Электронный научный журнал
 
Diagnostics, Resource and Mechanics 
         of materials and structures
ВыпускиО журналеАвторуРецензентуКонтактыНовостиРегистрация

2024 Выпуск 3

Все выпуски
 
2024 Выпуск 6
(в работе)
 
2024 Выпуск 5
 
2024 Выпуск 4
 
2024 Выпуск 3
 
2024 Выпуск 2
 
2024 Выпуск 1
 
2023 Выпуск 6
 
2023 Выпуск 5
 
2023 Выпуск 4
 
2023 Выпуск 3
 
2023 Выпуск 2
 
2023 Выпуск 1
 
2022 Выпуск 6
 
2022 Выпуск 5
 
2022 Выпуск 4
 
2022 Выпуск 3
 
2022 Выпуск 2
 
2022 Выпуск 1
 
2021 Выпуск 6
 
2021 Выпуск 5
 
2021 Выпуск 4
 
2021 Выпуск 3
 
2021 Выпуск 2
 
2021 Выпуск 1
 
2020 Выпуск 6
 
2020 Выпуск 5
 
2020 Выпуск 4
 
2020 Выпуск 3
 
2020 Выпуск 2
 
2020 Выпуск 1
 
2019 Выпуск 6
 
2019 Выпуск 5
 
2019 Выпуск 4
 
2019 Выпуск 3
 
2019 Выпуск 2
 
2019 Выпуск 1
 
2018 Выпуск 6
 
2018 Выпуск 5
 
2018 Выпуск 4
 
2018 Выпуск 3
 
2018 Выпуск 2
 
2018 Выпуск 1
 
2017 Выпуск 6
 
2017 Выпуск 5
 
2017 Выпуск 4
 
2017 Выпуск 3
 
2017 Выпуск 2
 
2017 Выпуск 1
 
2016 Выпуск 6
 
2016 Выпуск 5
 
2016 Выпуск 4
 
2016 Выпуск 3
 
2016 Выпуск 2
 
2016 Выпуск 1
 
2015 Выпуск 6
 
2015 Выпуск 5
 
2015 Выпуск 4
 
2015 Выпуск 3
 
2015 Выпуск 2
 
2015 Выпуск 1

 

 

 

 

 

N. V. Burmasheva, E. Yu. Prosviryakov

EXACT SOLUTIONS TO THE NAVIER–STOKES EQUATIONS FOR UNIDIRECTIONAL FLOWS OF MICROPOLAR FLUIDS IN A MASS FORCE FIELD

DOI: 10.17804/2410-9908.2024.3.041-063

The paper presents a family of exact solutions to the Navier-Stokes equation system used to describe inhomogeneous unidirectional flows of a viscous fluid taking into account couple stresses. Despite the presence of only one non-zero component of the velocity vector, this component depends on time and two spatial coordinates. In view of the incompressibility equation, which is a special case of the mass conservation law, there is no dependence on the third spatial coordinate. The resulting redefined system of equations is considered in a non-stationary formulation. The construction of a family of exact solutions for the resulting redefined equation system begins with the analysis of the homogeneous Couette-type solution as the simplest in this class. Further, the structure of the solution gradually becomes more complicated, i.e. the profile of the only non-zero component of the velocity vector is represented as a polynomial depending on one variable (horizontal coordinate). The polynomial coefficients functionally depend on the second (vertical) coordinate and time. It is shown that, due to the strong nonlinearity and heterogeneity of the equation under study, the sum of its individual solutions is not a solution. It is also shown that, in the linearly independent basis of the power functions of the horizontal coordinate, which determine the above-mentioned polynomial, the equation in question decomposes into a chain of the simplest homogeneous and inhomogeneous parabolic partial differential equations. These equations are integrated sequentially, the order of integration being described separately. The results reported in this study extend the family of previously presented exact solutions to describing unidirectional unsteady flows.

Keywords: exact solution, Navier–Stokes equation, inhomogeneous unidirectional flow, couple stresses, micropolar fluids

References:

  1. Couette, M. Etudes sur le frottement des liquids. Ann. Chim. Phys., 1890, 21, 433–510.
  2. Poiseuille, J. Recherches expérimenteles sur le mouvement des liquides dans les tubes de très-petits diamètres. Comptes rendus des séances de l’Académie des Sciences, 1840, 11, 961–967.
  3. Poiseuille, J. Recherches expérimenteles sur le mouvement des liquides dans les tubes de très-petits diamètres. Comptes rendus des séances de l’Académie des Sciences, 1840, 11, 1041–1048.
  4. Poiseuille, J. Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres (suite). Comptes rendus des séances de l’Académie des Sciences, 1841, 12, 112–115.
  5. Stokes, G.G. On the effect of the internal friction of fluid on the motion of pendulums. Camb. Philo. Trans., 1851, 9, 8–106.
  6. Nußelt, W. Die Abhängigkeit der Wärmeübergangszahl von der Rohrlänge. Zeitschrift des Vereins Deutscher Ingenieure, 1910, 54, 1154–1158.
  7. Nagel, Yu.A. Plane one-dimensional stationary flow of an ideal charged gas in its own electric field. Journal of Applied Mechanics and Technical Physics, 1971, 12, 19–26. DOI: 10.1007/BF00853977.
  8. Faskheev, I.O. One-dimensional flow of a fluid through a porous skeleton with consideration of the Darcy and frontal pressure interaction forces. Moscow University Mechanics Bulletin, 2013, 68, 21–24. DOI: 10.3103/S0027133013010044.
  9. Wallis, G. One-Dimensional Two-Phase Flow, Mcgraw-Hill, 1969, 408 p.
  10. Borzenko, E.I. and Shrager, G.R. Techeniya nenyutonovskoy zhidkosti so svobodnoy poverkhnostyu [Flows of a Non-Newtonian Liquid With a Free Surface]. Izd-vo Tomskogo Gosudarstvennogo Universiteta Publ., Tomsk, 2022, 210 p. (In Russian).
  11. Prokudin, S.V., Sofyin, A.S., and Agapov, A.A. Analytical solution of the problem of one-dimensional stationary flowing of the compressed liquid and gas. Bezopasnost Truda v Promyshlennosti, 2017, 5, 36–41. (In Russian). DOI: 10.24000/0409-2961-2017-5-36-41.
  12. Valiyev, Kh.F., Kraiko, A.N., and Tillyayeva, N.I. Stability of one-dimensional steady flows with detonation wave in a channel of variable cross-sectional area. Computational Mathematics and Mathematical Physics, 2020, 60 (4), 697–710. DOI: 10.1134/S096554252004017X.
  13. Burmasheva, N.V. and Prosviryakov, E.Yu. An exact solution to the description of a unidirectional Marangoni flow of a viscous incompressible fluid with the Navier boundary condition. Velocity field investigation. Diagnostics, Resource and Mechanics of materials and structures, 2019, 5, 23–39. DOI: 10.17804/2410-9908.2019.5.023-039. Available at: http://dream-journal.org/issues/2019-5/2019-5_259.html
  14. 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, pp. 61–75. DOI: 10.17804/2410-9908.2020.2.061-075. Available at: http://dream-journal.org/issues/2020-2/2020-2_288.html
  15. Bhaskar, B.S. and Chaudhary, S.K. Review of fluid flow and heat transfer through porous media heat exchangers. International Journal of New Innovations in Engineering and Technology, 2016, 6 (2), 28–42.
  16. Singh, H. and Myong, R.S. Critical review of fluid flow physics at micro- to nano‐scale porous media applications in the energy sector. Advances in Materials Science and Engineering, 2018, 2018, 9565240. DOI: 10.1155/2018/9565240.
  17. Coussot, P. Yield stress fluid flows: a review of experimental data. Journal of Non-Newtonian Fluid Mechanics, 2014, 211, 31–49. DOI: 10.1016/j.jnnfm.2014.05.006.
  18. Kiselev, A. Diffusion and mixing in fluid flow: a review. In: Ed., V. Sidoravičius, New Trends in Mathematical Physics, Springer, Dordrecht, 2009. DOI: 10.1007/978-90-481-2810-5_24.
  19. Taira, K., Brunton, S.L., Dawson, S.T.M., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V., and Ukeiley, L.S. Modal analysis of fluid flows: an overview. AIAA JOURNAL, 2017, 55 (12), 4013–4041. DOI: 10.2514/1.J056060.
  20. Ershkov, S.V., Prosviryakov, E.Yu., Burmasheva, N.V., and Christianto, V. Towards understanding the algorithms for solving the Navier-Stokes equations. Fluid Dynamics Research, 2021, 53 (4), 044501. DOI: 10.1088/1873-7005/ac10f0.
  21. Smagorinsky, J. History and progress. In: The Global Weather Experiment, Perspectives on Its Implementation and Exploitation: A Report of the FGGE Advisory Panel to the U.S. Committee for the Global Atmospheric Research Program, National Academy of Sciences, 1978, 4–12.
  22. Smagorinsky, J. The beginnings of numerical weather prediction and general circulation modeling: Early recollections. In: Ed., B. Zaltzman, Advances in Geophysics, Theory of Climate, vol. 25, Academic Press, 1983, 3–37.
  23. Smagorinsky, J., Phillips, N.A. Scientific problems of the global weather experiment. In: The Global Weather Experiment, Perspectives on Its Implementation and Exploitation: A Report of the FGGE Advisory Panel to the U.S. Committee for the Global Atmospheric Research Program (GARP), National Academy of Science, 1978, 13–21.
  24. Burmasheva, N.V., Prosviryakov, E.Yu. Exact solution of Navier–Stokes equations describing spatially inhomogeneous flows of a rotating fluid. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, 26 (2), 79–87. (In Russian). DOI: 10.21538/0134-4889-2020-26-2-79-87.
  25. Burmasheva, N.V. and Prosviryakov, E.Yu. A class of exact solutions for two–dimensional equations of geophysical hydrodynamics with two Coriolis parameters. Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya Matematika, 2020, 32, 33–48. (In Russian). DOI: 10.26516/1997-7670.2020.32.33.
  26. Burmasheva, N.V. and Prosviryakov, E.Yu. Isothermal layered flows of a viscous incompressible fluid with spatial acceleration in the case of three Coriolis parameters. Diagnostics, Resource and Mechanics of materials and structures, 2020, 3, 29–46. DOI: 10.17804/2410-9908.2020.3.029-046. Available at: http://dream-journal.org/issues/2020-3/2020-3_291.html
  27. Zyryanov, V.N. Teoriya ustanovivshikhsya okeanicheskikh techeniy [The Theory of Steady Ocean Currents]. Gidrometeoizdat Publ., Leningrad, 1985, 248 p. (In Russian).
  28. Korotaev, G.K., Mikhailova, E.N., and Shapiro, N.B. Teoriya ekvatorialnykh protivotecheniy v Mirovom okeane [Theory of Equatorial Countercurrents in the World's Oceans]. Naukova Dumka Publ., Kiev, 1986, 208 p. (In Russian).
  29. Monin, A.S. Teoreticheskie osnovy geofizicheskoy gidromekhaniki [Fundamentals of Geophysical Fluid Dynamics]. Gidrometeoizdat Publ., Leningrad, 1988, 424 p. (In Russian).
  30. Pedlosky, J. Geophysical fluid dynamics, Springer–Verlag, Berlin, New York, 1987, 710 p.
  31. Ostroumov, G.A. Free convection under the condition of the internal problem, NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, Washington, 1958.
  32. Birikh, R.V. Thermocapillary convection in a horizontal layer of liquid. Journal of Applied Mechanics and Technical Physics, 1966, 7 (3), 43–44. DOI: 10.1007/BF00914697.
  33. Burmasheva, N.V., Larina, E.A., and Prosviryakov, E.Yu. Unidirectional convective flows of a viscous incompressible fluid with slippage in a closed layer. AIP Conference Proceedings, 2019, 2176, 030023. DOI: 10.1063/1.5135147.
  34. Aristov, S.N. and Nycander, J. Convective flow in baroclinic vortices. Journal Physical Oceanography, 1994, 24 (9), 1841–1849. DOI: 10.1007/BF00914697.
  35. 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, 30 (2), 197–203. DOI: 10.1007/BF00852164.
  36. Aristov, S.N. Vikhrevye techeniya v tonkikh sloyakh zhidkosti [Eddy Currents in Thin Liquid Layers: Doctoral Thesis]. Vladivostok, 1990, 303 p. (In Russian).
  37. Burmasheva, N.V. and Prosviryakov, E.Yu. Thermocapillary convection of a vertical swirling liquid. Theoretical Foundations of Chemical Engineering, 2020, 54 (1), 230–239. DOI: 10.1134/S0040579519060034.
  38. Burmasheva, N.V. and Prosviryakov, E.Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation. Vestnik Samarskogo Gosudarstvennigo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2019, 23 (2), 341–360. DOI: 10.14498/vsgtu1670.
  39. Aristov, S.N., Knyazev, D.V., and 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, 43 (5), 642. DOI: 10.1134/S0040579509050066.
  40. Ershkov, S., Burmasheva, N., Leshchenko, D.D., and Prosviryakov, E.Yu. Exact solutions of the Oberbeck–Boussinesq equations for the description of shear thermal diffusion of Newtonian fluid flows. Symmetry, 2023, 15, 1730. DOI: 10.3390/sym15091730.
  41. Burmasheva, N.V. and Prosviryakov, E.Yu. Inhomogeneous Nusselt–Couette–Poiseuille flow. Theoretical Foundations of Chemical Engineering, 2022, 56 (5), 662–668. DOI: 10.1134/S0040579522050207.
  42. Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solution of the Couette–Poiseuille type for steady concentration flows. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2022, 164 (4), 285–301. (In Russian). DOI: 10.26907/2541-7746.2022.4.285-301.
  43. Burmasheva, N.V. and Prosviryakov, E.Yu. Influence of the Dufour effect on shear thermal diffusion flows. Dynamics, 2022, 2 (4), 367–379. DOI: 10.3390/dynamics2040021.
  44. Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions for steady convective layered flows with a spatial acceleration. Russian Mathematics, 2021, 65 (7), 8–16. DOI: 10.3103/S1066369X21070021.
  45. Burmasheva, N.V., Privalova, V.V., and Prosviryakov, E.Yu. Layered Marangoni convection with the Navier slip condition. Sādhanā, 2021, 46, 55. DOI: 10.1007/s12046-021-01585-5.
  46. Burmasheva, N.V. and 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, 32 (8), 3364–3371. DOI: 10.1016/j.jksus.2020.09.023.
  47. Bekezhanova, V.B. and Goncharova, O.N. Three-dimensional thermocapillary flow regimes with evaporation. Journal of Physics: Conference Series, 2017, 894, 012023. DOI: 10.1088/1742-6596/894/1/012023.
  48. Bekezhanova, V. and Goncharova, O. Problems of evaporative convection (review). Fluid Dynamics, 2018, 53, S69–S102. DOI:10.1134/S001546281804016X.
  49. Bekezhanova, V. and Goncharova, O. Theoretical analysis of the gravity impact on the parameters of flow regimes with inhomogeneous evaporation based on an exact solution of convection equations. Microgravity Science and Technology, 2022, 34, 88. DOI: 10.1007/s12217-022-10006-z.
  50. Dikanskii, Yu.I., Bedzhanyan, M.A., Kolesnikova, A.A., Gora, A.Yu., and Chernyshev, A.V. Dynamic effects in a magnetic fluid with microdrops of concentrated phase in a rotating magnetic field. Technical Physics, 2019, 64 (3), 337–341. DOI: 10.1134/S1063784219030113.
  51. Polunin, V.M., Ryapolov, P.A., Platonov, V.B., Sheldeshova, E.V., Karpova, G.V., and Arefev, I.M. Elasticity of a magnetic fluid in a strong magnetic field. Acoustical Physics, 2017, 63 (4), 416–423. DOI: 10.1134/S1063771017040108.
  52. Pivovarov, D.E. Numerical investigation of natural convection in slope longitudinal air layer. Trudy MAI, 2013, 68. (In Russian). Available at: http://www.mai.ru/science/trudy/
  53. Shablovsky, O.N. Spherical flow of an ideal fluid in a spatially nonuniform field of force. Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2020, 64, 146–155. (In Russian). DOI 10.17223/19988621/64/11.
  54. Kozlov, V.G., Kozlov, N.V., and Subbotin, S.V. Motion of fluid and a solid core in a spherical cavity rotating in an external force field. Doklady Physics, 2014, 59, 40–44. DOI: 10.1134/S1028335814010078.
  55. Burmasheva, N.V. and 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, 2, 30–51. DOI: 10.17804/2410-9908.2021.2.030-051. Available at: http://dream-journal.org/issues/2021-2/2021-2_316.html
  56. Burmasheva, N.V. and Prosviryakov, E.Yu. Unidirectional thermocapillary flows of a viscous incompressible fluid with the Navier boundary condition. AIP Conference Proceedings, 2019, 2176, 030002. DOI: 10.1063/1.5135126.
  57. Burmasheva, N.V., Larina, E.A., and Prosviryakov, E.Yu. A layered unidirectional flow of a viscous incompressible fluid induced in a closed layer by a nonuniform distribution of temperature and pressure fields, with allowance for the perfect slip condition. AIP Conference Proceedings, 2020, 2315, 020011. DOI: 10.1063/5.0036715.
  58. Burmasheva, N. and Prosviryakov, E. Exact solutions to Navier–Stokes equations describing a gradient nonuniform unidirectional vertical vortex fluid flow. Dynamics, 2022, 2 (2), 175–186. DOI: 10.3390/dynamics2020009.
  59. Burmasheva, N.V., Dyachkova, A.V., and Prosviryakov, E.Yu. Inhomogeneous Poiseuille flow. Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2022, 77, 68–85. (In Russian). DOI: 10.17223/19988621/77/6.
  60. Dinariyev, O.Yu. and Nikolayevskii, V.N. Defining relations for a viscoelastic medium with microrotation. Journal of Applied Mathematics and Mechanics, 1997, 61 (6), 987–994. DOI: 10.1016/S0021-8928(97)00127-5.
  61. DeSilva, C.N. and Kline, K.A. Nonlinear constitutive equations for directed viscoelastic materials with memory. Zeitschrift für angewandte Mathematik und Physik ZAMP, 1968, 19 (1), 128–139. DOI: 10.1007/BF0160328419.
  62. Allen, S.J., DeSilva, C.N., and Kline, K.A. Theory of simple deformable directed fluids. Phys. Fluids, 1967, 10 (12), 2551–2555. DOI: 10.1063/1.1762075.
  63. Eringen, A.C. Linear theory of micropolar viscoelasticity. International Journal of Engineering Science, 5, 191–204, DOI: 10.1016/0020-7225(67)90004-3.
  64. Stokes, V.K. Couple stresses in fluids. Phys. Fluids, 1966, 9 (9), 1709–1715. DOI: 10.1063/1.1761925.
  65. Stokes, V.K. Theories of Fluids with Microstructure. An Introduction. Springer, Berlin, Heidelberg, 1984, 212 p. DOI: 10.1007/978-3-642-82351-0.
  66. Stokes, V.K. Effects of couple stresses in fluids on hydromagnetic channel flows. Physics of Fluids, 1968, 11, 1131–1133. DOI: 10.1063/1.1692056.
  67. Stokes, V.K. On some effects of couple stresses in fluids on heat transfer. J. Heat Transfer, 1969, 91 (1), 182–184. DOI: 10.1115/1.3580094.
  68. Asibor, R.E., Omokhuale, E. Micropolar fluid behavior with constant pressure, permeability, heat and mass transfer. International Journal of Innovative Technology and Exploring Engineering, 2017, 6 (12), 36–43.
  69. Bég, O.A., Zueco, J., Takhar, H.S. Unsteady magnetohydrodynamic Hartmann–Couette flow and heat transfer in a Darcian channel with Hall current, ionslip, viscous and Joule heating effects: network numerical solutions. Communications in Nonlinear Science and Numerical Simulation, 2009, 14 (4), 1082–1097. DOI: 10.1016/j.cnsns.2008.03.015.
  70. Kocić, M., Stamenković, Ž., Petrović, J., and Bogdanović-Jovanović, J. Control of MHD flow and heat transfer of a micropolar fluid through porous media in a horizontal channel. Fluids, 2023, 8 (3), 93 (1–19). DOI: 10.3390/fluids8030093.
  71. El-Kabeir, S.M.M., Modather, M., Mansour, M.A. Effect of heat and mass transfer on free convection flow over a cone with uniform suction or injection in micropolar fluids. International Journal of Applied Mechanics and Engineering, 2006, 11 (1), 15–35.
  72. Rafique, Kh., Ibrar, N., Munir, A., Khalid A., Ijaz, A., and Asghar, A. Numerical analysis for energy transfer analysis of micropolar nanofluid by Keller box scheme. Acta Scientific Applied Physics, 2023, 3 (3), 36–44.
  73. Xu, Q., Zhong, X. Strong solutions to the three-dimensional barotropic compressible magneto-micropolar fluid equations with vacuum. Zeitschrift für angewandte Mathematik und Physik, 2021, 73 (1), 14. DOI: 10.1007/s00033-021-01642-3.
  74. Baranovskii, E.S., Burmasheva, N.V., and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations with couple stresses. Symmetry, 2021, 13, 1355 (1–12). DOI: 10.3390/sym13081355.
  75. Cosserat, E. and Cosserat, F. Théorie des Corps déformables, A. Hermann et Fils, Paris, 1909, 226.
  76. Aero, E.L., Bulygin, A.N. and Kuvshinski, E.V. Asymmetric hydromechanics. Journal of Applied Mathematics and Mechanics, 1965, 29, 333–346. DOI: 10.1016/0021-8928(65)90035-3.
  77. Eringen, A.C. Theory of micropolar fluids. J. Math. Mech., 1966, 16 (1), 1–18.
  78. Korn, G.A. and Korn, T.M. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Courier Corporation, North Chelmsford, 2013, 1152 p.

Н. В. Бурмашева, Е. Ю. Просвиряков

ТОЧНЫЕ РЕШЕНИЯ УРАВНЕНИЙ НАВЬЕ – СТОКСА ОДНОНАПРАВЛЕННЫХ ТЕЧЕНИЙ МИКРОПОЛЯРНЫХ ЖИДКОСТЕЙ В ПОЛЕ МАССОВЫХ СИЛ

В статье представлено семейство точных решений системы уравнений Навье – Стокса, используемой для описания неоднородных однонаправленных течений вязкой жидкости с учетом моментных напряжений. Несмотря на наличие только одной ненулевой компоненты вектора скорости, эта самая компонента зависит от времени и двух пространственных координат. Зависимость от третьей пространственной координаты отсутствует ввиду уравнения несжимаемости, являющегося частным случаем закона сохранения массы. Получающаяся переопределенная система уравнений рассматривается в нестационарной постановке. Построение семейства точных решений полученной переопределенной системы начинается с анализа однородного решения типа Куэтта как наиболее простого в этом классе. Далее структура решения постепенно усложняется: профиль единственной ненулевой компоненты вектора скорости представлен в виде полинома, зависящего от одной переменной (горизонтальной координаты). Коэффициенты полинома функционально зависят от второй (вертикальной) координаты и времени. Показано, что, ввиду сильной нелинейности и неоднородности исследуемого уравнения, сумма отдельных его решений не является решением. Также показано, что в линейно независимом базисе степенных функций горизонтальной координаты, определяющих вышеупомянутый полином, рассматриваемое уравнение распадается на цепочку простейших однородных и неоднородных уравнений в частных производных параболического типа. Данные уравнения интегрируются последовательно, порядок интегрирования отдельно описан. Результаты, изложенные в данной статье, обобщают ранее представленное авторами семейство точных решений для описания однонаправленных нестационарных течений.

Ключевые слова: точное решение, уравнение Навье – Стокса, неоднородное однонаправленное течение, моментные напряжения, микрополярные жидкости

Библиография:

  1. Couette M. Etudes sur le frottement des liquids // Ann. Chim. Phys. – 1890. – Vol. 21. – P. 433–510.
  2. Poiseuille J. Recherches exp´erimenteles sur le mouvement des liquides dans les tubes de tr`es petits diam`etres // Comptes rendus hebdomadaires des s´eances de l’Acadmemie des Sciences. – 1840. – Vol. 11. – P. 961–967.
  3. Poiseuille J. Recherches expérimenteles sur le mouvement des liquides dans les tubes de très-petits diamètres // Comptes rendus des séances de l’Académie des Sciences. – 1840. – Vol. 11. – P. 1041–1048.
  4. Poiseuille J. Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres (suite) // Comptes rendus des séances de l’Académie des Sciences. – 1841. – Vol. 12. – P. 112–115.
  5. 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.
  6. Nußelt W. Die Abhängigkeit der Wärmeübergangszahl von der Rohrlänge // Zeitschrift des Vereins Deutscher Ingenieure. – 1910. – Vol. 54. – P. 1154–1158.
  7. Nagel Yu. A. Plane one-dimensional stationary flow of an ideal charged gas in its own electric field // Journal of Applied Mechanics and Technical Physics. – 1971. – Vol. 12. – P. 19–26. – DOI: 10.1007/BF00853977.
  8. Faskheev I. O. One-dimensional flow of a fluid through a porous skeleton with consideration of the Darcy and frontal pressure interaction forces // Moscow University Mechanics Bulletin. – 2013. – Vol. 68. – P. 21–24. – DOI: 10.3103/S0027133013010044.
  9. Уоллис Г. Одномерные двухфазные течения / пер. с англ. – М. : Мир, 1972. – 440 c.
  10. Борзенко Е. И., Шрагер Г. Р. Течения неньютоновской жидкости со свободной поверхностью. – Томск : Издательство Томского государственного университета, 2022. – 210 с.
  11. Прокудин С. В., Софьин А. С., Агапов А. А. Аналитическое решение задачи одномерного стационарного течения сжимаемой жидкости и газа // Безопасность труда в промышленности. – 2017. – № 5. – C. 36–41. – DOI: 10.24000/0409-2961-2017-5-36-41.
  12. Valiyev Kh. F., Kraiko A. N., Tillyayeva N. I. Stability of one-dimensional steady flows with detonation wave in a channel of variable cross-sectional area // Computational Mathematics and Mathematical Physics. – 2020. – Vol. 60 (4). – P. 697–710. – DOI: 10.1134/S096554252004017XBurmasheva N. V., Prosviryakov E. Yu. An exact solution to the description of a unidirectional Marangoni flow of a viscous incompressible fluid with the Navier boundary condition. Velocity field investigation // Diagnostics, Resource and Mechanics of materials and structures. – 2019. – Iss. 5. – P. 23–39. – DOI: 10.17804/2410-9908.2019.5.023-039. – URL: http://dream-journal.org/issues/2019-5/2019-5_259.html
  13. 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. – P. 61–75. – DOI: 10.17804/2410-9908.2020.2.061-075. – URL: http://dream-journal.org/issues/2020-2/2020-2_288.html
  14. Bhaskar B. S., Chaudhary S. K. Review of fluid flow and heat transfer through porous media heat exchangers // International Journal of New Innovations in Engineering and Technology. – 2016. – Vol. 6 (2). – P. 28–42.
  15. Singh H., Myong R. S. Critical review of fluid flow physics at micro- to nano‐scale porous media applications in the energy sector // Advances in Materials Science and Engineering. – 2018. – Vol. 2018. – P. 9565240. – DOI: 10.1155/2018/9565240.
  16. Coussot P. Yield stress fluid flows: a review of experimental data // Journal of Non-Newtonian Fluid Mechanics. – 2014. – Vol. 211. – P. 31–49. – DOI: 10.1016/j.jnnfm.2014.05.006.
  17. Kiselev A. Diffusion and mixing in fluid flow: a review // New Trends in Mathematical Physics / ed by V. Sidoravičius. – Dordrecht : Springer, 2009. – DOI: 10.1007/978-90-481-2810-5_24.
  18. Modal analysis of fluid flows: an overview / K. Taira, S. L. Brunton, S. T. M. Dawson, C. W. Rowley, T. Colonius, B. J. McKeon, O. T. Schmidt, S. Gordeyev, V. Theofilis, L. S. Ukeiley // AIAA JOURNAL. – 2017. – Vol. 55 (12). – P. 4013–4041. – DOI: 10.2514/1.J056060.
  19. Towards understanding the algorithms for solving the Navier-Stokes equations / S. V. Ershkov, E. Yu. Prosviryakov, N. V. Burmasheva, V. Christianto // Fluid Dynamics Research. – 2021. – Vol. 53 (4). – P. 044501. – DOI: 10.1088/1873-7005/ac10f0.
  20. Smagorinsky J. History and progress // The Global Weather Experiment–Perspective on Its Implementation and Exploitation: A Report of the FGGE Advisory Panel to the U.S. Committee for the Global Atmospheric Research Program (GARP). – National Academy of Science, 1978. – P. 4–12.
  21. Smagorinsky, J. The beginnings of numerical weather prediction and general circulation modeling: early recollections // Advances in Geophysics. Vol. 25 : Theory of Climate / ed. by B. Zaltzman. – Academic Press, 1983. – P. 3–37.
  22. Smagorinsky J., Phillips N. A. Scientific problems of the global weather experiment // The Global Weather Experiment, Perspectives on Its Implementation and Exploitation: A Report of the FGGE Advisory Panel to the U.S. Committee for the Global Atmospheric Research Program (GARP). – National Academy of Science, 1978. – P. 13–21.
  23. 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, № 2. – С. 79–87. – DOI: 10.21538/0134-4889-2020-26-2-79-87.
  24. Бурмашева Н. В., Просвиряков Е. Ю. Класс точных решений для двумерных уравнений геофизической гидродинамики с двумя параметрами Кориолиса // Известия Иркутского государственного университета. Cерия «Математика». – 2020. – Т. 32. – С. 33–48. – DOI: 10.26516/1997-7670.2020.32.33.
  25. Burmasheva N. V., Prosviryakov E. Yu. Isothermal layered flows of a viscous incompressible fluid with spatial acceleration in the case of three Coriolis parameters // Diagnostics, Resource and Mechanics of materials and structures. – 2020. – Iss. 3. – P. 29–46. – DOI: 10.17804/2410-9908.2020.3.029-046. – URL: http://dream-journal.org/issues/2020-3/2020-3_291.html
  26. Зырянов В. Н. Теория установившихся океанических течений: Сращиваемые асимптотич. разложения: основы теории и приложения к задачам динамики океана. – Ленинград : Гидрометеоиздат, 1985. – 248 с.
  27. Коротаев Г. К., Михайлова Э. Н., Шапиро Н. Б. Теория экваториальных противотечений в Мировом океане. – Киев : Наукова думка, 1986. – 208 с.
  28. Монин А. С. Теоретические основы геофизической гидродинамики. – Ленинград : Гидрометеоиздат, 1988. – 424 с.
  29. Педлоски Дж. Геофизическая гидродинамика : в 2 т. – М. : Мир, 1984. – 398 с.
  30. Остроумов Г. А. Свободная конвекция в условиях внутренней задачи. – Москва ; Ленинград : Гос. изд-во техн.-теорет. лит., 1952. – 256 с.
  31. Бирих Р. В. О термокапиллярной конвекции в горизонтальном слое жидкости // Прикладная механика и техническая физика. – 1966. – № 3. – С. 69–72.
  32. Burmasheva N. V., Larina E. A., Prosviryakov E.Yu. Unidirectional convective flows of a viscous incompressible fluid with slippage in a closed layer. – AIP Conference Proceedings. – 2019. – Vol. 2176. – P. 030023. – DOI: 10.1063/1.5135147.
  33. Aristov S. N., Nycander J. Convective flow in baroclinic vortices // Journal Physical Oceanography. – 1994. – Vol. 24, No. 9. – C. 1841–1849.
  34. 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 (2). – P. 197–203. – DOI: 10.1007/BF00852164.
  35. Аристов С. Н. Вихревые течения в тонких слоях жидкости: дис. … докт. физ.-мат. наук. – Владивосток, 1990.
  36. Burmasheva N. V., Prosviryakov E. Yu. Thermocapillary convection of a vertical swirling liquid // Theoretical Foundations of Chemical Engineering. – 2020. – Vol. 54 (1). – P. 230–239. – DOI: 10.1134/S0040579519060034.
  37. Burmasheva N. V., Prosviryakov E. Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation // Vestnik Samarskogo Gosudarstvennigo Universiteta. Seriya Fiziko-Matematicheskie Nauki. – 2019. – Vol. 23 (2). – P. 341–360. – DOI: 10.14498/vsgtu1670.
  38. 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 (5). – P. 642. – DOI: 10.1134/S0040579509050066.
  39. Exact solutions of the Oberbeck–Boussinesq equations for the description of shear thermal diffusion of Newtonian fluid flows / S. Ershkov, N. Burmasheva, D. D. Leshchenko, E. Yu. Prosviryakov // Symmetry. – 2023. – Vol. 15. – 1730. – DOI: 10.3390/sym15091730.
  40. Burmasheva N. V., Prosviryakov E. Yu. Inhomogeneous Nusselt–Couette–Poiseuille flow // Theoretical Foundations of Chemical Engineering. – 2022. – Vol. 56 (5). – P. 662–668. – DOI: 10.1134/S0040579522050207.
  41. Бурмашева Н. В., Просвиряков Е. Ю. Точное решение типа Куэтта–Пуазейля для установившихся концентрационных течений // Ученые записки Казанского университета. Серия «Физико-математические науки». – 2022. – Т. 164, кн. 4. – С. 285–301. – DOI: 10.26907/2541-7746.2022.4.285-301.
  42. Burmasheva N. V., Prosviryakov E. Yu. Influence of the Dufour effect on shear thermal diffusion flows // Dynamics. – 2022. – Vol. 2, No. 4. – P. 367–379. – DOI: 10.3390/dynamics2040021.
  43. Burmasheva N. V., Prosviryakov E. Yu. Exact solutions for steady convective layered flows with a spatial // Russian Mathematics. – 2021. – Vol. 65 (7). – P. 8–16. – DOI: 10.3103/S1066369X21070021.
  44. Burmasheva N. V., Privalova V. V., Prosviryakov, E. Yu. Layered Marangoni convection with the Navier slip condition // Sādhanā. – 2021. – Vol. 46. – Art. No. 55. – DOI: 10.1007/s12046-021-01585-5.
  45. 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. – P. 3364–3371. – DOI: 10.1016/j.jksus.2020.09.023.
  46. Bekezhanova V. B., Goncharova O. N. Three-dimensional thermocapillary flow regimes with evaporation // Journal of Physics: Conference Series. – 2017. – Vol. 894. – DOI: 10.1088/1742-6596/894/1/012023.
  47. Bekezhanova V., Goncharova O. Problems of evaporative convection (review) // Fluid Dynamics. – 2018. – Vol. 53. – P. S69–S102. – DOI:10.1134/S001546281804016X.
  48. Bekezhanova V., Goncharova O. Theoretical analysis of the gravity impact on the parameters of flow regimes with inhomogeneous evaporation based on an exact solution of convection equations // Microgravity Science and Technology. – 2022. – Vol. 34. – P. 88. – DOI: 10.1007/s12217-022-10006-z.
  49. Dynamic effects in a magnetic fluid with microdrops of concentrated phase in a rotating magnetic field / Y. I. Dikanskii, M. A. Bedzhanyan, A. A. Kolesnikova, A. Yu. Gora, A. V. Chernyshev // Technical Physics. – 2019. – Vol. 64 (3). – P. 337–341. – DOI: 10.1134/S1063784219030113.
  50. Elasticity of a magnetic fluid in a strong magnetic field / V. M. Polunin, P. A. Ryapolov, V. B. Platonov, E. V. Sheldeshova, G. V. Karpova, I. M. Arefyev // Acoustical Physics. – 2017. – Vol. 63 (4). – P. 416–423. – DOI: 10.1134/S1063771017040108.
  51. Пивоваров Д. Е. Численное исследование конвективного теплообмена в наклонном продольном слое воздуха // Труды МАИ. – 2013. – № 68. – URL: http://www.mai.ru/science/trudy/
  52. Шабловский О. Н. Сферическое течение идеальной жидкости в пространственно-неоднородном силовом поле // Вестник Томского государственного университета. Математика и механика. – 2020. – № 64. – С. 146–155. – DOI 10.17223/19988621/64/11.
  53. Kozlov V. G., Kozlov N. V., Subbotin S. V. Motion of fluid and a solid core in a spherical cavity rotating in an external force field // Doklady Physics. – 2014. – Vol. 59. – P. 40–44. – DOI: 10.1134/S1028335814010078.
  54. 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/2021-2/2021-2_316.html
  55. Burmasheva N. V., Prosviryakov E. Yu. Unidirectional thermocapillary flows of a viscous incompressible fluid with the Navier boundary condition // AIP Conference Proceedings. – 2019. – Vol. 2176. – 030002. – DOI: 10.1063/1.5135126.
  56. Burmasheva N. V., Larina E. A., Prosviryakov E. Yu. A layered unidirectional flow of a viscous incompressible fluid induced in a closed layer by a nonuniform distribution of temperature and pressure fields, with allowance for the perfect slip condition // AIP Conference Proceedings. – 2020. – Vol. 2315. – 020011. – DOI: 10.1063/5.0036715.
  57. Burmasheva N., Prosviryakov E. Exact solutions to Navier–Stokes equations describing a gradient nonuniform unidirectional vertical vortex fluid flow // Dynamics. – 2022. – Vol. 2, No. 2. – P. 175–186. – DOI: 10.3390/dynamics2020009.
  58. Бурмашева Н. В., Дьячкова А. В., Просвиряков Е. Ю. Неоднородное течение Пуазейля // Вестник Томского государственного университета. Математика и механика. – 2022. – Т. 77. – С. 68–85. – DOI: 10.17223/19988621/77/6.
  59. Динариев О. Ю., Николаевский В. Н. Определяющие соотношения для вязкоупругой среды с микровращениями // Прикладная математика и механика. – 1997. – Т. 61, № 6. – С. 1023–1030.
  60. DeSilva C. N., Kline K. A. Nonlinear constitutive equations for directed viscoelastic materials with memory // Zeitschrift für angewandte Mathematik und Physik ZAMP. – 1968. – Vol. 19 (1). – P. 128–139. – DOI: 10.1007/BF0160328419.
  61. Allen S. J., DeSilva C. N., Kline K. A. Theory of simple deformable directed fluids // Phys. Fluids. – 1967. – Vol. 10 (12). – P. 551–2555. – DOI: 10.1063/1.1762075.
  62. Eringen A. C. Linear theory of micropolar viscoelasticity // International Journal of Engineering Science. – Vol. 5. – P. 191–204. – DOI: 10.1016/0020-7225(67)90004-3.
  63. Stokes V. K. Couple stresses in fluids // Phys. Fluids. – 1966. – Vol. 9, iss. 9. – P. 1709–1715. DOI: 10.1063/1.1761925.
  64. Stokes V. K. Theories of Fluids with Microstructure. An Introduction. – Berlin, Heidelberg : Springer, 1984. – 212 p. – DOI: 10.1007/978-3-642-82351-0.
  65. Stokes V. K. Effects of couple stresses in fluids on hydromagnetic channel flows // Physics of Fluids. – 1968. – Vol. 11. – P. 1131–1133. – DOI: 10.1063/1.1692056.
  66. Stokes V. K. On some effects of couple stresses in fluids on heat transfer // J. Heat Transfer. – 1969. – Vol. 91 (1). – P. 182–184. – DOI: 10.1115/1.3580094.
  67. Asibor R. E., Omokhuale E. Micropolar fluid behavior with constant pressure, permeability, heat and mass transfer // International Journal of Innovative Technology and Exploring Engineering. – 2017. – Vol. 6 (12). – P. 36–43.
  68. Bég O. A., Zueco J., Takhar H. S. Unsteady magnetohydrodynamic Hartmann–Couette flow and heat transfer in a Darcian channel with Hall current, ionslip, viscous and Joule heating effects: network numerical solutions // Communications in Nonlinear Science and Numerical Simulation. – 2009. – Vol. 14 (4). – P. 1082–1097. – DOI: 10.1016/j.cnsns.2008.03.015.
  69. Control of MHD Flow and heat transfer of a micropolar fluid through porous media in a horizontal channel / M. Kocić, Ž. Stamenković, J. Petrović, J. Bogdanović-Jovanović // Fluids. – 2023. – Vol. 8, iss. 3. – P. 93. – DOI: 10.3390/fluids8030093.
  70. El-Kabeir S. M. M., Modather M., Mansour M. A. Effect of heat and mass transfer on free convection flow over a cone with uniform suction or injection in micropolar fluids // International Journal of Applied Mechanics and Enginering. – 2006. – Vol. 11, No.1. – P. 15–35.
  71. Numerical analysis for energy transfer analysis of micropolar nanofluid by Keller box scheme / Kh. Rafique, N. Ibrar, A. Munir, A. Khalid, A. Ijaz, A. Asghar // Acta Scientific Applied Physics. – 2023. – Vol. 3 (3). – P. 36–44.
  72. Xu Q., Zhong X. Strong solutions to the three-dimensional barotropic compressible magneto-micropolar fluid equations with vacuum // Zeitschrift für angewandte Mathematik und Physik. – 2021. – Vol. 73 (1). – Art. No. 14. – DOI: 10.1007/s00033-021-01642-3.
  73. Baranovskii E. S., Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations with couple stresses // Symmetry. – 2021. – Vol. 13. – P. 1355. – DOI: 10.3390/sym13081355.
  74. Cosserat E., Cosserat F. Théorie des Corps déformables. – Paris : A. Hermann et Fils, 1909. – P. 226.
  75. Aero E. L., Bulygin A. N., Kuvshinski E. V. Asymmetric hydromechanics // Journal of Applied Mathematics and Mechanics. – 1965. – Vol. 29. – P. 333–346. – DOI: 10.1016/0021-8928(65)90035-3.
  76. Eringen A. C. Theory of micropolar fluids // J. Math. Mech. – 1966. – Vol. 16 (1). – P. 1–18.
  77. Корн Г., Корн Т. Справочник по математике для научных работников и инженеров. Определения. Теоремы. Формулы / пер. с англ. – 6-е изд., стер. – СПб. : Лань, 2003. – 831 с.

PDF      

Библиографическая ссылка на статью

Burmasheva N. V., Prosviryakov E. Yu. Exact Solutions to the Navier–stokes Equations for Unidirectional Flows of Micropolar Fluids in a Mass Force Field // Diagnostics, Resource and Mechanics of materials and structures. - 2024. - Iss. 3. - P. 41-63. -
DOI: 10.17804/2410-9908.2024.3.041-063. -
URL: http://dream-journal.org/issues/2024-3/2024-3_438.html
(accessed: 21.12.2024).

 

импакт-фактор
РИНЦ 0.42

категория К2
в перечне ВАК

МРДМК 2024
ЦКП Пластометрия
НЭБ РИНЦ
Google Scholar


РНБ
Лань

 

Учредитель:  Федеральное государственное бюджетное учреждение науки Институт машиноведения имени Э.С. Горкунова Уральского отделения Российской академии наук
Главный редактор:  С.В.Смирнов
При цитировании ссылка на Электронный научно-технический журнал "Diagnostics, Resource and Mechanics of materials and structures" обязательна. Воспроизведение материалов в электронных или иных изданиях без письменного разрешения редакции запрещено. Опубликованные в журнале материалы могут использоваться только в некоммерческих целях.
Контакты  
 
Главная E-mail 0+
 

ISSN 2410-9908 Регистрация СМИ в Роскомнадзоре Эл № ФС77-57355 от 24 марта 2014 г. © ИМАШ УрО РАН 2014-2024, www.imach.uran.ru