Name: laminar flow profile between parallel plates
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 17 s
Channel: JoVE

Устойчивый ламинарный поток между двумя параллельными пластинами описывает движение жидкости в узком ограниченном канале, часто используемом для моделирования простых жидкостных и газовых систем.
Для потока между двумя неподвижными пластинами уравнения Навье-Стокса упрощаются, так как скорость изменяется только перпендикулярно пластинам, без ускорения вдоль направления потока.
Это приводит к параболическому профилю скоростей, при котором жидкость движется быстрее всего в центре и замедляется до нуля на пластинах из-за трения, известного как условие отсутствия скольжения.
Скорость потока между пластинами зависит от градиента давления, вязкости жидкости и расстояния между пластинами; расширение зазора или увеличение градиента давления усиливает поток.
Средняя скорость потока также связана с перепадом давления, при этом пиковая скорость в центре канала в 1,5 раза превышает среднюю скорость.
Этот анализ остается справедливым только для низких чисел Рейнольдса, указывающих на ламинарные условия, и не работает для турбулентных потоков с высоким числом Рейнольдса.
Применение этого анализа включает в себя оросительные каналы, поток грунтовых вод и отстойники на водоочистных сооружениях, где существуют ламинарные условия.

Устойчивый ламинарный поток между параллельными пластинами

Steady, Laminar Flow Between Parallel Plates

Понимание поведения устойчивого ламинарного потока между параллельными пластинами необходимо для анализа и проектирования потока в узких прямоугольных каналах, которые обычно встречаются в различных системах транспортировки воды и дренажа. Уравнения Навье-Стокса управляют движением жидкости и, как правило, сложны для решения из-за их нелинейности. Однако, в некоторых случаях возможны упрощения, и к ним относится устойчивый ламинарный поток между параллельными пластинами. Для этого сценария мы предполагаем устойчивый несжимаемый ламинарный поток.

 Мы рассматриваем две горизонтальные пластины, выровненные вдоль оси x и расположенные на расстоянии «2h» друг от друга, с осью y, перпендикулярной пластинам, и движение жидкости предполагается параллельным. Здесь уравнение непрерывности указывает на отсутствие изменения скорости вдоль оси x, а уравнения Навье-Стокса сокращаются следующим образом:

<img alt="Equation 1" src="/files/ftp_upload/18063/18063_Equation_1.svg" style="text-align: center; height: 30px;">


 Уравнения в направлениях y и z не показывают градиента давления вдоль этих осей, указывая на то, что давление изменяется гидростатически в направлении y, что позволяет интегрировать:

<img alt="Equation 2" src="/files/ftp_upload/18063/18063_Equation_2.svg" style="text-align: center; height: 12px;">


 Для параллельных пластин с градиентом давления вдоль оси x уравнение упрощается:

<img alt="Equation 3" src="/files/ftp_upload/18063/18063_Equation_3.svg" style="text-align: center; height: 30px;">


 Интегрируя дважды и применяя граничные условия (u = 0 при y = +h и y = -h), получаем:

<img alt="Equation 4" src="/files/ftp_upload/18063/18063_Equation_4.svg" style="text-align: center; height: 30px;">

 Это выражение показывает, что профиль скорости через зазор между пластинами является параболическим, с максимальной скоростью при y = 0, посередине между пластинами. Объемный расход через единицу ширины пластин получается путем интегрирования профиля скорости по зазору.

 Понимание ламинарного потока между параллельными пластинами имеет практическое применение в таких областях, как потоки грунтовых вод, очистка сточных вод и смазка. Например, эта модель может аппроксимировать поток в слоях почвы, где вода движется между частицами почвы, как поток между пластинами. Знание расхода и распределения скорости помогает инженерам проектировать эффективные дренажные системы и прогнозировать движение грунтовых вод.

Устойчивый ламинарный поток между двумя параллельными пластинами описывает движение жидкости в узком ограниченном канале, часто используемом для моделирования простых жидкостных и газовых систем.
Для...

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JoVE Core

Civil Engineering

Unit 1

Раздел 1

Differential Analysis of Fluid Flow

Дифференциальный анализ потока жидкости

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

laminar flow profile between parallel plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings.

The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem becomes simpler because the flow is steady, incompressible, and laminar. Two horizontal plates are placed along the x-axis, and the gap between them is 2h. The y-axis is perpendicular to the plates, and the fluid moves parallel to the plates.

With these assumptions, the continuity equation shows that the velocity does not change along the x-axis. The Navier-Stokes equations also simplify in the y and z directions. They show no pressure gradient along those axes, so pressure changes hydrostatically in the y-direction. After integration, the x-direction equation includes the pressure gradient along the plates.

Solving that equation requires integrating twice and applying the no-slip boundary conditions, where u = 0 at y = +h and y = -h. The result is a parabolic velocity profile across the gap. The fluid moves fastest at y = 0, the midpoint between the plates. The volumetric flow rate for a unit width of the plates is then found by integrating the velocity profile across the space.

This flow model has practical uses in groundwater flow, wastewater treatment, and lubrication. It can also represent water moving through soil layers, where the flow resembles movement between plates. Engineers use the velocity distribution and flow rate to help design drainage systems and predict groundwater movement.

Laminar Flow Profile Between Parallel Plates

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steady, laminar flow between parallel plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings. The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem...

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key terms and concepts

Learn how steady laminar flow between parallel plates leads to a parabolic velocity profile, pressure-gradient behavior, and flow-rate calculations.

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JoVE Core Civil Engineering provides a foundational exploration of essential construction materials in civil engineering, focusing on bricks, stones, steel, wood, and concrete through easy-to-understand animated videos. This collection covers the properties, manufacturing processes, and practical applications of these materials. Additionally, the accompanied science-in-action videos demonstrate the research experiments performed in laboratories.

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Laminar Flow Profile Between Parallel Plates

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