N. V. Burmasheva, E. Yu. Prosviryakov
EXACT SOLUTIONS OF THE NAVIER–STOKES EQUATIONS FOR DESCRIBING AN ISOBARIC ONE-DIRECTIONAL VERTICAL VORTEX FLOW OF A FLUID
DOI: 10.17804/2410-9908.2021.2.030-051 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 parabolic-type 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 Bibliography:
- 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.
- 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 Navier-Stokes 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 Navier-Stokes equations-the 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 steady-state Navier-Stokes 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 Navier-Stokes equations. Appl. Mech. Rev. 1989, vol. 42 (11S), pp. 269–282. DOI: 10.1115/1.3152400.
- Pukhnachev V.V. Symmetries in the Navier-Stokes 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 steady-state 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. 449-456. 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/s10958-012-1003-0.
- Zhuk V.I., Protsenko I.G. Asymptotic model for the evolution of perturbations in the plane Couette-Poiseuille 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 two-phase 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. Physics-Uspekhi, 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 nano-scopic 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. 716-727. 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: 10.1088/0034-4885/68/12/R05.
- Beirão Da Veiga H. Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type 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/s10404-009-0466-x.
- 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/s12046-021-01585-5.
- 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: 10.7242/1999-6691/2019.12.2.20. (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.
- Ortiz-Pérez A.S., Dávalos-Orozco 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: 10.7242/1999-6691/2020.13.3.27. (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 three-dimensional 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 non-newtonian surface rheology under time-dependent longitudinal temperature gradient. Fluid Dynamics, 2015, vol. 50, No. 1, pp. 173–179. DOI: 10.1134/S0015462815010172.
- 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. 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 large-scale 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. Plane-parallel 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 two-layer 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. Non-stationary analogues of the Birikh solution. Izv. AltGU, 2011, Nos. 1–2 (69), pp. 62–69. (In Russian).
- Shvarz K.G. Plane-parallel 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. Unsteady-state 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: 10.7242/1999-6691/2015.8.4.38. (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´enard-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).
- 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: 10.21538/0134-4889-2017-23-2-32-41. (In Russian).
- Knyazev D.V. Two-dimensional 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 three-dimensional wind-induced flow of homogeneous fluid with geostrophic current. Computer Research and Modeling, 2009, vol. 1, No.1, pp. 57–66. DOI: 10.20537/2076-7633-2009-1-1-57-66. (In Russian).
- Burmasheva N.V., Prosviryakov E.Yu. Exact solution of Navier-Stokes equations describing spatially inhomogeneous flows of a rotating fluid. Trudy IMM UrO RAN, 2020, vol. 26, No. 2, pp. 79–87. DOI: 10.21538/0134-4889-2020-26-2-79-87. (In Russian).
- Burmasheva N.V., Prosviryakov E.Yu. A class of exact solutions for two-dimensional equations of geophysical hydrodynamics with two Coriolis parameters. The Bulletin of Irkutsk State University. Series «Mathematics», 2020, vol. 32, pp. 33–48. DOI: 10.26516/1997-7670.2020.32.33. (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: 10.17804/2410-9908.2020.1.006-023. Available at: https://dream-journal.org/issues/2020-1/2020-1_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: 10.17804/2410-9908.2020.2.061-075. Available at: https://dream-journal.org/DREAM_Issue_2_2020_Burmasheva_N.V._et_al._061_075.pdf
- Lin C.C. Note on a class of exact solutions in magneto-hydrodynamics. 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. Large-scale 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: 10.17804/2410-9908.2019.2.057-070. Available at: https://dream-journal.org/issues/2019-2/2019-2_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 three-dimensional 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 three-dimensional 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: 10.17804/2410-9908.2019.4.044-055. Available at: https://dream-journal.org/issues/2019-4/2019-4_262.html
- Privalova V.V., Prosviryakov E.Yu. Exact solutions for the convective creep Couette-Hiemenz 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: 10.17804/2410-9908.2018.2.092-109. Available at: https://dream-journal.org/DREAM_Issue_2_2018_Privalova_V.V._et_al._092_109.pdf
- Aristov S.N., Privalova V.V., Prosviryakov E.Yu. Planar linear Benard-Rayleigh 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: 10.17516/1997-1397-2016-9-4-404-415.
- 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 Flüssigkeit-sstrom 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, Izd-vo IPMekh RAN Publ., 2020, 384 p. (In Russian).
- Aristov S.N., Polyanin A.D. Exact solutions of unsteady three-dimensional Navier-Stokes equations. Doklady Physics, 2009, vol. 54, No. 7, pp. 316–321. DOI: 10.1134/S1028335809070039.
- Polyanin A.D. Exact generalized separable solutions of the Navier-Stokes 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: 10.18698/2309-3684-2019-1-6597. (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/S0869-56524863287-291. (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.
Н. В. Бурмашева, Е. Ю. Просвиряков
ТОЧНЫЕ РЕШЕНИЯ УРАВНЕНИЙ НАВЬЕ–СТОКСА ДЛЯ ОПИСАНИЯ ИЗОБАРИЧЕСКОГО ОДНОНАПРАВЛЕННОГО ВЕРТИКАЛЬНО ЗАВИХРЕННОГО ТЕЧЕНИЯ ЖИДКОСТИ
В статье предложено семейство точных решений уравнений Навье–Стокса для описания изобарических неоднородных однонаправленных движений жидкости. Из-за уравнения несжимаемости скорость неоднородного течения Куэтта зависит от двух координат и времени. Приведенное выражение для поля скоростей обладает широким функциональным произволом. Данное точное решение получено методом разделения переменных, причем используются обе алгебраические операции (аддитивность и мультипликативность) для обоснования важности модификации классического течения Куэтта. В статье собрана значительная библиографическая информация, позволяющая проследить изменение точного решения Куэтта для различных областей гидродинамики ньютоновской несжимаемой жидкости. Течение жидкости представлено полиномом, зависящим от одной переменной (горизонтальной координаты). Коэффициенты полинома функционально зависят от второй (вертикальной) координаты и времени и определяются цепочкой простейших однородных и неоднородных уравнений в частных производных параболического типа. Цепочка уравнений получена методом неопределенных коэффициентов после подстановки точного решения в уравнение Навье–Стокса. Приведен алгоритм интегрирования системы обыкновенных дифференциальных уравнений для изучения установившегося движения вязкой жидкости. В этом случае все функции, определяющие скорость, являются полиномами, топология вектора завихренности и касательных напряжений даже без конвективного перемешивания (ползущее течение) имеет сложную структуру.
Ключевые слова: точное решение, течение Куэтта, уравнение Навье–Стокса, неоднородное однонаправленное течение, метод разделения переменных, касательное напряжение Библиография:
- 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. – P. 642–662. DOI: 10.1134/S0040579509050066.
- Drazin P. G. and 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.
- Whitham G. B. The Navier-Stokes equations of motion / ed. L. Rosenhead. – Oxford, Clarendon : Laminar Boundary Layers, 1963. – P. 114–162.
- Dryden H. L., Murnaghan F. D., Bateman H. Report of the committee on hydrodynamics // Bull. Natl. Res. Counc. (US). – 1932. – Vol. 84 – P. 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 / ed. S. Flugge. – Berlin : Springer-Verlag. Handbuch der Physik, 1963. – VIII/2. – 384 p.
- Wang C. Y. Exact solution of the Navier-Stokes equations-the generalized Beltrami flows, review and extension // Acta Mech. – 1990. – Vol. 81. – P. 69–74. – DOI: 10.1007/BF01174556.
- Wang C. Y. Exact solutions of the steady-state Navier-Stokes equations // Annu. Rev. Fluid Mech. – 1991. – Vol. 23. – P. 159–177. – DOI: 10.1146/annurev.fl.23.010191.001111.
- Wang, C. Y. Exact solutions of the unsteady Navier-Stokes equations // Appl. Mech. Rev. – 1989. – Vol. 42 (11S). – P. 269–282. – DOI: 10.1115/1.3152400.
- Пухначёв В. В. Симметрии в уравнениях Навье-Стокса // Успехи механики. – 2006. – № 1. – С. 6–76.
- Couette M. Etudes sur le frottement des liquids // Ann. Chim. Phys. – 1890. – Vol. 21. – P. 433–510.
- Stokes G. G. On the effect of the internal friction of fluid on the motion of pendulums // Camb. Philo. Trans. – 1851. – Vol. 9. – P. 8–106.
- Taylor G. I. Stability of a Viscous Liquid Contained between Two Rotating Cylinders // Philosophical Transactions Royal Society of London. – 1923. – Vol. 223, No. 605–615. – P. 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 steady-state solutions // J. Fluid Mech. – 1981. – Vol. 108. – P. 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. – P. 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. – P. 452–460. – DOI: 10.1134/S0015462813040042.
- Wave number selection in the presence of noise: Experimental results / D. Zhilenko, O. Krivonosova, M. Gritsevich, P. Read // Chaos. – 2018. – Vol. 28. – 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. – P. 449–456. – DOI: 10.1134/S1063784210040031.
- Беляев Ю. Н., Монахов А. А., Яворская И. М. Устойчивость сферического течения Куэтта в толстых слоях при вращении внутренней сферы // Известия Академии наук СССР. Механика жидкости и газа. – 1978. – № 2. – С. 9–15.
- Pukhnachev V. V., Pukhnacheva T. P. The Couette problem for a Kelvin–Voigt medium // J. Math. Sci. – 2012. – Vol. 186. – P. 495–510. – DOI: 10.1007/s10958-012-1003-0.
- Zhuk V. I., Protsenko I. G. Asymptotic model for the evolution of perturbations in the plane Couette-Poiseuille flow // Doklady Mathematics. – 2006. – Vol. 74, No. 3. – P. 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. – P. 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. – P. 181–185. – DOI: 10.1134/S1064562412020068.
- Rudyak V., Isakov E., & Bord E. Instability of plane Couette flow of two-phase liquids // Tech. Phys. Lett. – 1998. – Vol. 24. – P. 199–200. – DOI: 10.1134/1.1262051.
- Shalybkov D. A. Hydrodynamic and hydromagnetic stability of the Couette flow // Physics-Uspekhi. – 2009. – Vol. 52, No. 9. – P. 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. – P. 64–71. – DOI: 10.1134/S0015462811010078.
- Кудинов В. А., Кудинов И. В. Получение точных аналитических решений гиперболических уравнений движения при разгонном течении Куэтта // Изв. РАН. Энергетика. – 2012. – № 1. – С. 119–133.
- Babkin V. A. Plane Turbulent Couette Flow // Journal of Engineering Physics and Thermophysics. – 2003. – Vol. 76. – P. 1251–1254. – DOI: 10.1023/B:JOEP.0000012026.19646.c6.
- Абрамян А. К., Миранцев Л. В., Кучмин А. Ю. Моделирование течения Куэтта простой жидкости в плоском канале наноразмерной высоты // Математическое моделирование. – 2012. – Т. 24, № 4. – С. 3–21.
- Malyshev V. and Manita A. Stochastic Micromodel of the Couette Flow // Theor. Prob. Appl. – 2009. – Vol. 53, No. 4. – P. 716–727. – 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. – P. 140–143. – DOI: 10.3103/S1062873810121044.
- Беляева Н. А., Кузнецов К. П. Анализ нелинейной динамической модели течения Куэтта структурированной жидкости в плоском зазоре // Вестник СамГТУ. Сер. Физико-математические науки. – 2012. – № 2 (27). – С. 85–92.
- Boundary slip in Newtonian liquids: a review of experimental studies / C. Neto, D. Evans, E. Bonaccurso, H.-J. Butt, V. S. J. Craig // Rep. Prog. Phys. – 2005. – Vol. 68. – P. 2859–2897. – DOI: 10.1088/0034-4885/68/12/R05.
- Beirao da Veiga H. Regularity for Stokes and general-ized Stokes systems under nonhomogeneous slip-typeboundary conditions // Adv. Different. – 2004. – 9 (9–10). – P. 1079–1114.
- Bocquet L., Charlaix E. Nanofluidics, from bulk to interfaces // Chem. Soc. Rev. – 2010. – Vol. 39. – P. 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. – P. 114503.
- Ng C. O., Wang C. Y. Apparent slip arising from Stokes shear flow over a bidimensional patterned surface // Microfluid Nanofluid. – 2010. – Vol. 8. – P. 361–371. – DOI: 10.1007/s10404-009-0466-x.
- Wang Y., Bhushan B. Boundary slip and nanobubble study in micro/nanofluidics using atomic force microscopy // Soft Matter. – 2010. – Vol. 6. – P. 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 Dynamics. – 2020. – Iss. 55. – P. 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. – 55. – DOI: 10.1007/s12046-021-01585-5.
- Привалова В. В., Просвиряков Е. Ю. Нелинейное изобарическое течение вязкой несжимаемой жидкости в тонком слое с проницаемыми границами // Вычислительная механика сплошных сред. – 2019. – Т. 12, № 2. – С. 230–242. – DOI: 10.7242/1999-6691/2019.12.2.20.
- Birikh R. V. Thermocapillary convection in a horizontal layer of liquid // J. Appl. Mech. Tech. Phys. – 1966. – No. 7. – P. 43–44. – DOI: 10.1007/BF00914697.
- Остроумов Г. А. Свободная конвекция в условиях внутренней задачи. – М. : Гостехиздат, 1952. – 256 с.
- Smith M. K., Davis S. H. Instabilities of dynamic thermocapillary liquid layers. Part. 1. Convective instabilities // J. Fluid Mech. – 1983. – Vol. 132. – Р. 119–144. – DOI: 10.1017/S0022112083001512.
- Ortiz-Pérez A. S., Dávalos-Orozco L. A. Convection in a horizontal fluid layer under an inclined temperature gradient // Phys. Fluid. – 2011. – Vol. 23. – 084107. – DOI: 10.1063/1.3626009.
- Бурмашева Н. В., Просвиряков Е. Ю. Точное решение для установившихся конвективных концентрационных течений типа Куэтта // Вычислительная механика сплошных сред. – 2020. – Т. 13, № 3. – С. 337–349. – DOI: 10.7242/1999-6691/2020.13.3.27.
- Аристов С. Н., Шварц К. Г. Вихревые течения адвективной природы во вращающемся слое жидкости. – Пермь : Изд-во Пермск. гос. ун-та, 2006.
- Aristov S. N., Prosviryakov E. Y. A new class of exact solutions for three-dimensional thermal diffusion equations // Theoretical Foundations of Chemical Engineering. – 2016. – Vol. 50, No. 3. – P. 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. – P. 230–239. – DOI: 10.1134/S0040579519060034.
- 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 Dynamics. – 2015. – Vol. 50, No. 1. – P. 173–179. – DOI: 10.1134/S0015462815010172.
- Бурмашева Н.В., Просвиряков Е.Ю. Крупномасштабная слоистая стационарная конвекция вязкой несжимаемой жидкости под действием касательных напряжений на верхней границе. Исследование поля скоростей // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. – 2017. – Т. 21, вып. 1. – С. 180–196. – DOI: 10.14498/vsgtu1527.
- Бурмашева Н. В., Просвиряков Е. Ю. Крупномасштабная слоистая стационарная конвекция вязкой несжимаемой жидкости под действием касательных напряжений на верхней границе. Исследование полей температуры и давления // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. – 2017. – Т. 21, вып. 4. – С. 736–751. – DOI: 10.14498/vsgtu1568.
- Burmasheva N. V., Prosviryakov E. Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation // Вестник СамГТУ. Сер. Физ.-мат. науки. – 2019. – Т. 23, вып. 2. – С. 341–360. – DOI: 10.14498/vsgtu1670.
- Аристов С. Н., Привалова В. В., Просвиряков Е. Ю. Стационарное неизотермическое течение Куэтта. Квадратичный нагрев верхней границы слоя жидкости // Нелинейная динамика. – 2016. – Т. 12, вып. 2. – С. 167–178. – DOI: 10.20537/nd1602001.
- Burmasheva N. V., Prosviryakov E. Yu. On Marangoni shear convective flows of inhomogeneous viscousincompressible fluids in view of the Soret effect // Journal of King Saud University - Science. – 2020. – Vol. 32, iss. 8. – P. 3364–3371. – DOI: 10.1016/j.jksus.2020.09.023.
- Schwarz K. G. Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries // Fluid Dynamics. – 2019. – Vol. 49, No. 4. – P. 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. – P. 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. – P. 59–67. – DOI: 10.1016/j.amc.2013.11.002.
- Bekezhanova V. B. Change of the types of instability of a steady two-layer flow in an inclined channel // Fluid Dynamics. – 2011. – Vol. 46 (525). – DOI: 10.1134/S001546281104003X.
- Горшков А. В., Просвиряков Е. Ю. Конвективное слоистое течение Экмана вязкой несжимаемой жидкости // Известия РАН. Физика атмосферы и океана. – 2018. – Т. 54, вып. 2. – С. 213–220. – DOI: 10.7868/S0003351518020101.
- Пухначёв В. В. Нестационарные аналоги решения Бириха // Изв. АлтГУ. – 2011. – № 1–2 (69). – С. 62–69.
- Shvarz K. G. Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries // Fluid Dynamics. – 2014. – Vol. 49, No. 4. – P. 438–442. – DOI: 10.1134/S0015462814040036.
- Андреев В. К. Решения Бириха уравнений конвекции и некоторые его обобщения : Препринт. – Красноярск : ИВМ СО РАН, 2010. – 24 с.
- Aristov S. N., Prosviryakov E. Y., Spevak L. F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows // Theoretical Foundations of Chemical Engineering. – 2016. – Vol. 50, No. 2. – P. 132–141. – DOI: 10.1134/S0040579516020019.
- Аристов С. Н., Просвиряков Е. Ю., Спевак Л. Ф. Нестационарная слоистая тепловая и концентрационная конвекция Марангони вязкой жидкости // Вычисл. мех. сплош. сред. – 2015. – Т. 8, № 4. – C. 445–456. – DOI: 10.7242/1999-6691/2015.8.4.38.
- Аристов С.Н., Просвиряков Е.Ю. О слоистых течениях плоской свободной конвекции // Нелин. динам. – 2013. – Т. 9. – № 4. – С. 651–657.
- Горшков А. В., Просвиряков Е. Ю. Слоистая конвекция Бенара-Марангони при теплообмене по закону Ньютона-Рихмана // Компьютерные исследования и моделирование. – 2016. – Т. 8, № 6. – С. 927–940.
- Горшков А. В., Просвиряков Е. Ю. Аналитические решения стационарной сложной конвекции, описывающие поле касательных напряжений разного знака // Тр. ИММ УрО РАН. – 2017. – Т. 23, № 2. – С. 32–41. – DOI: 10.21538/0134-4889-2017-23-2-32-41.
- Knyazev D. V. Two-dimensional flows of a viscous binary fluid between moving solid boundaries // Journal of Applied Mechanics and Technical Physics. – 2011. – Vol. 52, No. 2. – P. 212–217. – DOI: 10.1134/S0021894411020088.
- Компаниец Л. А., Питальская О. С. Точные решения модели Экмана трехмерного ветрового движения однородной жидкости с учетом геострофической составляющей // Компьютерные исследования и моделирование. – 2009. – Т. 1, вып. 1. – С. 57–66. – DOI: 10.20537/2076-7633-2009-1-1-57-66.
- Бурмашева Н. В., Просвиряков Е. Ю. Точное решение уравнений Навье–Стокса, описывающее пространственно неоднородные течения вращающейся жидкости // Труды Института математики и механики УрО РАН. – 2020. – Т. 26, вып. 2. – C. 79–87. – DOI: 10.21538/0134-4889-2020-26-2-79-87.
- Бурмашева Н. В., Просвиряков Е. Ю. Класс точных решений для двумерных уравнений геофизической гидродинамики с двумя параметрами Кориолиса // Известия Иркутского государственного университета. Серия Математика. – 2020. – Т. 32. – С. 33–48. – DOI: 10.26516/1997-7670.2020.32.33.
- 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. – P. 6–23. – DOI: 10.17804/2410-9908.2020.1.006-023. – URL: https://dream-journal.org/issues/2020-1/2020-1_278.html
- Burmasheva N. V. and 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. – P. 61–75. – DOI: 10.17804/2410-9908.2020.2.061-075. – URL: https://dream-journal.org/DREAM_Issue_2_2020_Burmasheva_N.V._et_al._061_075.pdf
- Lin C. C. Note on a class of exact solutions in magneto-hydrodynamics // Arch. Rational Mech. Anal. – 1957. – Vol. 1. – P. 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. Phys. – 1989. – Vol. 30, No. 2. – P. 197–203. – DOI: 10.1007/BF00852164.
- Аристов С. Н. Вихревые течения в тонких слоях жидкости : автореф. дис. ... докт. физ.-мат. наук: 01.02.05 / Аристов Сергей Николаевич. – Владивосток : ИАПУ, 1990. – 303 с.
- 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. – P. 107–114. – DOI: 10.1134/S0040579518060088.
- Аристов С. Н., Просвиряков Е. Ю. Крупномасштабные течения завихренной вязкой несжимаемой жидкости // Известия высших учебных заведений. Авиационная техника. – 2015. – Вып. 4. – С. 50–54.
- Аристов С. Н., Просвиряков Е. Ю. Неоднородные течения Куэтта // Нелинейная динамика. – 2014. – Т. 10, вып. 2. – C. 177–182. – DOI: 10.20537/nd1402004.
- 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. – P. 57–70. – DOI: 10.17804/2410-9908.2019.2.057-070. – URL: https://dream-journal.org/issues/2019-2/2019-2_249.html
- Aristov S. N., Prosviryakov E. Y. Unsteady layered vortical fluid flows // Fluid Dynamics. – 2016. – Vol. 51, No. 2. – P. 148–154. – DOI: 10.1134/S0015462816020034.
- Zubarev N. M., Prosviryakov E. Y. Exact solutions for layered three-dimensional nonstationary isobaric flows of a viscous incompressible fluid // Journal of Applied Mechanics and Technical Physics. – 2019. – Vol. 60, No. 6. – P. 1031–1037. – DOI: 10.1134/S0021894419060075.
- Просвиряков Е. Ю. Точные решения трехмерных потенциальных и завихренных течений Куэтта вязкой несжимаемой жидкости // Вестник Национального исследовательского ядерного университета МИФИ. – 2015. – Т. 4, вып. 6. – С. 501-506. – DOI: 10.1134/S2304487X15060127.
- Privalova V. V., Prosviryakov E. Yu., Simonov M. A. Nonlinear gradient flow of a vertical vortex fluid in a thin layer // Нелинейн. динам. – 2019. – Т. 15, вып. 3. – С. 271–283. – DOI: 10.20537/nd190306.
- Аристов С. Н., Просвиряков Е. Ю. Неоднородное конвективное течение Куэтта // Известия Российской академии наук. Механика жидкости и газа. – 2016. – № 5. – С. 3–9. – DOI: 10.7868/S0568528116050030.
- 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. – P. 44–55. – DOI: 10.17804/2410-9908.2019.4.044-055. – URL: https://dream-journal.org/issues/2019-4/2019-4_262.html
- Privalova V. V., Prosviryakov E. Yu. Exact solutions for the convective creep Couette-Hiemenz flow with the linear temperature distribution on the upper border // Diagnostics, Resource and Mechanics of materials and structures. – 2018. – Iss. 2. – P. 92–109. – DOI: 10.17804/2410-9908.2018.2.092-109. – URL: https://dream-journal.org/DREAM_Issue_2_2018_Privalova_V.V._et_al._092_109.pdf
- Аристов С. Н., Привалова В. В., Просвиряков Е. Ю. Плоская линейная конвекция Бенара-Рэлея при квадратичном нагреве верхней границы слоя вязкой несжимаемой жидкости // Вестник Казанского государственного технического университета им. А.Н. Туполева. – 2015. – Т. 71, № 2. – С. 69–75.
- Привалова В. В., Просвиряков Е. Ю. Стационарное конвективное течение Куэтта при линейном нагреве нижней границы слоя жидкости // Вестник Казанского государственного технического университета им. А.Н. Туполева. – 2015. – Т. 71, № 5. – С. 148–153.
- 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. – P. 167–178. – DOI: 10.20537/nd1602001.
- Аристов С. Н., Просвиряков Е. Ю. Об одном классе аналитических решений стационарной осесимметричной конвекции Бенара–Марангони вязкой несжимаемой жидкости // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. – 2013. – № 3 (32). – C. 110–118.
- Аристов С. Н., Просвиряков Е. Ю. Точные решения термокапиллярной конвекции при локализованном нагреве плоского слоя вязкой несжимаемой жидкости // Вестн. Казан. гос. техн. ун-та им. А. Н. Туполева. – 2014. – № 3. – С. 7–12.
- Andreev V. K., Cheremnykh E. N. 2D thermocapillary motion of three fluids in a flat chan_nel. Journal of Siberian Federal University // Mathematics and Physics. – 2016. – Vol. 9, iss. 4. – P. 404–415. – DOI: 10.17516/1997-1397-2016-9-4-404-415.
- 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. – P. 7–20. – DOI: 10.1134/S1990478916010026.
- Андреев В. К., Собачкина Н. Л. Движение бинарной смеси в плоских и цилиндриче_ских областях. – Красноярск : СФУ, 2012. – 187 с.
- Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeit-sstrom eingetauchten geraden Kreiszylinder // Dingler’s Politech. J. – 1911. – Vol. 326. – P. 321–324.
- Riabouchinsky D. Quelques considerations sur les mouvements plans rotationnels d’ un liquide // C. R. Hebdomadaires Acad. Sci. – 1924. – Vol. 179. – P. 1133–1136.
- Ландау Л. Д., Лифшиц Е. М. Гидродинамика. – 6-е. изд. – М. : Физматлит, 2006.
- Полянин А. Д., Журов А. И. Методы разделения переменных и точные решения нелинейных уравнений математической физики. – М. : Издатель-ство «ИПМех РАН», 2020. – 384 с.
- Aristov S. N., Polyanin A. D. Exact solutions of unsteady three-dimensional Navier-Stokes equations // Doklady Physics. – 2009. – Vol. 54, No. 7. – P. 316–321. – DOI: 10.1134/S1028335809070039.
- Полянин А. Д. Точные решения уравнений Навье-Стокса с обобщенным разделением переменных // Доклады АН. – 2001. – Т. 380, № 4. – С. 491–496.
- Полянин А. Д. Методы функционального разделения переменных и их применение в математической физике // Мат. моделирование и численные методы. – 2019. – № 1. – С. 65–97. – DOI: 10.18698/2309-3684-2019-1-6597.
- Polyanin A. D., Aristov S. N. Systems of hydrodynamic type equations: exact solutions, transformations, and nonlinear stability // Doklady Physics. – 2009. – Vol. 54, No. 9. – P. 429–434. – DOI: 10.1134/S1028335809090079.
- Полянин А. Д., Журов А. И. Решения с функциональным разделением переменных двух классов нелинейных уравнений математической физики // Доклады АН. – 2019. – Т. 486, № 3. – С. 287–291. – DOI: 10.31857/S0869-56524863287-291.
- 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. – P. 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. – P. 47–68. – DOI: 10.1023/B:NODY.0000034646.18621.73.
Библиографическая ссылка на статью
Burmasheva N. V., Prosviryakov E. Yu. Exact Solutions of the Navier–stokes Equations for Describing An Isobaric One-Directional Vertical Vortex Flow of a Fluid // Diagnostics, Resource and Mechanics of materials and structures. -
2021. - Iss. 2. - P. 30-51. - DOI: 10.17804/2410-9908.2021.2.030-051. -
URL: http://dream-journal.org/issues/content/article_316.html (accessed: 10.11.2024).
|