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N. V. Burmasheva, E. Yu. Prosviryakov

EXACT SOLUTIONS TO THE NAVIER–STOKES EQUATIONS FOR UNIDIRECTIONAL FLOWS OF MICROPOLAR FLUIDS IN A MASS FORCE FIELD

DOI: 10.17804/2410-9908.2024.3.041-063

The paper presents a family of exact solutions to the Navier-Stokes equation system used to describe inhomogeneous unidirectional flows of a viscous fluid taking into account couple stresses. Despite the presence of only one non-zero component of the velocity vector, this component depends on time and two spatial coordinates. In view of the incompressibility equation, which is a special case of the mass conservation law, there is no dependence on the third spatial coordinate. The resulting redefined system of equations is considered in a non-stationary formulation. The construction of a family of exact solutions for the resulting redefined equation system begins with the analysis of the homogeneous Couette-type solution as the simplest in this class. Further, the structure of the solution gradually becomes more complicated, i.e. the profile of the only non-zero component of the velocity vector is represented as a polynomial depending on one variable (horizontal coordinate). The polynomial coefficients functionally depend on the second (vertical) coordinate and time. It is shown that, due to the strong nonlinearity and heterogeneity of the equation under study, the sum of its individual solutions is not a solution. It is also shown that, in the linearly independent basis of the power functions of the horizontal coordinate, which determine the above-mentioned polynomial, the equation in question decomposes into a chain of the simplest homogeneous and inhomogeneous parabolic partial differential equations. These equations are integrated sequentially, the order of integration being described separately. The results reported in this study extend the family of previously presented exact solutions to describing unidirectional unsteady flows.

Keywords: exact solution, Navier–Stokes equation, inhomogeneous unidirectional flow, couple stresses, micropolar fluids

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Н. В. Бурмашева, Е. Ю. Просвиряков

ТОЧНЫЕ РЕШЕНИЯ УРАВНЕНИЙ НАВЬЕ – СТОКСА ОДНОНАПРАВЛЕННЫХ ТЕЧЕНИЙ МИКРОПОЛЯРНЫХ ЖИДКОСТЕЙ В ПОЛЕ МАССОВЫХ СИЛ

В статье представлено семейство точных решений системы уравнений Навье – Стокса, используемой для описания неоднородных однонаправленных течений вязкой жидкости с учетом моментных напряжений. Несмотря на наличие только одной ненулевой компоненты вектора скорости, эта самая компонента зависит от времени и двух пространственных координат. Зависимость от третьей пространственной координаты отсутствует ввиду уравнения несжимаемости, являющегося частным случаем закона сохранения массы. Получающаяся переопределенная система уравнений рассматривается в нестационарной постановке. Построение семейства точных решений полученной переопределенной системы начинается с анализа однородного решения типа Куэтта как наиболее простого в этом классе. Далее структура решения постепенно усложняется: профиль единственной ненулевой компоненты вектора скорости представлен в виде полинома, зависящего от одной переменной (горизонтальной координаты). Коэффициенты полинома функционально зависят от второй (вертикальной) координаты и времени. Показано, что, ввиду сильной нелинейности и неоднородности исследуемого уравнения, сумма отдельных его решений не является решением. Также показано, что в линейно независимом базисе степенных функций горизонтальной координаты, определяющих вышеупомянутый полином, рассматриваемое уравнение распадается на цепочку простейших однородных и неоднородных уравнений в частных производных параболического типа. Данные уравнения интегрируются последовательно, порядок интегрирования отдельно описан. Результаты, изложенные в данной статье, обобщают ранее представленное авторами семейство точных решений для описания однонаправленных нестационарных течений.

Ключевые слова: точное решение, уравнение Навье – Стокса, неоднородное однонаправленное течение, моментные напряжения, микрополярные жидкости

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Burmasheva N. V., Prosviryakov E. Yu. Exact Solutions to the Navier–stokes Equations for Unidirectional Flows of Micropolar Fluids in a Mass Force Field // Diagnostics, Resource and Mechanics of materials and structures. - 2024. - Iss. 3. - P. 41-63. -
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