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N. V. Burmasheva, E. Yu. Prosviryakov

EXACT SOLUTIONS OF THE NAVIER–STOKES EQUATIONS FOR DESCRIBING AN ISOBARIC ONE-DIRECTIONAL VERTICAL VORTEX FLOW OF A FLUID

DOI: 10.17804/2410-9908.2021.2.030-051

The article proposes a family of exact solutions to the Navier–Stokes equations for describing isobaric inhomogeneous unidirectional fluid motions. Due to the incompressibility equation, the velocity of the inhomogeneous Couette flow depends on two coordinates and time. The expression for the velocity field has a wide functional arbitrariness. This exact solution is obtained by the method of separation of variables, and both algebraic operations (additivity and multiplicativity) are used to substantiate the importance of modifying the classical Couette flow. The article contains extensive bibliographic information that makes it possible to trace a change in the exact Couette solution for various areas of the hydrodynamics of a Newtonian incompressible fluid. The fluid flow is described by a polynomial depending on one variable (horizontal coordinate). The coefficients of the polynomial functionally depend on the second (vertical) coordinate and time; they are determined by a chain of the simplest homogeneous and inhomogeneous partial differential parabolic-type equations. The chain of equations is obtained by the method of undetermined coefficients after substituting the exact solution into the Navier–Stokes equation. An algorithm for integrating a system of ordinary differential equations for studying the steady motion of a viscous fluid is presented. In this case, all the functions defining velocity are polynomials. It is shown that the topology of the vorticity vector and shear stresses has a complex structure even without convective mixing (creeping flow).

Keywords: exact solution, Couette flow, Navier–Stokes equation, inhomogeneous unidirectional flow, method of separation of variables, shear stress

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Н. В. Бурмашева, Е. Ю. Просвиряков

ТОЧНЫЕ РЕШЕНИЯ УРАВНЕНИЙ НАВЬЕ–СТОКСА ДЛЯ ОПИСАНИЯ ИЗОБАРИЧЕСКОГО ОДНОНАПРАВЛЕННОГО ВЕРТИКАЛЬНО ЗАВИХРЕННОГО ТЕЧЕНИЯ ЖИДКОСТИ

В статье предложено семейство точных решений уравнений Навье–Стокса для описания изобарических неоднородных однонаправленных движений жидкости. Из-за уравнения несжимаемости скорость неоднородного течения Куэтта зависит от двух координат и времени. Приведенное выражение для поля скоростей обладает широким функциональным произволом. Данное точное решение получено методом разделения переменных, причем используются обе алгебраические операции (аддитивность и мультипликативность) для обоснования важности модификации классического течения Куэтта. В статье собрана значительная библиографическая информация, позволяющая проследить изменение точного решения Куэтта для различных областей гидродинамики ньютоновской несжимаемой жидкости. Течение жидкости представлено полиномом, зависящим от одной переменной (горизонтальной координаты). Коэффициенты полинома функционально зависят от второй (вертикальной) координаты и времени и определяются цепочкой простейших однородных и неоднородных уравнений в частных производных параболического типа. Цепочка уравнений получена методом неопределенных коэффициентов после подстановки точного решения в уравнение Навье–Стокса. Приведен алгоритм интегрирования системы обыкновенных дифференциальных уравнений для изучения установившегося движения вязкой жидкости. В этом случае все функции, определяющие скорость, являются полиномами, топология вектора завихренности и касательных напряжений даже без конвективного перемешивания (ползущее течение) имеет сложную структуру.

Ключевые слова: точное решение, течение Куэтта, уравнение Навье–Стокса, неоднородное однонаправленное течение, метод разделения переменных, касательное напряжение

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Библиографическая ссылка на статью

Burmasheva N. V., Prosviryakov E. Yu. Exact Solutions of the Navier–stokes Equations for Describing An Isobaric One-Directional Vertical Vortex Flow of a Fluid // Diagnostics, Resource and Mechanics of materials and structures. - 2021. - Iss. 2. - P. 30-51. -
DOI: 10.17804/2410-9908.2021.2.030-051. -
URL: http://dream-journal.org/issues/content/article_316.html
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