An Exact Solution with Inhomogeneous Boundary Conditions for a Steady Non-Uniform Couette Flow between Permeable Plates

K. V. Gubareva; E. Yu. Prosviryakov; A. V. Eremin

doi:10.17804/2410-9908.2025.5.066-086

K. V. Gubareva, E. Yu. Prosviryakov, A. V. Eremin

AN EXACT SOLUTION WITH INHOMOGENEOUS BOUNDARY CONDITIONS FOR A STEADY NON-UNIFORM COUETTE FLOW BETWEEN PERMEABLE PLATES

DOI: 10.17804/2410-9908.2025.5.066-086

The paper offers an analytical solution for a non-uniform isobaric disturbed unidirectional stationary flow of a viscous incompressible fluid in an infinitely long horizontal layer. The boundaries of the infinite fluid layer are represented by permeable plates. A generalized Couette flow is discussed, where the longitudinal (horizontal) velocity is the linear form of the other longitudinal (horizontal) coordinate. The coefficients of the linear form depend on the transverse (vertical) coordinate. The effect of the normal component of the velocity vector (uniform permeability) via the inertial terms of the Navier–Stokes equation is taken into account. The velocity, vorticity, and shear stress fields are computed from the exact solution to the Navier–Stokes equations. This exact solution belongs to the Lin–Sidorov–Aristov family. The hydrodynamic fields are parametrically visualized in the Matlab software. It is shown that permeability results in the nonlinear deformation of the velocity profile and that the inhomogeneity of the boundary conditions causes the spatial variability of the hydrodynamic fields. The solution can be used to verify numerical methods and to model flows in channels with complex boundary conditions.

Keywords: Couette flow, permeable boundaries, inhomogeneous boundary conditions, analytical solution, viscous incompressible fluid, vorticity, shear stresses

References:

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К. В. Губарева, Е. Ю. Просвиряков, А. В. Еремин

ТОЧНОЕ РЕШЕНИЕ С НЕОДНОРОДНЫМИ ГРАНИЧНЫМИ УСЛОВИЯМИ ДЛЯ УСТАНОВИВШЕГОСЯ НЕОДНОРОДНОГО ТЕЧЕНИЯ КУЭТТА МЕЖДУ ПРОНИЦАЕМЫМИ ПЛАСТИНАМИ

В работе получено аналитическое решение для неоднородного изобарического возмущенного однонаправленного стационарного течения вязкой несжимаемой жидкости в бесконечно протяженном горизонтальном слое. Границами бесконечного слоя жидкости являются проницаемые пластины. Рассмотрено обобщенное течение Куэтта, в котором продольная (горизонтальная) скорость является линейной формой от другой продольной (горизонтальной) координаты. Коэффициенты линейной формы зависят от поперечной (вертикальной) координаты. Учтено влияние нормальной компоненты вектора скорости (равномерной проницаемости) через инерционные члены уравнения Навье – Стокса. На основе точного решения уравнений Навье – Стокса вычислены поля скорости, завихренности, касательных напряжений. Данное точное решение принадлежит семейству Линя – Сидорова – Аристова. Проведена параметрическая визуализация гидродинамических полей в компьютерной программе Matlab. Показано, что проницаемость приводит к нелинейной деформации профиля скорости, а неоднородность граничных условий вызывает пространственную изменчивость гидродинамических полей. Решение может использоваться для верификации численных методов и моделирования течений в каналах со сложными граничными условиями.

Ключевые слова: течение Куэтта, проницаемые границы, неоднородные граничные условия, аналитическое решение, вязкая несжимаемая жидкость, завихренность, касательные напряжения

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Bogoyavlenskij O. The new effect of oscillations of the total kinematic momentum vector of viscous fluid // Physics of Fluids. – 2022. – Vol. 34 (12). – P. 123104. – DOI: 10.1063/5.0127990.
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Библиографическая ссылка на статью

Gubareva K. V., Prosviryakov E. Yu., Eremin A. V. An Exact Solution with Inhomogeneous Boundary Conditions for a Steady Non-Uniform Couette Flow between Permeable Plates // Diagnostics, Resource and Mechanics of materials and structures. - 2025. - Iss. 5. - P. 66-86. -
DOI: 10.17804/2410-9908.2025.5.066-086. -
URL: http://dream-journal.org/issues/content/article_522.html
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