Точные решения Куэтта-Хименца для описания установившегося ползущего конвективного течения вязкой несжимаемой жидкости с учетом теплообмена



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Аннотация

Изучается установившееся ползущее конвективное течение вязкой несжимаемой жидкости в тонком бесконечном слое. Исследование течения жидкости основано на использовании класса точных решений для уравнений Обербека-Буссинеска в приближении Стокса. Поле скоростей описывается точным решением Хименца. Поле температуры и поле давление линейно зависят от горизонтальной (продольной) координаты, что соответствует классу точных решений Остроумова-Бириха. Конвективное движение вязкой несжимаемой жидкости индуцировалось касательными напряжениями на верхней проницаемой (пористой) границе и заданием теплового источника на нижней границе. Кроме того, на верхней границе учитывался теплообмен по закону Ньютона-Рихмана. Полученные точные решения описывают противотечения в жидкости, у которых количество застойных точек не превышает трех. Формирование противотечений в жидкости сопровождается отсосом (sucking) и вдувом (injection) жидкости через проницаемую границу. Наличие большего числа застойных точек формирует ячеистую структуру линий тока. Кроме того, поле скоростей, полученное при решении краевой задачи, характеризуется локализацией течения вблизи границ слоя жидкости (пограничный слой). Полученные в статье точные решения могут использоваться для решения нелинейной системы Обербека-Буссинеска. Показано, что при линеаризации системы Обербека-Буссинеска число Грасгофа может принимать большие значения, зависящие от показателя геометрической анизотропии.

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Introduction. The system consisting of the Boussinesq approximation of the Navier-Stokes equations (Oberbeck-Boussinesq equation system) and the incompressibility equation is one of the most common mathematical systems of equations which describes convective flows of a viscous incompressible fluid [1- 7]. These equations are extremely useful tools for the modeling of convective processes in a fluid [1-7]. The complexity of obtaining exact solutions for the Navier-Stokes equations and their modifications consists in the presence of a substantial (quadratic) nonlinearity of the equations. This nonlinearity is generated by the presence of a convective term in the total derivative. By now, a certain stock of exact solutions to the Oberbeck-Boussinesq equations has been accumulated [3-18]. Exact solutions describing convective flows of a viscous incompressible fluid allow the qualitative and quantitative characteristics of a moving flow to be re-evaluated. With certain assumptions for highly viscous fluids, the Oberbeck-Boussinesq system can be simplified by neglecting inertial effects due to the predominance of viscous forces (the Stokes approximation) [8, 10-19]. The first classes of exact solutions for the natural convection equations were considered by G. A. Ostroumov [20], R. V. Birich [21], and M. I. Schliomis [22]. Later on, the exact solutions presented in [20-22] were generalized in papers and monographs [3, 6-12, 18, 19, 23-30]. The classes of the exact solutions, based on the representation of velocities in the form of linearly dependent functions, were proposed by C. C. Lin for magnetic hydrodynamics [31]. The most complete list of exact solutions obtained in Lin’s class for isothermal fluids is found in the review [32]. The Hiemenz exact solution [33] belongs to the class of velocities, linear in part of the coordinates. It is characterized by the existence of a stagnant point in the flow of a viscous incompressible fluid. By now, there have appeared Hiemenz exact solution modifications for isothermal flows [34-36] and convective flows [37-40] of a viscous incompressible fluid. The proposed generalizations of the Hiemenz exact solution family allow one to investigate counterflows in a fluid that are induced by kinematic, dynamic, and thermal perturbations of the fluid flow (boundary conditions) [8, 9, 11-17, 37-40]. Additionally, papers studying the existence of stagnant points in the flow and flows near such points are worthy of note. The assignment of relations between hydrodynamic fields ignoring fluid evaporation and suction predominates among the diversity of boundary conditions on the free (interphase) boundary of the fluid layer [3, 12, 26-30]. Convective flows with fluid evaporation from a free surface can be simulated by replacing it with a permeable non-deformable boundary [7, 37]. This approach is implemented for the description of convection in magnetic fluids [37-40] and nanofluids with various properties [37-40] by numerical integration of the Oberbeck-Boussinesq equations. To eliminate the deficiency of exact solutions for the Oberbeck-Boussinesq 533 P r i v a l o v a V. V., P r o s v i r y a k o v E. Yu. system, an exact solution for the stationary creeping convective flow of a horizontal infinite layer of a viscous incompressible fluid is obtained in this paper. Tangential stresses are given on the upper thermally insulated permeable layer boundary, constant pressure and the Newton-Richman law of heat exchange being taken into account at the upper boundary. The no-slip condition is satisfied and the heat source is given at the lower boundary. The obtained exact solution is studied in detail for the diagnostics of the stratification of hydrodynamic fields describing the modification of the Couette-Hiemenz flow [15-17]. 1. Problem statement. The plane motion equations of a viscous incompressible fluid in an infinite layer with plane boundaries (Fig. 1), which describe the effect of temperature on the distribution of hydrodynamic fields, are considered in the Boussinesq approximation [1, 15-17] as (︁
×

Об авторах

Валентина Викторовна Привалова

Институт машиноведения УрО РАН

Email: valentprival@gmail.com
кандидат физико-математических наук; научный сотрудник; сектор нелинейной вихревой гидродинамики Россия, 620049, Екатеринбург, ул. Комсомольская, 34

Евгений Юрьевич Просвиряков

Институт машиноведения УрО РАН; Уральский федеральный университет им. первого Президента России Б. Н. Ельцина

Email: evgen_pros@mail.ru
доктор физико-математических наук; заведующий сектором; сектор нелинейной вихревой гидродинамики Россия, 620049, Екатеринбург, ул. Комсомольская, 34; Россия, 620002, Екатеринбург, ул. Мира, 19

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