Inhomogeneous Couette Poiseuille Flow of a Viscous Incompressible Fluid in An Infinite Horizontal Layer with Permeable Boundaries

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

doi:10.17804/2410-9908.2025.5.006-028

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

INHOMOGENEOUS COUETTE POISEUILLE FLOW OF A VISCOUS INCOMPRESSIBLE FLUID IN AN INFINITE HORIZONTAL LAYER WITH PERMEABLE BOUNDARIES

DOI: 10.17804/2410-9908.2025.5.006-028

This paper presents an exact analytical solution to the boundary value problem for a steady flow of a viscous incompressible fluid in a plane channel formed by two parallel permeable plates. Particular emphasis is placed on the combined effect of three physical key factors, namely a constant longitudinal pressure gradient, uniform normal fluid flow through permeable boundaries, and an inhomogeneous boundary condition on the upper wall, where streamwise velocity is specified as a linear function of the transverse coordinate. The classical full no-slip condition is implemented on the lower boundary. The system of steady-state Navier–Stokes equations is reduced to two independent 2nd-order ordinary differential equations, which are solved by the method of superposition. The obtained closed-form analytical formulas for the velocity profiles allow a detailed analysis of the effect of the pressure gradient, penetration rate, and the parameters of the inhomogeneous boundary condition on the flow structure. The solution strictly satisfies all the specified boundary conditions and the continuity equation. It extends the classical Poiseuille and Couette problems to the case of permeable boundaries and non-uniform slip, thus providing a valuable theoretical tool for modeling processes in microfluidics, filtration, and lubrication.

Keywords: Couette–Poiseuille flow, viscous incompressible fluid, permeable boundaries, pressure gradient, inhomogeneous boundary condition, exact analytical solution, Navier–Stokes equations, steady flow, plane channel

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

НЕОДНОРОДНОЕ ТЕЧЕНИЕ КУЭТТА – ПУАЗЕЙЛЯ ВЯЗКОЙ НЕСЖИМАЕМОЙ ЖИДКОСТИ В БЕСКОНЕЧНОМ ГОРИЗОНТАЛЬНОМ СЛОЕ С ПРОНИЦАЕМЫМИ ГРАНИЦАМИ

В работе представлено точное аналитическое решение краевой задачи для установившегося течения вязкой несжимаемой жидкости в плоском канале, образованном двумя параллельными проницаемыми пластинами. Особое внимание уделено совместному влиянию трех ключевых физических факторов: постоянного градиента давления в продольном направлении канала, равномерного нормального потока жидкости через проницаемые границы и неоднородного граничного условия на верхней стенке, где продольная компонента скорости задана в виде линейной функции поперечной координаты. На нижней границе реализуется классическое условие полного прилипания. Система уравнений Навье – Стокса для стационарного режима сводится к двум независимым обыкновенным дифференциальным уравнениям второго порядка, которые решаются методом суперпозиции. Полученные замкнутые аналитические формулы для профилей скорости позволяют провести детальный анализ влияния градиента давления, скорости проницания и параметров неоднородного граничного условия на структуру течения. Решение строго удовлетворяет всем заданным граничным условиям и уравнению неразрывности. Решение обобщает классические задачи Пуазейля и Куэтта на случай проницаемых границ и неоднородного скольжения, предоставляя ценный теоретический инструмент для моделирования процессов в микрофлюидике, фильтрации и смазке.

Ключевые слова: течение Куэтта – Пуазейля, вязкая несжимаемая жидкость, проницаемые границы, градиент давления, неоднородное граничное условие, точное аналитическое решение, уравнения Навье – Стокса, установившееся течение, плоский канал

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Библиографическая ссылка на статью

Gubareva K. V., Prosviryakov E. Yu., Eremin A. V. Inhomogeneous Couette Poiseuille Flow of a Viscous Incompressible Fluid in An Infinite Horizontal Layer with Permeable Boundaries // Diagnostics, Resource and Mechanics of materials and structures. - 2025. - Iss. 5. - P. 6-28. -
DOI: 10.17804/2410-9908.2025.5.006-028. -
URL: http://dream-journal.org/issues/content/article_523.html
(accessed: 18.04.2026).