A new exact solution is found for the plane convection of a viscous incompressible fluid describable by the Oberbeck–Boussinesq equation system in an infinite thin layer. The solution of the boundary-value problem is obtained for the fluid flow arising in the case of inhomogeneous velocity distribution and a linear heat source at the upper boundary of an infinite layer of a viscous incompressible fluid. A creeping convective flow is studied by the generalization of the isothermal class of the Hiemenz exact solutions. The temperature and pressure fields are linear with respect to the horizontal coordinate in this class of solutions. The analysis of polynomial solutions describing natural fluid convection in the Stokes approximation is presented. The paper shows the existence of points where the velocity field vanishes inside the fluid layer. These points are termed stagnation points and indicate the presence of counterflows in the fluid. Similar investigations are carried out for the temperature and pressure fields. The isotherms and isobars are shown to be always locally parabolic or hyperbolic, i.e. to have an infinitely distant point, due to the structure of the discussed exact solution.
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Найдено новое точное решение плоской конвекции вязкой несжимаемой жидкости, описываемой системой уравнений Обербека–Буссинеска в бесконечном тонком слое.
Решение краевой задачи получено для течения жидкости, возникающее при неоднородном распределении скоростей и линейного источника тепла на верхней границе бесконечного слоя вязкой несжимаемой жидкости. Исследование ползущего конвективного течения осуществляется посредством обобщения изотермического класса точных решений Хименца, в котором поля температуры и давления линейны относительно горизонтальной координаты. Приведен анализ полиномиальных решений, описывающих естественную конвекцию жидкости в приближении Стокса. Показано существование точек, в которых поле скоростей обращается в нуль внутри слоя жидкости, что определяет существование застойных точек и противотечений в жидкости. Аналогичные исследования проведены для полей температуры и давления. Показано, что изотермы и изобары из-за структуры рассматриваемого точного решения всегда будут локально параболическими или гиперболическими (иметь бесконечно удаленную точку).
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DOI: 10.1134/S0015462811010078.
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