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

Un écoulement laminaire stable entre deux plaques parallèles décrit le mouvement des fluides dans un canal étroit et confiné, souvent utilisé pour modéliser des systèmes fluides simples.
Pour l’écoulement entre deux plaques fixes, les équations de Navier-Stokes simplifient car la vitesse ne varie que perpendiculairement aux plaques, sans accélération dans le sens de l’écoulement.
Cela conduit à un profil de vitesse parabolique, où le fluide se déplace le plus rapidement au centre et ralentit jusqu’à zéro au niveau des plaques en raison de la friction, connue sous le nom de condition de non-dérapage.
Le débit entre les plaques dépend du gradient de pression, de la viscosité du fluide et de la séparation des plaques ; l’élargissement de l’écart ou l’augmentation du gradient de pression améliore l’écoulement.
La vitesse moyenne d’écoulement est également liée à la chute de pression, la vitesse de pointe au centre du canal étant 1,5 fois la vitesse moyenne.
Cette analyse ne reste valable que pour les nombres de Reynolds faibles, indiquant des conditions laminaires, et échoue pour les écoulements turbulents à nombre de Reynolds élevé.
Les applications de cette analyse comprennent les canaux d’irrigation, l’écoulement des eaux souterraines et les bassins de sédimentation dans les usines de traitement de l’eau, où des conditions laminaires existent.

Écoulement laminaire stable entre plaques parallèles

Steady, Laminar Flow Between Parallel Plates

Il est essentiel de comprendre l'écoulement laminaire stable entre plaques parallèles pour analyser et concevoir l'écoulement dans des canaux rectangulaires étroits, que l'on trouve couramment dans divers systèmes de transport et de drainage de l'eau. Les équations de Navier-Stokes régissent le mouvement des fluides et sont généralement difficiles à résoudre en raison de leur non-linéarité. Cependant, des simplifications sont possibles dans certains cas, comme l'écoulement laminaire stable entre plaques parallèles. Pour ce scénario, nous supposons un écoulement laminaire stable et incompressible.

En considérant deux plaques horizontales alignées le long de l'axe des x et espacées d'une distance de « 2h », avec l'axe des y perpendiculaire aux plaques, le mouvement du fluide est supposé être parallèle. Ici, l'équation de continuité n'indique aucun changement de vitesse le long de l'axe des x, et les équations de Navier-Stokes se réduisent comme suit:

<img alt="Equation 1" src="/files/ftp_upload/18063/18063_Equation_1.svg" style="text-align: center; height: 30px;">


Les équations dans les directions y et z ne montrent aucun gradient de pression le long de ces axes, ce qui indique que la pression varie de manière hydrostatique dans la direction y, ce qui permet à l'intégration de donner:

<img alt="Equation 2" src="/files/ftp_upload/18063/18063_Equation_2.svg" style="text-align: center; height: 12px;">


Pour des plaques parallèles avec un gradient de pression le long de l'axe des x, l'équation se simplifie:

<img alt="Equation 3" src="/files/ftp_upload/18063/18063_Equation_3.svg" style="text-align: center; height: 30px;">


En intégrant deux fois et en appliquant des conditions aux limites (u = 0 à y = +h et y = -h), nous obtenons:

<img alt="Equation 4" src="/files/ftp_upload/18063/18063_Equation_4.svg" style="text-align: center; height: 30px;">

Cette expression révèle que le profil de vitesse à travers l'espace entre les plaques est parabolique, avec une vitesse maximale à y = 0, à mi-chemin entre les plaques. Le débit volumétrique à travers une largeur unitaire des plaques est obtenu en intégrant le profil de vitesse à travers l'espace.

La compréhension de l'écoulement laminaire entre des plaques parallèles a des applications pratiques dans l'écoulement des eaux souterraines, le traitement des eaux usées et la lubrification. Par exemple, ce modèle peut approximer l'écoulement dans les couches de sol, où l'eau se déplace entre les particules du sol comme un écoulement entre des plaques. La connaissance du débit et de la distribution de la vitesse aide les ingénieurs à concevoir des systèmes de drainage efficaces et à prévoir le mouvement des eaux souterraines.

Un écoulement laminaire stable entre deux plaques parallèles décrit le mouvement des fluides dans un canal étroit et confiné, souvent utilisé pour modéliser des systèmes fluides simples.
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Enseignement

JoVE Core

Civil Engineering

Génie civil

Unit 1

Unité 1

Differential Analysis of Fluid Flow

Analyse différentielle de l'écoulement des fluides

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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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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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Laminar Flow Profile Between Parallel Plates

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