Exact Solutions for the Description of Nonuniform Unidirectional Flows of Magnetic Fluids in the Lin–sidorov–aristov Class

L. S. Goruleva; E. Yu. Prosviryakov

doi:10.17804/2410-9908.2023.5.039-052

L. S. Goruleva, E. Yu. Prosviryakov

EXACT SOLUTIONS FOR THE DESCRIPTION OF NONUNIFORM UNIDIRECTIONAL FLOWS OF MAGNETIC FLUIDS IN THE LIN–SIDOROV–ARISTOV CLASS

DOI: 10.17804/2410-9908.2023.5.039-052

The paper considers the exact integration of magnetic hydrodynamic equations for describing nonuniform unidirectional flows of viscous incompressible fluids. The construction of an exact solution is based on the well-known representation of hydrodynamic fields as the Lin–Sidorov–Aristov class. The 3d magnetic field is described by linear forms with respect to two spatial coordinates (longitudinal, or horizontal). The coefficients of the linear forms depend on the third coordinate and time. In view of the incompressibility condition, the 1D velocity field depends on two coordinates and time. The pressure is shown to be determined by a quadratic form with constant coefficients. These coefficients are determined by pressure distribution on the known (free) boundary. The exact solution is illustrated by the integration of non-1D hydrodynamic fields in the case of the steady motion of a conducting viscous incompressible fluid. This solution is polynomial, and it will be useful for the formulation of new problems of hydrodynamic stability.

Acknowledgement: The work was performed under the state assignment, theme No. AAAA-A18-118020790140-5.

Keywords: exact solution, Navier–Stokes equation, magnetic hydrodynamics, conducting fluid, nonuniform flow

References:

Drazin, P.G. and Riley, N. The Navier–Stokes Equations: A Classification of Flows and Exact Solutions, Cambridge University Press, Cambridge, 2006, 196 p. DOI: 10.1017/cbo9780511526459.
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.
Pukhnachev, V.V. Symmetries in the Navier–Stokes equations. Uspekhi Mekhaniki, 2006, 1, 6–76. (In Russian).
Wang, C.Y. Exact solutions of the unsteady Navier–Stokes equations. Applied Mechanics Review, 1989, 42 (11S), 269–282. DOI: 10.1115/1.3152400.
Wang, C.Y. Exact solutions of the steady-state Navier–Stokes equations. Annual Review of Fluid Mechanics, 1991, 23, 159–177. DOI: 10.1146/annurev.fl.23.010191.001111.
Ostroumov, G.A. Free convection under the condition of the internal problem, 1958, NASA Technical Memorandum 1407.
Birikh, R.V. Thermocapillary convection in a horizontal layer of liquid. Journal of Applied Mechanics and Technical Physics, 1966, 7, 43–44. DOI: 10.1007/bf00914697.
Hartmann, J. and Lazarus, F. Theory of the laminar flow of an electrically conducting liquid in a homogeneous magnetic field. Kongelige Danske Videnskabernes Selskab Mathematisk-fysiske Meddelelser, 1937, 15 (6), 1–28.
Lin, C.C. Note on a class of exact solutions in magnetohydrodynamics. Archive for Rational Mechanics and Analysis, 1958, 1, 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. Journal of Applied Mechanics and Technical Physics, 1989, 30 (2), 197–203. DOI: 10.1007/BF00852164.
Aristov, S.N. Vikhrevye techeniya v tonkikh sloyakh zhidkosti [Eddy Currents in Thin Liquid Layers. Optimization of Boundary and Distributed Controls in Semilinear Hyperbolic Systems: Synopsis of Doctoral Thesis]. Vladivostok, 1990, 303 p. (In Russian).
Aristov, S.N. and Prosviryakov, E.Yu. A new class of exact solutions for three-dimensional thermal diffusion equations. Theoretical Foundations of Chemical Engineering, 2016, 50 (3), 286–293. DOI: 10.1134/S0040579516030027.
Baranovskii, E.S., Burmasheva, N.V., and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations with couple stresses. Symmetry, 2021, 13 (8), 1355. DOI: 10.3390/sym13081355.
Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations for describing the convective flows of multilayer fluids. Russian Journal of Nonlinear Dynamics, 2022, 18 (3), 397–410. DOI: 10.20537/nd220305.
Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations describing stratified fluid flows. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya: Fiziko-Matematicheskie Nauki, 2021, 25 (3), 491–507. DOI: 10.14498/vsgtu1860.
Prosviryakov, E.Yu. Exact solutions to generalized plane Beltrami–Trkal and Ballabh flows. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya: Fiziko-Matematicheskie Nauki, 2020, 24 (2), 319–330. DOI: 10.14498/vsgtu1766.
Kovalev, V.P. and Prosviryakov, E.Yu. A new class of non-helical exact solutions of the Navier–Stokes equations. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya: Fiziko-Matematicheskie Nauki, 2020, 24 (4), 762–768. DOI: 10.14498/vsgtu1814.
Kovalev, V.P., Prosviryakov, E.Yu., and Sizykh, G.B. Obtaining examples of exact solutions of the Navier–Stokes equations for helical flows by the method of summation of velocities. Trudy MFTI, 2017, 9 (1), 71–88. (In Russian).
Zubarev, N.M. and Prosviryakov, E.Yu. Exact solutions for layered three-dimensional nonstationary isobaric flows of a viscous incompressible fluid. Journal of Applied Mechanics and Technical Physics, 2019, 60 (6), 1031–1037. DOI: 10.1134/S0021894419060075.
Prosviryakov, E.Yu. New class of exact solutions of Navier–Stokes equations with exponential dependence of velocity on two spatial coordinates. Theoretical Foundations of Chemical Engineering, 2019, 53 (1), 107–114. DOI: 10.1134/S0040579518060088.
Privalova, V.V. and Prosviryakov, E.Yu. A new class of exact solutions of the Oberbeck–Boussinesq equations describing an incompressible fluid. Theoretical Foundations of Chemical Engineering, 2022, 56 (3), 331–338. DOI: 10.1134/S0040579522030113.
Goruleva, L.S. and Prosviryakov, E.Yu. A new class of exact solutions to the Navier-Stokes equations with allowance made for internal heat release. Chemical Physics and Mesoscopy, 2022, 24 (1), 82–92. DOI: 10.15350/17270529.2022.1.7. (In Russian).
Goruleva, L.S. and Prosviryakov, E.Yu. Unidirectional steady-state inhomogeneous Couette flow with a quadratic velocity profile along a horizontal coordinate. Diagnostics, Resource and Mechanics of materials and structures, 2022, 3, 47–60. DOI: 10.17804/2410-9908.2022.3.047-060. Available at: http://dream-journal.org/issues/2022-3/2022-3_367.html
Bogoyavlenskij, O. The new effect of oscillations of the total angular momentum vector of viscous fluid. Physics of Fluids, 2022, 34, 083108. DOI: 10.1063/5.0101870.
Bogoyavlenskij, O. The new effect of oscillations of the total kinematic momentum vector of viscous fluid. Physics of Fluids, 2022, 34, 123104. DOI: 10.1063/5.0127990.
Goruleva, L.S. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations for describing inhomogeneous isobaric vertical vortex fluid flows in regions with permeable boundaries. Diagnostics, Resource and Mechanics of materials and structures, 2023, 1, 1–53. DOI: 10.17804/2410-9908.2023.1.041-053. Available at: http://dream-journal.org/issues/2023-1/2023-1_393.html
Okatev, R.S., Frick, P.G., and Kolesnichenko, I.V. Hartmann flow in a fluid layer with spatially inhomogeneous properties. Vestnik YuUrGU, Seriya Matematika, Mekhanika, Fizika, 2023, 15 (3), 34–42. DOI: 10.14529/mmph230304. (In Russian).

Л. С. Горулева, Е. Ю. Просвиряков

ТОЧНЫЕ РЕШЕНИЯ ДЛЯ ОПИСАНИЯ НЕОДНОРОДНЫХ ОДНОНАПРАВЛЕННЫХ ТЕЧЕНИЙ МАГНИТНЫХ ЖИДКОСТЕЙ В КЛАССЕ ЛИНЯ – СИДОРОВА – АРИСТОВА

В статье рассматривается точное интегрирование уравнений магнитной гидродинамики для описания неоднородных однонаправленных течений вязкой несжимаемой жидкости. Построение точного решения базируется на известном представлении гидродинамических полей как семейства полей Линя – Сидорова – Аристова. Трехмерное магнитное поле описывается линейными формами относительно двух пространственных координат (продольных, или горизонтальных). Коэффициенты линейных форм зависят от третьей координаты и времени. Одномерное поле скоростей зависит, в силу условия несжимаемости, от двух координат и времени. Давление, как показано в статье, определяется квадратичной формой с постоянными коэффициентами. Эти коэффициенты определяются распределением давления на известной (свободной) границе. Для иллюстрации точного решения проведено интегрирование неодномерных гидродинамических полей в случае установившегося движения вязкой проводящей несжимаемой жидкости. Данное решение является полиномиальным и будет полезным для постановки новых задач гидродинамической устойчивости.

Благодарность: Работа выполнена в рамках государственного задания по теме № АААА-А18- 118020790140-5.

Ключевые слова: точное решение, уравнение Навье – Стокса, магнитная гидродинамика, проводящая жидкость, неоднородное течение

Библиография:

Drazin P. G., Riley N. The Navier–Stokes Equations: A classification of flows and exact solutions. – Cambridge : Cambridge University Press, 2006. – 196 p. – DOI: 10.1017/cbo9780511526459.
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, No. 4. – P. 044501. – DOI: 10.1088/1873-7005/ac10f0.
Пухначев В. В. Симметрии в уравнениях Навье–Стокса // Успехи механики. – 2006. –Т. 4, № 1. – С. 6–76.
Wang C. Y. Exact solutions of the unsteady Navier–Stokes equations // Applied Mechanics Review. – 1989. – Vol. 42 (11S). – P. 269–282. – DOI: 10.1115/1.3152400.
Wang C. Y. Exact solutions of the steady-state Navier–Stokes equations // Annual Review of Fluid Mechanics. – 1991. – Vol. 23. – P. 159–177. – DOI: 10.1146/annurev.fl.23.010191.001111.
Ostroumov G. A. Free convection under the condition of the internal problem. – 1958. – NASA Technical Memorandum 1407.
Birikh R. V. Thermocapillary convection in a horizontal layer of liquid // Journal of Applied Mechanics and Technical Physics. – 1966. – Vol. 7 (3). – P. 43–44. – DOI: 10.1007/bf00914697.
Hartmann J., Lazarus F. Theory of the laminar ow of an electrically conducting liquid in an homogeneous magnetic field // Kongelige Danske Videnskabernes Selskab Mathematisk-fysiske Meddelelser. – 1937. – 15 (6). – P. 1–28.
Lin C. C. Note on a class of exact solutions in magnetohydrodynamics // Archive for Rational Mechanics and Analysis. – 1958. – 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 // Journal of Applied Mechanics and Technical Physics. – 1989. – Vol. 30 (2). – 197–203. – DOI: 10.1007/BF00852164.
Аристов С. Н. Вихревые течения в тонких слоях жидкости : автореф. дис. … докт. физ.-мат. наук. – Владивосток, 1990. – 303 с.
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 (3). – 286–293. – DOI: 10.1134/S0040579516030027.
Baranovskii E. S., Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations with couple stresses // Symmetry. – 2021. – Vol. 13 (8). – P. 1355. – DOI: 10.3390/sym13081355.
Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations for describing the convective flows of multilayer fluids // Russian Journal of Nonlinear Dynamics. – 2022. – Vol. 18, No. 3. – P. 397–410. – DOI: 10.20537/nd220305.
Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations describing stratified fluid flows // Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya «Fiziko-Matematicheskie Nauki». – 2021. – Vol. 25, No. 3. – P. 491–507. – DOI: 10.14498/vsgtu1860.
Prosviryakov E. Yu. Exact solutions to generalized plane Beltrami–Trkal and Ballabh flows // Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya «Fiziko-Matematicheskie Nauki» – 2020. – Vol. 24, No. 2. – P. 319–330. – DOI: 10.14498/vsgtu1766.
Kovalev V. P., Prosviryakov E. Yu. A new class of non-helical exact solutions of the Navier–Stokes equations // Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya «Fiziko-Matematicheskie Nauki». – 2020. – Vol. 24, No. 4. – P. 762–768. – DOI: 10.14498/vsgtu1814.
Ковалёв В. П., Просвиряков Е. Ю., Сизых Г. Б. Получение примеров точных решений уравнений Навье–Стокса для винтовых течений методом суммирования скоростей // Труды МФТИ. – 2017. – Т. 9, № 1. – С. 71–88.
Zubarev N. M., Prosviryakov E. Yu. Exact solutions for layered three-dimensional nonstationary isobaric flows of a viscous incompressible fluid // Journal of Applied Mechanics and Technical Physics. – 2019. – Vol. 60 (6). – P. 1031–1037. – DOI: 10.1134/S0021894419060075.
Prosviryakov E. Yu. 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 (1). – P. 107–114. – DOI: 10.1134/S0040579518060088.
Privalova V. V., Prosviryakov E. Yu. A new class of exact solutions of the Oberbeck–Boussinesq equations describing an incompressible fluid // Theoretical Foundations of Chemical Engineering. – 2022. – Vol. 56 (3). – P. 331–338. – DOI: 10.1134/S0040579522030113.
Goruleva L. S., Prosviryakov E. Yu. A new class of exact solutions to the Navier-Stokes equations with allowance made for internal heat release // Chemical Physics and Mesoscopy. – 2022. – Vol. 24, No. 1. – P. 82–92. – DOI: 10.15350/17270529.2022.1.7. (In Russian).
Goruleva L. S., Prosviryakov E. Yu. Unidirectional steady-state inhomogeneous Couette flow with a quadratic velocity profile along a horizontal coordinate // Diagnostics, Resource and Mechanics of materials and structures. – 2022. – Iss. 3. – P. 47–60. – DOI: 10.17804/2410-9908.2022.3.047-060. – URL: http://dream-journal.org/issues/2022-3/2022-3_367.html
Bogoyavlenskij O. The new effect of oscillations of the total angular momentum vector of viscous fluid // Physics of Fluids. – 2022. – Vol. 34. – 083108. – DOI: 10.1063/5.0101870.
Bogoyavlenskij O. The new effect of oscillations of the total kinematic momentum vector of viscous fluid // Physics of Fluids. – 2022. – Vol. 34. – 123104. – DOI: 10.1063/5.0127990.
Goruleva L. S. and Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations for describing inhomogeneous isobaric vertical vortex fluid flows in regions with permeable boundaries // Diagnostics, Resource and Mechanics of materials and structures. – 2023. – Iss. 1. – P. 41–53. – DOI: 10.17804/2410-9908.2023.1.041-053. – URL: http://dream-journal.org/issues/2023-1/2023-1_393.html
Окатьев Р. С., Фрик П. Г., Колесниченко И. В. Течение Гартмана в слое жидкости с пространственно неоднородными свойствами // Вестник ЮУрГУ. Серия «Математика. Механика. Физика». – 2023. – T. 15, № 3. – С. 34–42. – DOI: 10.14529/mmph230304.

Библиографическая ссылка на статью

Goruleva L. S., Prosviryakov E. Yu. Exact Solutions for the Description of Nonuniform Unidirectional Flows of Magnetic Fluids in the Lin–sidorov–aristov Class // Diagnostics, Resource and Mechanics of materials and structures. - 2023. - Iss. 5. - P. 39-52. -
DOI: 10.17804/2410-9908.2023.5.039-052. -
URL: http://dream-journal.org/issues/2023-5/2023-5_415.html
(accessed: 04.04.2025).