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

K. V. Gubareva; E. Yu. Prosviryakov; A. V. Eremin

doi:10.17804/2410-9908.2026.3.101-115

K. V. Gubareva, E. Yu. Prosviryakov, A. V. Eremin

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

DOI: 10.17804/2410-9908.2026.3.101-115

An exact analytical solution is derived for the system of equations describing an unsteady convective flow of a viscous incompressible fluid in a plane layer, taking into account the buoyancy force and viscous dissipation of energy. A class of shear flows with isothermal boundaries is considered, for which the temperature depends only on the transverse coordinate and time. The solution is constructed in the form of series in eigenfunctions, allowing one to describe the time evolution of velocity, temperature, and pressure fields. The flow characteristics are analyzed with water as an example for various time scales. The solution can be used to study transient processes in microfluidic systems and heat exchangers.

Keywords: exact solution, unsteady convection, viscous dissipation, Boussinesq approximation, energy equation

References:

Landau, L.D. and Lifshitz, E.M. Teoreticheskaya fizika. T. 6. Gidrodinamika [Theoretical Physics. Hydrodynamics: vol. 6]. Fizmatlit Publ., Moscow, 2006, 736 p. (In Russian).
Schlichting, H. and Gersten, K. Boundary-Layer Theory, 9th ed., Springer, Berlin, Heidelberg, 2017, 805 p. DOI: 10.1007/978-3-662-52919-5.
Bird, R.B. and Stewart, W.E. Lightfoot EN Transport Phenomena, 2nd ed., Wiley, 2002, 895 р.
Schlichting, H. Teoriya pogranichnogo sloya [Boundary-Layer Theory]. Nauka Publ., Moscow, 1974, 712 p. (In Russian).
Loitsyanskiy, L.G. Mekhanika zhidkosti i gaza [Mechanics of Liquids and Gases]. Drofa Publ., Moscow, 2003, 840 p. (In Russian).
Levich, V.G. Fiziko-khimicheskaya gidrodinamika [Physicochemical Hydrodynamics]. Fizmatgiz Publ., Moscow, 1959, 699 p. (In Russian).
Gershuni, G.Z. and Zhukhovitsky, E.M. Konvektivnaya ustoichivost neszhimaemoy zhidkosti [Convective Stability of Incompressible Fluid]. Nauka Publ., Moscow, 1972, 392 p. (In Russian).
Bejan, A. Convection Heat Transfer, 4^th ed., Wiley, 2013, 704 p.
Shah, R.K. and London, A.L. Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data, Academic Press, 1978, 477 p.
Pukhnachev, V.V. Group properties of the Navier–Stokes equations in the two-dimensional case. Prikl. Mekh. Tekh. Fiz., 1960, 1, 83–90. (In Russian).
Prosviryakov, E.Yu., Mikhailov, S.A., Ledyankina, O.A., and Goruleva, L.S. Exact solutions to the Navier–Stokes equations with the Boussinesq approximation for describing binary fluid flows. Russian Aeronautics, 2023, 66, 500–509. DOI: 10.3103/S106879982303011X.
Aristov, S.N. and Prosviryakov, E.Yu. Nonuniform convective Couette flow. Fluid Dynamics, 2016, 51 (5), 581–587. DOI: 10.1134/S001546281605001X.
Goncharova, O.N. and Rezanova, E.V. Example of an exact solution of the stationary problem of two-layer flows with evaporation at the interface. Journal of Applied Mechanics and Technical Physics, 2014, 55, 247–257. DOI: 10.1134/S0021894414020072.
Batchelor, G.K. An Introduction to Fluid Dynamics, Cambridge University Press, 1967, 615 p.
Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, Oxford University Press, 1961.
Gebhart, B. Effects of viscous dissipation in natural convection. Journal of Fluid Mechanics, 1962, 14 (2), 225–232. DOI: 10.1017/S0022112062001196.
Barletta, A. and Nield, D.A. Combined forced and free convective flow in a vertical porous channel: the effects of viscous dissipation and pressure work. Transport in Porous Media, 2009, 79, 319–334. DOI: 10.1007/s11242-008-9320-y.
Gubareva, K.V. and 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, 6, 23–38. DOI: 10.17804/2410-9908.2025.6.023-038. Available at: http://dream-journal.org/issues/2025-6/2025-6_528.html
Gubareva, K.V., Prosviryakov, E.Yu., and Eremin, A.V. Inhomogeneous Couette–Poiseuille flow of a viscous incompressible fluid in an infinite horizontal layer with permeable boundaries. Diagnostics, Resource and Mechanics of materials and structures, 2025, 5, 6–28. DOI: 10.17804/2410-9908.2025.5.006-028. Available at: http://dream-journal.org/issues/2025-5/2025-5_523.html
Gubareva, K.V., Prosviryakov, E.Yu., and Eremin, A.V. An exact solution with inhomogeneous boundary conditions for a steady non-uniform Couette flow between permeable plates. Diagnostics, Resource and Mechanics of materials and structures, 2025, 5, 66–86. DOI: 10.17804/2410-9908.2025.5.066-086. Available at: http://dream-journal.org/issues/2025-5/2025-5_522.html

К. В. Губарева, Е. Ю. Просвиряков, А. В. Еремин

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

В работе получено точное аналитическое решение системы уравнений, описывающих нестационарное конвективное течение вязкой несжимаемой жидкости в плоском слое с учетом силы Архимеда и вязкой диссипации энергии. Рассматривается класс сдвиговых течений с изотермическими границами, при котором температура зависит только от поперечной координаты и времени. Построено решение в виде рядов по собственным функциям, позволяющее описывать эволюцию полей скорости, температуры и давления во времени. Проведен анализ характеристик течения на примере воды при различных временных масштабах. Решение может быть использовано для анализа переходных процессов в микрофлюидных системах и теплообменных устройствах.

Ключевые слова: точное решение, нестационарная конвекция, вязкая диссипация, приближение Буссинеска, уравнение энергии

Библиография:

Ландау Л. Д., Лифшиц Е. М. Теоретическая физика. Гидродинамика. – 5-е изд. – М. : Физматлит, 2006. – 736 с. – Т. 6.
Schlichting H., Gersten K. Boundary-Layer Theory. – 9th ed. – Berlin; Heidelberg : Springer, 2017. – 805 p. – DOI: 10.1007/978-3-662-52919-5.
Bird R. B., Stewart W. E., Lightfoot E. N. Transport Phenomena. – 2nd ed. – Wiley, 2002. – 895 р.
Шлихтинг Г. Теория пограничного слоя. – М. : Наука, 1974. – 712 с.
Лойцянский Л. Г. Механика жидкости и газа. – М. : Дрофа, 2003. – 840 с.
Левич В. Г. Физико-химическая гидродинамика. – 2-е изд., доп. и перераб. – Москва : Физматгиз, 1959. – 699 с.
Гершуни Г. З., Жуховитский Э. М. Конвективная устойчивость несжимаемой жидкости. – М. : Наука, 1972. – 392 с.
Bejan A. Convection Heat Transfer. – 4th ed. – Wiley, 2013. – 704 p.
Shah R. K., London A. L. Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data. – Academic Press, 1978. – 477 p.
Пухначев В. В. Групповые свойства уравнений Навье–Стокса в двумерном случае // Прикладная механика и техническая физика. – 1960. – Т. 1 (1). – С. 83–90.
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.
Aristov S. N., Prosviryakov E. Yu. Nonuniform convective Couette flow // Fluid Dynamics. – 2016. – Vol. 51 (5). – P. 581–587. – DOI: 10.1134/S001546281605001X.
Goncharova O. N., Rezanova E. V. Example of an exact solution of the stationary problem of two-layer flows with evaporation at the interface // Journal of Applied Mechanics and Technical Physics. – 2014. – Vol. 55 (2). – P. 247–257. – DOI: 10.1134/S0021894414020072.
Batchelor G. K. An Introduction to Fluid Dynamics. – Cambridge University Press, 1967. – 615 p.
Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. – Oxford University Press, 1961.
Gebhart B. Effects of viscous dissipation in natural convection // Journal of Fluid Mechanics. – 1962. – Vol. 14 (2). – P. 225–232. – DOI: 10.1017/S0022112062001196.
Barletta A., Nield D. A. Combined forced and free convective flow in a vertical porous channel: the effects of viscous dissipation and pressure work // Transport in Porous Media. – 2009. – Vol. 79. – P. 319–334. – DOI: 10.1007/s11242-008-9320-y.
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
Gubareva K. V., Prosviryakov E. Yu., Eremin A. V. Inhomogeneous Couette–Poiseuille flow of a viscous incompressible fluid in an infinite horizontal layer with permeable boundaries // Diagnostics, Resource and Mechanics of materials and structures. – 2025. – Iss. 5. – P. 6–28. – DOI: 10.17804/2410-9908.2025.5.006-028. – URL: http://dream-journal.org/issues/2025-5/2025-5_523.html
Gubareva K. V., Prosviryakov E. Yu., Eremin A. V. An exact solution with inhomogeneous boundary conditions for a steady non-uniform Couette flow between permeable plates // Diagnostics, Resource and Mechanics of materials and structures. – 2025. – Iss. 5. – P. 66–86. – DOI: 10.17804/2410-9908.2025.5.066-086. – URL: http://dream-journal.org/issues/2025-5/2025-5_522.html

Библиографическая ссылка на статью

Gubareva K. V., Prosviryakov E. Yu., Eremin A. V. Exact Analytical Solution for the Problem of Unsteady Convection in the Boussinesq Approximation with Account for Viscous Dissipation // Diagnostics, Resource and Mechanics of materials and structures. - 2026. - Iss. 3. - P. 101-115. -
DOI: 10.17804/2410-9908.2026.3.101-115. -
URL: http://dream-journal.org/issues/2026-3/2026-3_552.html
(accessed: 26.09.2026).