N. V. Burmasheva, E. Yu. Prosviryakov
A CLASS OF EXACT SOLUTIONS WITH SPATIAL ACCELERATION FOR THE DESCRIPTION OF VISCOUS INCOMPRESSIBLE FLUID FLOWS IN THE FIELD OF MASS FORCES
DOI: 10.17804/2410-9908.2021.1.006-025 The article presents a new class of exact solutions to the system of Navier–Stokes equations, which allows one to take into account the nonlinear distribution of the pressure field and the influence of external volumetric forces, as well as the possibility of horizontal fluid outflow/inflow when modeling its vertical motion. This class is a generalization of the Lin–Sidorov–Aristov class, which assumes the linear distribution of two projections of the fluid flow velocity vector along a part of the coordinates and the independence of the third projection of the velocity vector from these coordinates.
Keywords: class of exact solutions, Navier–Stokes equation, overdetermined system, field of mass (volumetric) forces References:
- Shablovsky O.N. Spherical flow of an ideal fluid in a spatially nonuniform field of force. Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika [Tomsk State University Journal of Mathematics and Mechanics], 2020, vol. 64. pp. 146–155. DOI: 10.17223/19988621/64/11. (In Russian).
- Manuylovich S.V. Longitudinal-periodic flow of a viscous fluid generated by a wall volumetric force. Izvestiya RAN. Mekhanika Zhidkosti i Gaza, 2015, no. 4, pp. 59–67. (In Russian).
- Manuylovich S.V. Cross-flow control in the boundary layer on a sliding wing using a wall volumetric force. Izvestiya RAN. Mekhanika Zhidkosti i Gaza, 2020, no. 1, pp. 45–56. DOI: 10.31857/S0568528120010090. (In Russian).
- Rosensweig R.E. Magnetic fluids. Ann. Rev. Fluid Mech., 1987, vol. 19, no. 1, pp. 437–463. DOI: 10.1146/annurev.fl.19.010187.002253.
- Rosensweig R.E. Ferrohydrodynamics, Cambridge, Cambridge University Press, 1985, 344 p.
- Yerin C.V., Vivchar V.I. Ellipsometry of magnetic fluid in a magnetic field. Journal of Magnetism and Magnetic Materials, 2020, vol. 498, pp. 166144. DOI: 10.1016/j.jmmm.2019.166144.
- Wei Pan, Jianghong Gong. Effects of magnetic fluid on magnetic fluid damper. Key Engineering Materials, 2012, vols. 512–515, pp. 1479–1483. DOI: 10.4028/www.scientific.net/KEM.512-515.1479.
- Safarik I., Prochazkova J., Baldikova E., Timko M., Kopcansky P., Rajnak M., Torma N., Pospiskova K. Modification of diamagneticmaterials using magnetic fluids. Ukr. J. Phys., 2020, vol. 65, no. 9, pp. 751–760. DOI: 10.15407/ujpe65.9.751.
- Horváth B., Szalai I. Nonlinear magnetic properties of magnetic fluids for automotive applications. Hungarian Journal of Industry and Chemistry, 2020, vol. 48, no. 1, pp. 61–65. DOI: 10.33927/hjic-2020-09.
- Bratsun D.A., Vyatkin V.A., Mukhamatullin A.R. On exact nonstationary solutions of equations of vibrational convection. Computational Continuum Mechanics, 2017, vol. 10, no. 4, pp. 433–444. DOI: 10.7242/1999-6691/2017.10.4.35. (In Russian).
- Polezhaev V.I., Bello M.S., Verezub N.A., Dubovik K.G., Lebedev A.P., Nikitin S.A., Pavlovskiy D.S., Fedyushkin A.I. Konvektivnye protsessy v nevesomosti [Convective Processes under Microgravity]. Moscow, Nauka Publ., 1991, 242 p. (In Russian).
- Polezhaev V.I., Sazonov V.V. Mechanics of Weightlessness and Gravity Sensitive Systems. Annotatsii dokl. nauch.-issled. seminara [Abstracts of Scientific Research Workshop, Preprint no. 898]. Moscow, IPM im. M.V. Keldysha Publ, 2009, 36 p. (In Russian).
- Surzhikov S.T. Giperzvukovoe obtekanie razrezhennym gazom poverkhnostnogo tleyushchego razryada s vneshnim magnitnym polem [Hypersonic Rarefied Gas Flow Past a Surface Glow Discharge with an External Magnetic Field]. Moscow, Institut Problem Mekhaniki RAN Publ., 2011, 274 p. (In Russian).
- Kedrinskiy V.K. Role of nuclei density as a “hidden” parameter in the formation of anomalous zones in a heavy cavitating magma. Journal of Applied Mechanics and Technical Physics, 2014, vol. 55, pp. 276–281. DOI: 10.1134/S0021894414020102.
- Afonin V.V., Petrov V.Y. On exact solutions for a Luttinger liquid with one impurity. Jetp Lett., 2019, vol. 109, pp. 762–767. DOI: 10.1134/S0021364019110031.
- Kouwn S., Moon T., Oh P. Exact solution and exotic fluid in cosmology. Entropy, 2012, vol. 14, pp. 1771–1783. DOI: 10.3390/e14091771.
- Aristov S.N. Stationary cylindrical vortex in a viscous fluid. Dokl. Akad. Nauk, 2001, vol. 377, pp. 477–480. (In Russian).
- Aristov S.N. Exact solution of the point source problem. Doklady Mathematics, 1995, vol. 40, no. 7, pp. 346–348. (In Russian).
- Aristov S.N. Three-Dimensional Conical Viscous Incompressible Fluid Flows. Fluid Dynamics, 1998, vol. 33, no. 929. DOI: 10.1007/BF02698663.
- Aristov S.N., Grabovskii V.I. Self-similar solution of the Navier-Stokes equations governing gas flows in rotary logarithmically spiral two-dimensional channels. Fluid Dyn., 1995, vol. 30, pp. 838–844. DOI: 10.1007/BF02078198.
- Temam R. Navier-Stokes Equations and Nonlinear Functional Analysis, Sоciety for Industriаl Mаthematics, 1987, 155 p.
- Landau L.D., Lifshitz E.M. Fluid Mechanics: Vol. 6 (Course of Theoretical Physics S), 2nd Edition. Butterworth-Heinemann, 1987, 560 p.
- Khmelnik S.I. Navier-Stokes equations. On the existence and the search method for global solutions. Israel, Mathematics in Computer Comp., 2010, 106 p. (In Russian).
- Rouse H. Advanced mechanics of fluids, New York, Wiley, 1959, 474 p.
- Boussinesq J. Théorie de l'écoulement tourbillonnant et tumultueux des liquides dans les lits rectilignes a grande section, Paris, Gauthier-Villars et fils, 1897, 90 p.
- Gershuni G.Z., Zhukhovitskii E.M. Convective stability of incompressible fluids. Israel Program for Scientific Translations. Jerusalem: Keter Publishing House, 1976, 330 p.
- Müller P. Boussinesq approximation. In: The Equations of Oceanic Motions, Cambridge, Cambridge University Press, 2006, pp. 119–126. DOI: 10.1017/CBO9780511617843.012.
- Burmasheva N.V., Prosviryakov E.Y. 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).
- Burmasheva N.V., Prosviryakov E.Y. 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, no. 8, pp. 3364–3371. DOI: 10.1016/j.jksus.2020.09.023.
- Martyushev S.G., Sheremet M.A. Conjugate natural convection combined with surface thermal radiationin a three-dimensional enclosure with a heat source. International Journal of Heat and Mass Transfer, 2014, vol. 73, pp. 340–353. DOI: 10.1016/j.ijheatmasstransfer.2014.02.009.
- Couette M. Etudes sur le frottement des liquids. Ann. Chim. Phys., 1890, vol. 21, pp. 433–510.
- Poiseuille J. Recherches experimentelles sur le mouvement des liquides dans les tubes de tres petits diametres. Comptes Rendus, 1840, vol. 11, pp. 961–967.
- Stokes G.G. On the effct of the internal friction of fluid on the motion of pendulums. Camb. Philo. Trans., 1851, vol. 9, pp. 8–106.
- Lin C.C. Note on a class of exact solutions in magneto-hydrodynamics. Arch. Rational Mech. Anal., 1958, vol. 1, pp. 391–395.
- Shmiglevskiy Yu.D. Analiticheskie issledovaniya dinamiki gaza i zhidkosti [Analytical Studies of Fluid and Gas Dynamics]. Moscow, Editorial URSS, 1999, 231 p.
- 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, pp. 197–203. DOI: 10.1007/BF00852164.
- Aristov S.N. Eddy currents in thin liquid layers. Dr. Phys. & Math. Sci. Thesis. Vladivostok, 1990, 330 pp.
- Aristov S.N., Prosviryakov E.Yu. 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.
- Prosviryakov E.Yu. A sufficient condition for the absence of strong and weak discontinuities in gas flows in flat channels. Diagnostics, Resource and Mechanics of materials and structures, 2019, iss. 3, pp. 25–40. DOI: 10.17804/2410-9908.2019.3.025-040.
- Aristov S.N., Prosviryakov E.Yu. Large–scale flows of viscous incompressible vortical fluid. Russian Aeronautics, 2015, vol. 58, no. 4, pp. 413–418. DOI: 10.3103/S1068799815040091.
- 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.
- 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.
- Privalova V.V. and Prosviryakov E.Yu. The effect of tangential boundary stresses on the convective unidirectional flow of a viscous fluid layer under the lower boundary heating condition. Diagnostics, Resource and Mechanics of materials and structures, 2019, iss. 4, pp. 44–55. DOI: 10.17804/2410-9908.2019.4.044-055.
- Burmasheva 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, pp. 23–39. DOI: 10.17804/2410-9908.2019.5.023-039.
- 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.
- 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.
- Burmasheva N.V., Prosviryakov E.Yu. Temperature field investigation in layered flows of a vertically swirling viscous incompressible fluid under two thermocapillar forces at a free boundary. Diagnostics, Resource and Mechanics of materials and structures, 2019, iss. 1, pp. 6–42. DOI: 10.17804/2410-9908.2019.1.006-042.
- Gorshkov A.V., Prosviryakov E.Y. Complex stationary convection with third-kind boundary conditions at the boundaries of a fluid layer. Diagnostics, Resource and Mechanics of materials and structures, 2016, iss. 2, pp. 34–47. DOI: 10.17804/2410-9908.2016.2.034-047.
- Burmasheva N.V., Prosviryakov E.Yu. Exact solutions for natural convection of layered flows of a viscous incompressible fluid with Specified tangential forces and the linear distribution of temperature on the layer boundaries. Diagnostics, Resource and Mechanics of materials and structures, 2017, iss. 4, pp. 16–31. DOI: 10.17804/2410-9908.2017.4.016-031.
- Petrov A.G. Exact solution of the Navier-Stokes equations in a fluid layer between the moving parallel plates. Journal of Applied Mechanics and Technical Physics, 2012, vol. 53, no. 5, pp. 642–646. DOI: 10.1134/S0021894412050021.
- Zyryanov V.N. Teoriya ustanovivshikhsya okeanicheskikh techeniy [Theory of steady ocean currents]. Leningrad, Gidrometeoizdat Publ., 1985. (In Russian).
- 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, 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. Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya “Matematika” [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. 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, pp. 29–46. DOI: 10.17804/2410-9908.2020.3.029-046.
- Meleshko S.V., Pukhnachev V.V. On a class of partially invariant solutions of the Navier-Stokes equations. Journal of Applied Mechanics and Technical Physics, 1999, vol. 40, no. 2, pp. 208–216.
- Pukhnachev V.V. Symmetries in the Navier-Stokes equations. Uspekhi Mekhaniki, 2006, vol. 1, pp. 3–76. (In Russian).
- Aristov S.N., Gitman I.M. Viscous flow between two moving parallel disk: exact solutions and stability analysis. J. Fluid Mech., 2002, vol. 464, p. 209–215. DOI: 10.1017/S0022112002001003.
- Berker R. An exact solution of the Navier-Stokes equation - the vortex with curvilinear axis. Int. J. Eng. Sci. 1982, vol. 20, p. 217–230.
- Berker R. A new solution of the Navier-Stokes equation for the motion of a fluid contained between two parallel plates rotating about the same axis. Arch. Mech., 1979, vol. 31, pp. 265–280.
- Dorrepaal J.M. An exact solution of the Navier-Stokes equation which describes non-orthogonal stagnation point flow in two dimensions. J. Fluid Mech., 1986, vol. 163, p. 141–147.
- Ekman V.W. On the influence of the earth's rotation on ocean currents. Ark. Mat. Astron. Fys., 1905, vol. 2, pp. 1–52.
- Gupta A.S. Ekman layer on a porous plate. Phys. F1uids, 1972, vol. 15, pp. 930–931.
- Marques F., Sanchez J., Weidman P.D. Generalized Couette–Poiseuille flow with boundary mass transfer. J. Fluid Mech., 1998, vol. 374, pp. 221–249.
- Rajagopal K.R. A class of exact solutions to the Navier-Stokes equations. Int. J. Eng. Sci., 1984, vol. 22, pp. 451–458.
- Rasmussen H. Steady flow between two porous disks. ZAMP, 1970, vol. 21, pp. 187–195.
- 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.
- Wang C.Y. Exact solutions of the steady-state Navier-Stokes equations. Annu. Rev. Fluid Mech., 1991, vol. 23, pp. 159–177.
- Wang C.Y. Stagnation flow with slip: Exact solution of the Navier-Stokes equations. ZAMP, 2003, vol. 54, pp. 184–189.
- Wang, C.Y. On a class of exact solutions of the Navier-Stokes equations. J. Appl. Mech., 1966, vol. 33, pp. 696–698.
Н. В. Бурмашева, Е. Ю. Просвиряков
КЛАСС ТОЧНЫХ РЕШЕНИЙ С ПРОСТРАНСТВЕННЫМ УСКОРЕНИЕМ ДЛЯ ОПИСАНИЯ ТЕЧЕНИЙ ВЯЗКОЙ НЕСЖИМАЕМОЙ ЖИДКОСТИ В ПОЛЕ МАССОВЫХ СИЛ
В статье представлен новый класс точных решений системы уравнений Навье–Стокса, позволяющий учитывать нелинейное распределение поля давления и влияние внешних объемных сил, а также принимать во внимание возможность горизонтального оттока/притока жидкости при моделировании ее движения в вертикальном направлении. Данный класс является обобщением класса Линя–Сидорова–Аристова, предполагающего линейное распределение двух проекций вектора скорости течения жидкости по части координат и независимость от этих координат третьей проекции вектора скорости.
Ключевые слова: класс точных решений, уравнение Навье–Стокса, переопределенная система, поле массовых (объемных) сил Библиография:
- Шабловский О. Н. Сферическое течение идеальной жидкости в пространственнонеоднородном силовом поле // Вестн Том. гос. ун-та. Математика и механика. – 2020. – № 64. – С. 146–155. – DOI: 10.17223/19988621/64/11.
- Мануйлович С. В. Продольно-периодическое течение вязкой жидкости, порождаемые пристеночной объемной силой // Известия РАН. Механика жидкости и газа. – 2015. – № 4. – С. 59–67.
- Мануйлович С. В. Управление поперечным течением в пограничном слое на скользящем крыле с помощью пристеночной объемной силы // Известия РАН. Механика жидкости и газа. – 2020. – № 1. – С. 45–56. – DOI: 10.31857/S0568528120010090.
- Rosensweig R. E. Magnetic fluids // Ann. Rev. Fluid Mech. – 1987. – Vol. 19, no. 1. – P. 437–463. – DOI: 10.1146/annurev.fl.19.010187.002253.
- Розенцвейг Р. Феррогидродинамика / пер. с англ. – М. : Мир, 1989. – 352 с.
- Yerin C. V., Vivchar V. I. Ellipsometry of magnetic fluid in a magnetic field // Journal of Magnetism and Magnetic Materials. – 2020. – Vol. 498. – P. 166144. – DOI: 10.1016/j.jmmm.2019.166144.
- Wei Pan, Jianghong Gong. Effects of magnetic fluid on magnetic fluid damper // Key Engineering Materials. – 2012. – Vols. 512–515. – P. 1479–1483. – DOI: 10.4028/www.scientific.net/KEM.512-515.1479.
- Modification of diamagneticmaterials using magnetic fluids / I. Safarik, J. Prochazkova, E. Baldikova, M. Timko, P. Kopcansky, M. Rajnak, N. Torma, K. Pospiskova // Ukr. J. Phys. – 2020. – Vol. 65, no. 9. – P. 751-760. – DOI: 10.15407/ujpe65.9.751.
- Horváth B., Szalai I. Nonlinear magnetic properties of magnetic fluids for automotive applications // Hungarian Journal of Industry and Chemistry. – 2020. – Vol. 48, no. 1. – P. 61–65. – DOI: 10.33927/hjic-2020-09.
- Брацун Д. А., Вяткин В. А., Мухаматуллин А. Р. О точных нестационарных решениях уравнений вибрационной конвекции // Вычислительная механика сплошных сред. – 2017. – Т. 10, № 4. – С. 433–444. – DOI: 10.7242/1999-6691/2017.10.4.35.
- Конвективные процессы в невесомости / В. И. Полежаев, М. С. Белло, Н. А. Верезуб, К. Г. Дубовик, А. П. Лебедев, С. А. Никитин, Д. С. Павловский, А. И. Федюшкин. – М. : Наука, 1991. – 242 с.
- Полежаев В. И., Сазонов В. В. Механика невесомости и гравитационно-чувствительные системы. Аннотации докладов научно-исследовательского семинара. – М. : Институт проблем механики РАН им. В. М. Келдыша (препринт № 898), 2009. – 76 с.
- Суржиков С. Т. Гиперзвуковое обтекание разреженным газом поверхностного тлеющего разряда с внешним магнитным полем. – М. : Ин-т проблем механики РАН, 2011. – 274 с.
- Kedrinskiy V. K. Role of nuclei density as a “hidden” parameter in the formation of anomalous zones in a heavy cavitating magma // Journal of Applied Mechanics and Technical Physics. – 2014. – Vol. 55. – P. 276–281. – DOI: 10.1134/S0021894414020102.
- Afonin V. V., Petrov V. Y. On exact solutions for a Luttinger liquid with one impurity // Jetp Lett. – 2019. – Vol. 109. – P. 762–767. – DOI: 10.1134/S0021364019110031.
- Kouwn S., Moon T., Oh P. Exact solution and exotic fluid in cosmology // Entropy. – 2012. – Vol. 14. – P. 1771–1783. – DOI: 10.3390/e14091771.
- Аристов С. Н. Стационарный цилиндрический вихрь в вязкой жидкости // Доклады Академии наук. – 2001. – Т. 377. – С. 477–480.
- Аристов С. Н. Точное решение задачи о точечном источнике // Доклады Академии наук. – 1995. – Т. 343, № 1. – С. 50–52.
- Aristov S. N. Three-Dimensional Conical Viscous Incompressible Fluid Flows // Fluid Dynamics. – 1998. – Vol. 33, no. 929. – DOI: 10.1007/BF02698663.
- Aristov S. N., Grabovskii V. I. Self-similar solution of the Navier-Stokes equations governing gas flows in rotary logarithmically spiral two-dimensional channels // Fluid Dyn. – 1995 – Vol. 30. – P. 838–844. – DOI: 10.1007/BF02078198.
- Темам Р. Уравнения Навье-Стокса. Теория и численный анализ / пер. с англ. – М. : Мир, 1981. – 408 с.
- Ландау Л. Д., Лифшиц Е. М. Гидродинамика. Теоретическая физика : том VI. – Изд. 4-е, стереотипное. – М. : Наука, 1988. – 736 с.
- Хмельник С. И. Уравнения Навье-Стокса. Существование и метод поиска глобального решения. – Израиль, Mathematics in Computer Comp., 2010. – 106 с.
- Рауз Х. Механика жидкости. – М. : Изд. литературы по строительству, 1967. – 392 с.
- Boussinesq J. Théorie de l'écoulement tourbillonnant et tumultueux des liquides dans les lits rectilignes a grande section. – Paris, Gauthier-Villars et fils, 1897. – 90 p.
- Гершуни Г. З., Жуховицкий Е. М. Конвективная устойчивость несжимаемой жидкости. – М. : Наука, 1972. – 392 с.
- Müller P. Boussinesq approximation. In the equations of oceanic motions. – Cambridge : Cambridge University Press, 2006. – DOI:10.1017/CBO9780511617843.012.
- 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.
- 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, no. 8. – P. 3364–3371. – DOI: 10.1016/j.jksus.2020.09.023.
- Martyushev S. G., Sheremet M. A. Conjugate natural convection combined with surface thermal radiationin a three-dimensional enclosure with a heat source // International Journal of Heat and Mass Transfer. – 2014. – Vol. 73. – P. 340–353. – DOI: 10.1016/j.ijheatmasstransfer.2014.02.009.
- Couette M. Etudes sur le frottement des liquides // Ann. Chim. Phys.. – 1890. – Vol. 21. – P. 433–510.
- Poiseuille J. Recherches experimentelles sur le mouvement des liquides dans les tubes de tres petits diametres // Comptes Rendus. – 1840. – Vol. 11. – P. 961–967.
- Stokes G. G. On the effct of the internal friction of fluid on the motion of pendulums / G.G. Stokes // Camb. Philo. Trans. – 1851. – Vol. 9. – P. 8–106.
- Lin C. C. Note on a class of exact solutions in magneto-hydrodynamics // Arch. Rational Mech. Anal. – 1958. – Vol. 1. – P. 391–395.
- Шмыглевский Ю. Д. Аналитические исследования динамики газа и жидкости. – М. : Эдиториал УРСС, 1999. – 231 с.
- Сидоров А. Ф. О двух классах решений уравнений механики жидкости и газа и их связи с теорией бегущих волн // ПМТФ. – 1989. – Вып. 2. – С. 34–40.
- Аристов С. Н. Вихревые течения в тонких слоях жидкости : дис. ... докт. физ.-мат. наук: 01.02.05/ – Владивосток : ИАПУ, 1990. – 330 с.
- Аристов С. Н., Просвиряков Е. Ю. Новый класс точных решений трехмерных уравнений термодиффузии // Теоретические основы химической технологии. – 2016. – Т. 50, № 3. – С. 294–301. – DOI: 10.7868/S0040357116030027.
- Prosviryakov E. Yu. A sufficient condition for the absence of strong and weak discontinuities in gas flows in flat channels // Diagnostics, Resource and Mechanics of materials and structures. – 2019. – Iss. 3. – P. 25–40. – DOI: 10.17804/2410-9908.2019.3.025-040.
- Аристов С. Н., Просвиряков Е. Ю. Крупномасштабные течения завихренной вязкой несжимаемой жидкости // Известия высших учебных заведений. Авиационная техника. – 2015. – Вып. 4. – С. 50–54.
- 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. – Вып. 2. – С. 57–70. – DOI: 10.17804/2410-9908.2019.2.057-070.
- 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.
- Privalova V.V. and Prosviryakov E.Yu. The effect of tangential boundary stresses on the convective unidirectional flow of a viscous fluid layer under the lower boundary heating condition // Diagnostics, Resource and Mechanics of materials and structures. – 2019. – Iss. 4. – P. 44–55. – DOI: 10.17804/2410-9908.2019.4.044-055.
- Burmasheva 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.
- 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.
- 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. – Вып. 2. – С. 61–75. – DOI: 10.17804/2410-9908.2020.2.061-075.
- Burmasheva N.V., Prosviryakov E.Yu. Temperature field investigation in layered flows of a vertically swirling viscous incompressible fluid under two thermocapillar forces at a free boundary // Diagnostics, Resource and Mechanics of materials and structures. – 2019. – Iss. 1. – P. 6–42. – DOI: 10.17804/2410-9908.2019.1.006-042.
- Gorshkov A.V., Prosviryakov E.Y. Complex stationary convection with third-kind boundary conditions at the boundaries of a fluid layer // Diagnostics, Resource and Mechanics of materials and structures. – 2016. – Iss. 2. – P. 34–47. – DOI: 10.17804/2410-9908.2016.2.034-047.
- Burmasheva N.V., Prosviryakov E.Yu. Exact solutions for natural convection of layered flows of a viscous incompressible fluid with Specified tangential forces and the linear distribution of temperature on the layer boundaries // Diagnostics, Resource and Mechanics of materials and structures. – 2017. – Iss. 4. – P. 16–31. – DOI: 10.17804/2410-9908.2017.4.016-031.
- Petrov A. G. Exact solution of the Navier-Stokes equations in a fluid layer between the moving parallel plates // Journal of Applied Mechanics and Technical Physics. – 2012 – Vol. 53, no. 5 – P. 642–646. – DOI: 10.1134/S0021894412050021.
- Зырянов В. Н. Теория установившихся океанических течений. – Ленинград : Гидрометеоиздат, 1985.
- 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. 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.
- Meleshko S. V., Pukhnachev V. V. On a class of partially invariant solutions of the Navier-Stokes equations // Journal of Applied Mechanics and Technical Physics. – 1999. – Vol. 40, no. 2. – P. 208–216.
- Пухначёв В. В. Симметрии в уравнениях Навье-Стокса // Успехи механики. – 2006. – Т. 1. – С. 6–76.
- Aristov S. N., Gitman I. M. Viscous flow between two moving parallel disk: exact solutions and stability analysis // J. Fluid Mech. – 2002. – Vol. 464. – P. 209–215. – DOI: 10.1017/S0022112002001003.
- Berker R. An exact solution of the Navier-Stokes equation - the vortex with curvilinear axis // Int. J. Eng. Sci. – 1982. – Vol. 20. – P. 217–230.
- Berker R. A new solution of the Navier-Stokes equation for the motion of a fluid contained between two parallel plates rotating about the same axis // Arch. Mech. – 1979. – Vol. 31. – P. 265–280.
- Dorrepaal J. M. An exact solution of the Navier-Stokes equation which describes non-orthogonal stagnation point flow in two dimensions // J. Fluid Mech. – 1986. – Vol. 163. – P. 141–147.
- Ekman V. W. On the influence of the earth's rotation on ocean currents // Ark. Mat. Astron. Fys. – 1905. – Vol. 2. – P. 1–52.
- Gupta A. S. Ekman layer on a porous plate // Phys. F1uids. – 1972. – Vol. 15. – P. 930–931.
- Marques F., Sanchez J., Weidman P.D. Generalized Couette – Poiseuille flow with boundary mass transfer // J. Fluid Mech. – 1998. – Vol. 374. – P. 221–249.
- Rajagopal K. R. A class of exact solutions to the Navier-Stokes equations // Int. J. Eng. Sci. – 1984. – Vol. 22. – P. 451–458.
- Rasmussen H. Steady flow between two porous disks // ZAMP. – 1970. – Vol. 21. – P. 187–195.
- 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.
- Wang C. Y. Exact solutions of the steady-state Navier-Stokes equations // Annual Review of Fluid Mechanics – 1991. – Vol. 23. – P. 159–177.
- Wang C. Y. Stagnation flow with slip: Exact solution of the Navier-Stokes equations // ZAMP. – 2003. – Vol. 54. – P. 184–189.
- Wang C. Y. On a class of exact solutions of the Navier-Stokes equations // J. Appl. Mech. – 1966. – Vol. 33. – P. 696–698.
Библиографическая ссылка на статью
Burmasheva N. V., Prosviryakov E. Yu. A Class of Exact Solutions with Spatial Acceleration for the Description of Viscous Incompressible Fluid Flows in the Field of Mass Forces // Diagnostics, Resource and Mechanics of materials and structures. -
2021. - Iss. 1. - P. 6-25. - DOI: 10.17804/2410-9908.2021.1.006-025. -
URL: http://dream-journal.org/issues/content/article_310.html (accessed: 10.12.2024).
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