Analytical Study of the Ekman Angle for the Benard–marangoni Convective Flow of Viscous Incompressible Fluid

A. V. Gorshkov; E. Yu. Prosviryakov

doi:10.17804/2410-9908.2021.4.34-49

A. V. Gorshkov, E. Yu. Prosviryakov

ANALYTICAL STUDY OF THE EKMAN ANGLE FOR THE BENARD–MARANGONI CONVECTIVE FLOW OF VISCOUS INCOMPRESSIBLE FLUID

DOI: 10.17804/2410-9908.2021.4.34-49

The paper considers the convective flow of a viscous incompressible fluid over a rotating surface. It studies the angle between the fluid velocity vector in the upper layer and the temperature gradient vector on the free surface. For the study, an analytical solution to the Oberbeck–Boussinesq equations is constructed, which describes the stratified Ekman flow taking into account two components of the Coriolis force. The temperature gradient and the conditions of Marangoni thermocapillary convection are set at the upper (free) boundary, and the condition of fluid adhesion is set on the lower (solid) boundary. The representation of velocities in the form of linear functions of horizontal coordinates is used. It is shown that, when the flow depth tends to infinity, the angle between the upper layer fluid velocity vector and the temperature gradient vector tends to π/2.

Keywords: analytical solution, thermocapillary convection, Ekman flow

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А. В. Горшков, Е.Ю. Просвиряков

АНАЛИТИЧЕСКОЕ ИССЛЕДОВАНИЕ УГЛА ЭКМАНА ДЛЯ КОНВЕКТИВНОГО ТЕЧЕНИЯ БЕНАРА-МАРАНГОНИ ВЯЗКОЙ НЕСЖИМАЕМОЙ ЖИДКОСТИ

В работе рассмотрено конвективное течение вязкой несжимаемой жидкости по вращающейся поверхности. Исследован угол между вектором скорости жидкости в верхнем слое и вектором градиента температуры на свободной поверхности. Для исследования построено аналитическое решение уравнений Обербека – Буссинеска, описывающее слоистое течение Экмана с учетом двух компонент силы Кориолиса. На верхней (свободной) границе задан градиент температуры и условия термокапиллярной конвекции Марангони. На нижней (твердой) границе задано условие прилипания жидкости. Использовано представление скоростей в виде линейных функций горизонтальных координат. Показано, что при стремлении глубины потока к бесконечности угол между вектором скорости жидкости в верхнем слое и вектором градиента температуры стремится к π/2.

Ключевые слова: аналитическое решение, термокапиллярная конвекция, течение Экмана

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Gorshkov A. V. Prosviryakov E. Yu. Analytical Study of the Ekman Angle for the Isothermal Flow of a Viscous Incompressible Fluid in View of the Navier Boundary Condition // AIP Conference Proceedings. – 2020. – Vol. 2315. – 020018. – DOI: 10.1063/5.0036889.
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Shrira V. I., Almelah R. B. Upper-ocean Ekman current dynamics: a new perspective // Journal of Fluid Mechanics. – 2020. – Vol. 887. – A 24. – DOI: 10.1017/jfm.2019.1059.
Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows / M. Fečkan, Y. Guan, D. O’Regan, J. R. Wang // Monatshefte für Mathematik. – 2020. – 193. – P. 623–636. – DOI: 10.1007/s00605-020-01414-7.
Global modes and large-scale structures in an Ekman boundary layer / J. L. Ortiz-Tarin, S. Lee, O. Flores, S. Sarkar // Journal of Physics: Conference Series. – 2020. – Vol. 1522. – P. 012011. – DOI: 10.1088/1742-6596/1522/1/012011.
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.
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. 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
Burmasheva N. V., Privalova V. V., Prosviryakov E. Yu. Layered Marangoni convection with the Navier slip condition // SADHANA-ACADEMY PROCEEDINGS IN ENGINEERING SCIENCES. – 2021. – Vol. 46, iss. 1, No. 55. – DOI: 10.1007/s12046-021-01585-5.
Towards understanding the algorithms for solving the Navier–Stokes equations / Sergey V. Ershkov, Evgeniy Yu. Prosviryakov, Natalya V. Burmasheva, and Victor Christianto // Fluid Dynamics Research. – 2021. – Vol. 53, No. 4. – P. 044501. – DOI:10.1088/1873-7005/ac10f0.

Библиографическая ссылка на статью

Gorshkov A. V., Prosviryakov E. Yu. Analytical Study of the Ekman Angle for the Benard–marangoni Convective Flow of Viscous Incompressible Fluid // Diagnostics, Resource and Mechanics of materials and structures. - 2021. - Iss. 4. - P. 34-48. -
DOI: 10.17804/2410-9908.2021.4.34-49. -
URL: http://dream-journal.org/issues/2021-4/2021-4_340.html
(accessed: 30.03.2025).