika, 2016, vol. 12, No. 2, pp. 167–178. DOI: 10.20537/nd1602001. (In Russian).
 Burmasheva N.V., Prosviryakov E.Yu. On Marangoni shear convective flows of inhomogeneous viscous incompressible fluids in view of the Soret effect. Journal of King Saud University – Science, 2020, vol. 32, iss. 8, pp. 3364–3371. DOI: 10.1016/j.jksus.2020.09.023.
 Schwarz K.G. Planeparallel advective flow in a horizontal incompressible fluid layer with rigid boundaries. Fluid Dynamics, 2019, vol. 49, No. 4, pp. 438–442. DOI: 10.1134/S0015462819110016.
 Andreev V.K., Stepanova I.V. Unidirectional flows of binary mixtures within the framework of the Oberbeck–Boussinesq model. Fluid Dynamics, 2016, vol. 51, No. 2, pp. 136–147. DOI: 10.1134/S0015462816020022.
 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.
 Bekezhanova V.B. Change of the types of instability of a steady twolayer flow in an inclined channel. Fluid Dynamics, 2011, vol. 46 (525). DOI: 10.1134/S001546281104003X.
 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. (In Russian).
 Pukhnachev V.V. Nonstationary analogues of the Birikh solution. Izv. AltGU, 2011, Nos. 1–2 (69), pp. 62–69. (In Russian).
 Shvarz K.G. Planeparallel advective flow in a horizontal incompressible fluid layer with rigid boundaries. Fluid Dynamics, 2014, vol. 49, No. 4, pp. 438–442. DOI: 10.1134/S0015462814040036.
 Andreev V.K. Resheniya Birikha uravneniy konvektsii i nekotoryye ego obobshcheniya [Birich's solutions of convection equations and some of its generalizations: preprint]. Krasnoyarsk, IVM SO RAN Publ., 2010, 24 p. (In Russian).
 Aristov S.N., Prosviryakov E.Y., Spevak L.F. Unsteadystate Bénard–Marangoni convection in layered viscous incompressible flows. Theoretical Foundations of Chemical Engineering, 2016, vol. 50, No. 2, pp. 132–141. DOI: 10.1134/S0040579516020019.
 Aristov S.N., Prosviryakov E.Yu., 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: 10.7242/19996691/2015.8.4.38. (In Russian).
 Aristov S.N., Prosviryakov E.Yu. On laminar ﬂows of planar free convection. Nelineynaya Dinamika, 2013, vol. 9, No. 4, pp. 651–657. (In Russian).
 Gorshkov A.V., Prosviryakov E.Y. Layered B´enardMarangoni 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).
 Gorshkov A.V., Prosviryakov E.Y. Analytic solutions of stationary complex convection describing a shear stress field of different signs. Trudy IMM UrO RAN, 2017, vol. 23, No. 2, pp. 32–41. DOI: 10.21538/0134488920172323241. (In Russian).
 Knyazev D.V. Twodimensional flows of a viscous binary fluid between moving solid boundaries. Journal of Applied Mechanics and Technical Physics, 2011, vol. 52, No. 2, pp. 212–217. DOI: 10.1134/S0021894411020088.
 Kompaniets L.A., Pitalskaya O.S. Exact solutions of Ekman's model for threedimensional windinduced flow of homogeneous fluid with geostrophic current. Computer Research and Modeling, 2009, vol. 1, No.1, pp. 57–66. DOI: 10.20537/207676332009115766. (In Russian).
 Burmasheva N.V., Prosviryakov E.Yu. Exact solution of NavierStokes equations describing spatially inhomogeneous flows of a rotating fluid. Trudy IMM UrO RAN, 2020, vol. 26, No. 2, pp. 79–87. DOI: 10.21538/0134488920202627987. (In Russian).
 Burmasheva N.V., Prosviryakov E.Yu. A class of exact solutions for twodimensional equations of geophysical hydrodynamics with two Coriolis parameters. The Bulletin of Irkutsk State University. Series «Mathematics», 2020, vol. 32, pp. 33–48. DOI: 10.26516/19977670.2020.32.33. (In Russian).
 Burmasheva N.V., Prosviryakov E.Yu. An exact solution for describing the unidirectional Marangoni flow of a viscous incompressible fluid with the Navier boundary condition. Temperature field investigation. Diagnostics, Resource and Mechanics of materials and structures, 2020, iss. 1, pp. 6–23. DOI: 10.17804/24109908.2020.1.006023. Available at: https://dreamjournal.org/issues/20201/20201_278.html
 Burmasheva N.V., Prosviryakov E.Yu. Exact solution for describing a unidirectional Marangoni flow of a viscous incompressible fluid with the Navier boundary condition. Pressure field investigation. Diagnostics, Resource and Mechanics of materials and structures, 2020, iss. 2, pp. 61–75, DOI: 10.17804/24109908.2020.2.061075. Available at: https://dreamjournal.org/DREAM_Issue_2_2020_Burmasheva_N.V._et_al._061_075.pdf
 Lin C.C. Note on a class of exact solutions in magnetohydrodynamics. Arch. Rational Mech. Anal., 1957, vol. 1, pp. 391–395. DOI: 10.1007/BF00298016.
 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. Phy., 1989, vol. 30, No. 2, pp. 197–203. DOI: 10.1007/BF00852164.
 Aristov S.N. Eddy currents in thin liquid layers: Synopsis of a Dr. Phys. & Math. Sci. Thesis, Vladivostok, 1990, 303 p. (In Russian).
 Prosviryakov E.Y. New class of exact solutions of navier–stokes equations with exponential dependence of velocity on two spatial coordinates. Theoretical Foundations of Chemical Engineering, 2019, vol. 53, No. 1, pp. 107–114. DOI: 10.1134/S0040579518060088.
 Aristov S.N., Prosviryakov E.Y. Largescale flows of viscous incompressible vortical fluid. Russian Aeronautics, 2015, vol. 58, No. 4, pp. 413–418. DOI: 10.3103/S1068799815040091.
 Aristov S.N., Prosviryakov E.Y. Inhomogeneous Couette ﬂow. Nelineynaya Dinamika, 2014, vol. 10, No. 2, pp. 177–182. DOI: 10.20537/nd1402004. (In Russian).
 Privalova V.V., Prosviryakov E.Yu. Vortex flows of a viscous incompressible fluid at constant vertical velocity under perfect slip conditions. Diagnostics, Resource and Mechanics of materials and structures, 2019, iss. 2, pp. 57–70. DOI: 10.17804/24109908.2019.2.057070. Available at: https://dreamjournal.org/issues/20192/20192_249.html
 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.
 Zubarev N.M., Prosviryakov E.Y. Exact solutions for layered threedimensional nonstationary isobaric flows of a viscous incompressible fluid. Journal of Applied Mechanics and Technical Physics, 2019, vol. 60, No. 6, pp. 1031–1037. DOI: 10.1134/S0021894419060075.
 Prosviryakov E.Yu. Exact solutions of threedimensional potential and vortical Couette flows of a viscous incompressible fluid. Bulletin of the National Research Nuclear University MIFI, 2015, vol. 4, No. 6, pp. 501–506. DOI: 10.1134/S2304487X15060127. (In Russian).
 Privalova V.V., Prosviryakov E.Yu., Simonov M.A. Nonlinear gradient flow of a vertical vortex fluid in a thin layer. Nelineynaya Dinamika, 2019, vol. 15, No. 3, pp. 271–283. DOI: 10.20537/nd190306. (In Russian).
 Aristov S.N., Prosviryakov E.Y. Nonuniform convective Couette flow. Fluid Dynamics, 2016, vol. 51, No. 5, pp. 581–587. DOI: 10.7868/S0568528116050030. (In Russian).
 Privalova V.V., Prosviryakov E.Yu. The effect of tangential boundary stresses on the convective unidirectional flow of viscous fluid layer under the heating the lower boundary condition. Diagnostics, Resource and Mechanics of materials and structures, 2019, iss. 4, pp. 44–55. DOI: 10.17804/24109908.2019.4.044055. Available at: https://dreamjournal.org/issues/20194/20194_262.html
 Privalova V.V., Prosviryakov E.Yu. Exact solutions for the convective creep CouetteHiemenz flow with the linear temperature distribution on the upper border. Diagnostics, Resource and Mechanics of materials and structures, 2018, iss. 2, pp. 92–109. DOI: 10.17804/24109908.2018.2.092109. Available at: https://dreamjournal.org/DREAM_Issue_2_2018_Privalova_V.V._et_al._092_109.pdf
 Aristov S.N., Privalova V.V., Prosviryakov E.Yu. Planar linear BenardRayleigh convection with quadratic heating of the upper boundary of a layer of a viscous incompressible fluid. Vestnik Kazanskogo gos. tekhn. Universiteta im. A.M. Tupoleva, 2015, vol. 71, No. 2, pp. 69–75. (In Russian).
 Privalova V.V., Prosviryakov E.Yu. Stationary convective Couette flow with linear heating of the lower boundary of the liquid layer. Vestnik Kazanskogo gos. tekhn. Universiteta im. A.M. Tupoleva, 2015, vol. 71, No. 2, pp. 148–153. (In Russian).
 Aristov S.N., Privalova V.V., Prosviryakov E.Y. Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer. Russian Journal of Nonlinear Dynamics, 2016, vol. 12, No. 2, pp. 167–178. DOI: 10.20537/nd1602001. (In Russian).
 Aristov S.N., Prosviryakov E.Y. On one class of analytic solutions of the stationary axisymmetric convection Bénard–Maragoni viscous incompreeible fluid. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2013, No. 3 (32), pp. 110–118. (In Russian).
 Aristov S.N., Prosviryakov E.Y. Exact solutions of thermocapillary convection with localized heating of a flat layer of a viscous incompressible fluid. Vestnik Kazanskogo gos. tekhn. Universiteta im. A.M. Tupoleva, 2014, No. 3, pp. 7–12. (In Russian).
 Andreev V.K., Cheremnykh E.N. 2D thermocapillary motion of three fluids in a flat channel. Journal of Siberian Federal University. Mathematics and Physics, 2016, vol. 9, iss. 4, pp. 404–415. DOI: 10.17516/19971397201694404415.
 Andreev V.K., Cheremnykh E.N. The joint creeping motion of three viscid liquids in a plane layer: A priori estimates and convergence to steady flow. Journal of Applied and Industrial Mathematics, 2016, vol. 10, No. 1, pp. 7–20. DOI: 10.1134/S1990478916010026.
 Andreev V.K., Sobachkina N.L. Dvizhenie binarnoy smesi v ploskikh i cilindricheskikh oblastyazkh [Movement of a binary mixture in flat and cylindrical regions]. Krasnoyarsk, SFU Publ., 2012, 187 p. (In Russian).
 Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingler’s Politech. J., 1911, vol. 326, pp. 321–324.
 Riabouchinsky D. Quelques considerations sur les mouvements plans rotationnels d’un liquid. C. R. Hebdomadaires Acad. Sci., 1924, vol. 179, pp. 1133–1136.
 Landau L.D., Lifshitz E.M. Fluid Mechanics: Volume 6 (Course of Theoretical Physics), 2nd ed., Elsevier, 558 p.
 Polyanin A.D., Zhurov A.I. Metody razdeleniya peremennykh i tochnye resheniya nelineynykh uravneniy matematicheskoy fiziki [Methods of separation of variables and exact solutions of nonlinear equations of mathematical physics]. Moscow, Izdvo IPMekh RAN Publ., 2020, 384 p. (In Russian).
 Aristov S.N., Polyanin A.D. Exact solutions of unsteady threedimensional NavierStokes equations. Doklady Physics, 2009, vol. 54, No. 7, pp. 316–321. DOI: 10.1134/S1028335809070039.
 Polyanin A.D. Exact generalized separable solutions of the NavierStokes equations. Doklady RAN, 2001, vol. 380, No. 4, pp. 491–496. (In Russian).
 Polyanin A.D. Methods of functional separation of variables and their application in mathematical physics. Matematicheskoe Modelirovanie i Chislennye Metody, 2019, No. 1, pp. 65–97. DOI: 10.18698/23093684201916597. (In Russian).
 Polyanin A.D., Aristov S.N. Systems of hydrodynamic type equations: exact solutions, transformations, and nonlinear stability. Doklady Physics, 2009, vol. 54, No. 9, pp. 429–434. DOI: 10.1134/S1028335809090079.
 Polyanin A.D., Zhurov A.I. Functional separable solutions of two classes of nonlinear mathematical physics equations. Doklady AN, 2019, vol. 486, No. 3, pp. 287–291. DOI: 10.31857/S086956524863287291. (In Russian).
 Aristov S.N., Polyanin A.D. New classes of exact solutions and some transformations of the Navier–Stokes equations. Russian J. Math. Physics, 2010, vol. 17, No. 1, pp. 1–18. DOI: 10.1134/S1061920810010012.
 Meleshko S.V. A particular class of partially invariant solutions of the Navier–Stokes equations. Nonlinear Dynamics, 2004, vol. 36, No. 1, pp. 47–68, DOI: 10.1023/B:NODY.0000034646.18621.73.
Article reference
Burmasheva N. V., Prosviryakov E. Yu. Exact Solutions of the Navier–stokes Equations for Describing An Isobaric OneDirectional Vertical Vortex Flow of a Fluid // Diagnostics, Resource and Mechanics of materials and structures. 
2021.  Iss. 2.  P. 3051.  DOI: 10.17804/24109908.2021.2.030051. 
URL: http://eng.dreamjournal.org/issues/20212/20212_316.html (accessed: 10/11/2024).
