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    Periodic flow of a viscous fluid with a predetermined pressure and temperature gradient

    2018, Vol. 14, no. 1, pp.  81-97

    Author(s): Deryabina M. S., Martynov S. I.

    A procedure is proposed for constructing an approximate periodic solution to the equations of motion of a viscous fluid in an unbounded region in the class of piecewise smooth functions for a given gradient of pressure and temperature for small Reynolds numbers. The procedure is based on splitting the region of the liquid into cells, and finding a solution with boundary conditions corresponding to the periodic function. The cases of two- and three-dimensional flows of a viscous fluid are considered. It is shown that the solution obtained can be regarded as a flow through a periodic system of point particles placed in the cell corners. It is found that, in a periodic flow, the fluid flow rate per unit of cross-sectional area is less than that in a similar Poiseuille flow.
    Keywords: viscous fluid, periodic solution, piecewise function, gradient, pressure, temperature
    Citation: Deryabina M. S., Martynov S. I., Periodic flow of a viscous fluid with a predetermined pressure and temperature gradient, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp.  81-97

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    [1] Хаппель Дж., Бреннер Г., Гидродинамика при малых числах Рейнольдса, Мир, Москва, 1976, 632 с.; Happel J., Brenner H., Low Reynolds number hydrodynamics with special applications to particulate media, Prentice-Hall, Englewood Cliffs, N.J., 1965
    [2] Gray W. G., Miller C. T., Introduction to the thermodynamically constrained averaging theory for porous medium systems, Springer, Cham, 2014, 582 pp.
    [3] Дерябина М. С., Мартынов С. И., “Моделирование течения вязкой жидкости с частицами через ячейки пористой среды”, Вычислительная механика сплошных сред, 9:4 (2016), 420–429 [Deryabina M. S., Martynov S. I., “Simulation of the viscous flow with particles through cell porous medium”, Vychisl. Mekh. Sploshn. Sred, 9:4 (2016), 420–429 (Russian)]
    [4] Brady J. F., Durlofsky L. J., “The sedimentation rate of disordered suspensions”, Phys. Fluids, 31 (1988), 717–727  crossref
    [5] Hasimoto H., “On the periodic fundamental solutions of the Stokes' equations and their application to viscous flow past a cubic array of spheres”, J. Fluid Mech., 5 (1959), 317–328  crossref
    [6] Zick A. A., Homsy G. M., “Stokes flow through periodic array of spheres”, J. Fluid Mech., 115 (1982), 13–26  crossref
    [7] Mo G., Sangani A. S., “A method for computing flow interactions among spherical objects and its application to suspensions of drops and porous particles”, Phys. Fluids, 6:5 (1994), 1637–1652  crossref
    [8] Мартынов С. И., “Взаимодействие частиц в течении с параболическим профилем скорости”, Изв. РАН. Механика жидкости и газа, 2000, № 1, 84–91; Martynov S. I., “Particle interaction in a flow with a parabolic velocity profile”, Fluid Dynam., 35:1 (2000), 68–74  crossref
    [9] Мартынов С. И., “Гидродинамическое взаимодействие частиц”, Изв. РАН. Механика жидкости и газа, 1998, № 2, 112–119; Martynov S. I., “Hydrodynamic interaction of particles”, Fluid Dynam., 33:2 (1998), 245–251  crossref
    [10] Мартынов С. И., “Движение вязкой жидкости через периодическую решетку сфер”, Изв. РАН. Механика жидкости и газа, 2002, № 6, 48–54; Martynov S. I., “Viscous flow past a periodic array of spheres”, Fluid Dynam., 37:6 (2002), 889–895  crossref
    [11] Мартынов С. И., Сыромясов А. О., “Симметрия периодической решетки частиц и потока вязкой жидкости в приближении Стокса”, Изв. РАН. Механика жидкости и газа, 2007, № 3, 7–20; Martynov S. I., Syromyasov A. O., “Symmetry of a periodic array of particles and a viscous fluid flow in the Stokes approximation”, Fluid Dynam., 42:3 (2007), 340–353  crossref
    [12] Аристов С. Н., Полянин А. Д., “Точные решения трехмерных нестационарных уравнений Навье – Стокса”, Докл. РАН, 427:1 (2009), 35–40  mathnet [Aristov S. N., Polyanin A. D., “Exact solutions of three-dimensional nonstationary Navier – Stokes equations”, Dokl. Akad. Nauk, 427:1 (2009), 35–40 (Russian)]
    [13] Аристов С. Н., Просвиряков Е. Ю., “Неоднородные течения Куэтта”, Нелинейная динамика, 10:2 (2014), 177–182  mathnet [Aristov S. N., Prosviryakov E. Yu., “Non-uniform currents of Couette”, Nelin. Dinam., 10:2 (2014), 177–182 (Russian)]
    [14] Князев Д. В., Колпаков И. Ю., “Точные решения задачи о течении вязкой жидкости в цилиндрической области с меняющимся радиусом”, Нелинейная динамика, 11:1 (2015), 89–97  mathnet [Knyazev D. V., Kolpakov I. Yu., “Exact solutions of the problem of the flow of a viscous fluid in a cylindrical region with a varying radius”, Nelin. Dinam., 11:1 (2015), 89–97 (Russian)]
    [15] Дерябина М. С., Мартынов С. И., “Построение периодического решения уравнений движения вязкой жидкости с заданным градиентом давления”, Журн. СВМО, 18:3 (2016), 187–193  mathnet [Deryabina M. S., Martynov S. I., “Construction of periodic solutions equations of motion of a viscous fluid with a predetermined pressure gradient”, Zh. SVMO, 18:3 (2016), 187–193 (Russian)]
    [16] Ландау Л. Д., Лифшиц Е. М., Теоретическая физика: В 10 тт., т. 6, Гидродинамика, 5-е изд., Физматлит, Москва, 2006, 736 с.; Landau L. D., Lifshitz E. M., Course of theoretical physics: Vol. 6. Fluid mechanics, 2nd ed., Butterworth/Heinemann, Oxford, 2003, 552 pp.
    [17] Свешников А. Г., Боголюбов А. Н., Кравцов В. В., Лекции по математической физике, МГУ, Москва, 1993, 352 с. [Sveshnikov A. G., Bogolyubov A. N., Kravtsov V. V., Lectures on mathematical physics, MGU, Moscow, 1993 (Russian)]

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