Exact Analytical Solution to the Problem of Stationary Convection in the Boussinesq Approximation with Account for Viscous Dissipation

K. V. Gubareva; E. Yu. Prosviryakov

doi:10.17804/2410-9908.2025.6.023-038

K. V. Gubareva, E. Yu. Prosviryakov

EXACT ANALYTICAL SOLUTION TO THE PROBLEM OF STATIONARY CONVECTION IN THE BOUSSINESQ APPROXIMATION WITH ACCOUNT FOR VISCOUS DISSIPATION

DOI: 10.17804/2410-9908.2025.6.023-038

An exact analytical solution is obtained for the system of equations governing stationary convection of a viscous incompressible fluid, accounting for the buoyancy force (within the Boussinesq approximation) and viscous dissipation. The flow in a plane layer between two parallel plates is considered. It is found that the system admits two mutually exclusive classes of solutions. One describes a thermogravitational flow with a linear dependence of temperature on the longitudinal coordinate and a velocity dependent only on the transverse coordinate. The other class represents a generalized shear flow, combining the Couette and Poiseuille profiles with temperature depending solely on the transverse coordinate and explicitly accounting for dissipative heating. It is shown that the energy equation prohibits the simultaneous existence of a transverse velocity shear and a longitudinal temperature gradient. For the latter class, an explicit closed-form solution is constructed for a particular case, namely a generalized Couette–Poiseuille flow with a constant pressure gradient and isothermal boundaries, which includes closed-form expressions for velocity, temperature, and pressure. All the solutions strictly satisfy the equations of motion and energy, as well as the specified boundary conditions.

Acknowledgement: The research was performed under the state assignment of the Russian Ministry of Science and Higher Education for the IES UB RAS, theme No. 124020600042-9.

Keywords: exact solution, Couette–Poiseuille flow, viscous dissipation, Boussinesq approximation, stationary convection

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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, 41–53. DOI: 10.17804/2410-9908.2023.1.041-053. Available at: http://dream-journal.org/issues/2023-1/2023-1_393.html
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К. В. Губарева, Е. Ю. Просвиряков

ТОЧНОЕ АНАЛИТИЧЕСКОЕ РЕШЕНИЕ ЗАДАЧИ О СТАЦИОНАРНОЙ КОНВЕКЦИИ В ПРИБЛИЖЕНИИ БУССИНЕСКА С УЧЕТОМ ВЯЗКОЙ ДИССИПАЦИИ

В работе получено точное аналитическое решение системы уравнений стационарной конвекции для вязкой несжимаемой жидкости с учетом силы Архимеда (в приближении Буссинеска) и вязкой диссипации. Рассматривается течение в плоском слое между двумя параллельными пластинами. Установлено, что система допускает два взаимоисключающих класса решений. Первый описывает термогравитационное течение с линейной зависимостью температуры от продольной координаты и скоростью, зависящей только от поперечной координаты. Второй класс представляет собой обобщенное сдвиговое течение, сочетающее профили Куэтта и Пуазейля с температурой, зависящей лишь от поперечной координаты, и явным учетом диссипативного нагрева. Установлено, что уравнение энергии запрещает одновременное существование поперечного сдвига скорости и продольного градиента температуры. Для второго класса построено явное замкнутое решение частного случая (обобщенного течения Куэтта – Пуазейля с постоянным градиентом давления и изотермическими границами), включающее замкнутые выражения для скорости, температуры и давления. Все решения строго удовлетворяют уравнениям движения и энергии, а также заданным граничным условиям.

Благодарность: Исследования выполнены в рамках государственного задания Минобрнауки РФ для ИМАШ УрО РАН, тема № 124020600042-9.

Ключевые слова: точное решение, течение Куэтта – Пуазейля, вязкая диссипация, приближение Буссинеска, стационарная конвекция

Библиография:

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Burmasheva N. V. Prosviryakov E. Yu. Thermocapillary convection of a vertical swirling liquid // Theoretical Foundations of Chemical Engineering. – 2020. – Vol. 54. – P. 230–239. – DOI: 10.1134/S0040579519060034.
Goruleva L. S., 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
Exact solutions to the Navier–Stokes equations with the Boussinesq approximation for describing binary fluid flows / E. Yu. Prosviryakov, S. A. Mikhailov, O. A. Ledyankina, L. S. Goruleva // Russian Aeronautics. – 2023. – Vol. 66. – P. 500–509. – DOI: 10.3103/S106879982303011X.
Prosviryakov E. Yu., Ledyankina O. A., Goruleva L. S. Exact solutions to the Navier–Stokes equations for describing the flow of multicomponent fluids with internal heat generation // Russian Aeronautics. – 2024. – Vol. 67. – P. 60–69. – DOI: 10.3103/S1068799824010070.
Goruleva L. S., Obabkov I. I., Prosviryakov E. Yu. Exact solutions to the Oberbeck–Boussinesq equations for convective Stokes flows // Diagnostics, Resource and Mechanics of materials and structures. – 2024. – Iss. 2. – P. 36–49. – DOI: 10.17804/2410-9908.2024.2.036-049. – URL: http://dream-journal.org/issues/2024-2/2024-2_437.html
Bashurov V. V., Burmasheva N. V., Prosviryakov E. Yu. Exact solution to the velocity field description for Couette–Poiseulle flows of binary liquids // Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki. – 2024. – Vol. 28 (4). – P. 759–772. – DOI: 10.14498/vsgtu2098.
Prosviryakov E. Yu., Ledyankina O. A., Goruleva L. S. A class of exact solutions of magnetic hydrodynamics equations for describing convective flows of multilayer fluids // Russian Aeronautics. – 2024. – Vol. 67. – P. 550–563. – DOI: 10.3103/S1068799824030103.
Goruleva L. S., Obabkov I. I., Prosviryakov E. Yu. Exact solutions to the Oberbeck–Boussinesq equations for describing multilayer fluid flows in the stokes approximation // Diagnostics, Resource and Mechanics of materials and structures. – 2025. – Iss. 2. – P. 6–27. – DOI: 10.17804/2410-9908.2025.2.006-027. – URL: http://dream-journal.org/issues/2025-2/2025-2_504.html
Prosviryakov E., Goruleva L., Ledyankina O. A new exact solution of the Navier–Stokes equations for describing the unsteady flows of vertically swirled fluid // Russian Aeronautics. – 2025. – Vol. 68 (2). – P. 393–401. – DOI: 10.3103/S1068799825020151.
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Просвиряков Е. Ю., Горулева Л. С., Альес М. Ю. Класс точных решений уравнений Обербека–Буссинеска с учетом диссипативной функции Рэлея // Химическая физика и мезоскопия. – 2024. – Т. 26 (2). – С. 164–178. – DOI: 10.62669/17270227.2024.2.15.

Библиографическая ссылка на статью

Gubareva K. V., Prosviryakov E. Yu. Exact Analytical Solution to the Problem of Stationary Convection in the Boussinesq Approximation with Account for Viscous Dissipation // Diagnostics, Resource and Mechanics of materials and structures. - 2025. - Iss. 6. - P. 23-38. -
DOI: 10.17804/2410-9908.2025.6.023-038. -
URL: http://dream-journal.org/issues/2025-6/2025-6_528.html
(accessed: 18.04.2026).