Name: parabolic flow in circular tubes
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 23 s
Channel: JoVE

Considérons un écoulement stable et incompressible à travers un tube cylindrique droit de rayon constant R, connu sous le nom d’écoulement de Hagen-Poiseuille.
La géométrie cylindrique permet d’utiliser les coordonnées r, θ et z pour les mesures radiales, angulaires et axiales.
Le fluide s'écoulant parallèlement à l'axe du tube a une composante de vitesse purement axiale et ne varie qu'avec la distance radiale.
Ces conditions simplifient les équations de Navier-Stokes, révélant un profil de vitesse parabolique montrant que la vitesse est la plus élevée au centre et diminue vers les parois, atteignant zéro à la frontière.
Cela démontre l’écoulement laminaire, où les particules de fluide se déplacent en douceur en couches organisées.
L'intégration de ce profil sur la section transversale du tube donne la loi de Poiseuille, qui exprime le débit Q comme dépendant de la quatrième puissance du rayon du tube, de la viscosité du fluide et du gradient de pression.
La vitesse moyenne V, calculée à partir de Q, est la moitié de la vitesse maximale observée au centre du tube.
Cette distribution de vitesse confirme que l’écoulement est le plus élevé au centre et diminue paraboliquement jusqu’à zéro au niveau de la paroi.

Écoulement laminaire stable dans des tubes circulaires

Steady, Laminar Flow in Circular Tubes

L'écoulement de Hagen-Poiseuille décrit l'écoulement stable et incompressible d'un fluide visqueux à travers un tube cylindrique de rayon constant R. Ce profil d'écoulement est souvent appliqué pour comprendre le transport de fluides dans des canaux étroits, tels que des capillaires. Il sert d'exemple fondamental d'écoulement laminaire. Dans ce modèle, les coordonnées cylindriques (r,θ,z) sont utilisées pour décrire les dimensions radiales (r), angulaires (θ) et axiales (z) à l'intérieur du tube. Pour l'écoulement de Hagen-Poiseuille, le profil de vitesse est purement axial, ce qui signifie que le vecteur de vitesse pointe uniquement le long de l'axe z et varie uniquement avec la distance radiale r par rapport au centre du tube.

Les équations régissant le mouvement des fluides, les équations de Navier-Stokes, se simplifient dans ces conditions. Sans composantes de vitesse angulaire ou radiale, la vitesse axiale v_z(r) apparaît comme une fonction de r seul. La résolution des équations de Navier-Stokes réduites donne un profil de vitesse parabolique:

<img alt="Equation 1" src="/files/ftp_upload/18065/18065_Equation 1.svg" style="text-align: center; height: 27px;">


Ce profil démontre que la traînée visqueuse diminue la vitesse du fluide du maximum au centre à zéro au niveau de la paroi du tube. Cette vitesse nulle à la limite, connue sous le nom de condition de non-glissement, résulte du frottement entre le fluide et la paroi du tube. Par conséquent, l'écoulement est laminaire, les particules de fluide se déplaçant en couches parallèles sans mélange latéral.

L'intégration du profil de vitesse sur la section transversale du tube donne le débit volumétrique Q, connu sous le nom de loi de Poiseuille:

<img alt="Equation 2" src="/files/ftp_upload/18065/18065_Equation 2.svg" style="text-align: center; height: 27px;">


Où ΔP est la différence de pression sur la longueur L du tube et μ est la viscosité dynamique du fluide. Cette relation montre que Q est très sensible au rayon du tube, augmentant avec la quatrième puissance de R. Il s'ensuit que même une légère augmentation du rayon augmente considérablement le débit, un principe essentiel dans les applications de transport de fluides.

D'après la loi de Poiseuille, la vitesse moyenne V du fluide peut être exprimée comme suit :

<img alt="Equation 3" src="/files/ftp_upload/18065/18065_Equation 3.svg" style="text-align: center; height: 27px;">

Cette vitesse moyenne est précisément la moitié de la vitesse maximale au centre du tube, ce qui confirme la distribution de vitesse parabolique et démontre la nature prévisible et stratifiée de l'écoulement laminaire dans un tube cylindrique.

Considérons un écoulement stable et incompressible à travers un tube cylindrique droit de rayon constant R, connu sous le nom d’écoulement de Hagen-Poiseuille.
Le cylindrique ...

Education

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

parabolic flow in circular tubes

Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries.

Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves only along the z-axis and changes with distance r from the center of the tube.

Under these conditions, the Navier-Stokes equations become simpler. With no angular or radial velocity components, the axial velocity vz(r) depends only on r. Solving the reduced equations gives a parabolic velocity profile. The fluid moves fastest at the center and slows toward the wall because viscous drag reduces the speed.

At the tube wall, the velocity is zero. This is the no-slip condition, which comes from friction between the fluid and the wall. The result is laminar flow, where fluid particles travel in parallel layers with no lateral mixing.

Integrating the velocity profile over the tube cross section gives the volumetric flow rate Q. This is Poiseuille’s law. In this relation, ΔP is the pressure difference across tube length L, and μ is the fluid’s dynamic viscosity. The flow rate depends strongly on the tube radius and increases with the fourth power of R, so even a small increase in radius can greatly raise Q.

Poiseuille’s law also gives the mean velocity V of the fluid. The average velocity is exactly half of the maximum velocity at the tube center. This matches the parabolic profile and shows the predictable, layered nature of flow in a circular tube.

Parabolic Flow in Circular Tubes

euler&#39;s equations of motion

Pressure and Velocity in Inviscid Flow

Pressure and velocity in inviscid flow are linked by Euler's equations of motion. In fluid mechanics, viscosity creates shear stress, which is a force that acts along a surface. For low-viscosity fluids like water, these shear stresses are very small. That lets the fluid be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent. Only normal stresses remain, and these act perpendicular to a fluid element. Pressure is the negative of the normal stress. It stays the...

stream function

Stream Function for Incompressible Flow

The stream function helps describe two-dimensional incompressible flow. In this type of flow, the continuity equation keeps mass conserved. That means fluid entering a region must be balanced by fluid leaving it. For incompressible flow, the density stays constant. The continuity requirement then becomes a zero-divergence condition for the velocity field. In two dimensions, the horizontal and vertical velocity components are usually written as u and v. A stream function, written as ψ, is used...

irrotational flow

Irrotational Flow Near Boundaries

Irrotational flow is fluid motion in which particles do not spin around their own axes. It has zero vorticity, which means the curl of the velocity field is zero. That condition places limits on the velocity gradients in the flow. A simple example is uniform flow. In this case, the velocity u stays constant, while v and w are both zero. This satisfies the irrotational conditions and gives a basic model of non-rotating flow. When a solid object is added, irrotational flow usually remains only...

velocity potential

Velocity Potential in Pipe Flow

Velocity potential in pipe flow describes the motion of fluid in the central region of a long, straight pipe with a uniform cross-section. In this region, the flow is steady, incompressible, and irrotational. Irrotational means the fluid particles do not spin around their own axes. A scalar function called the velocity potential, written as φ, is used to describe this motion. The velocity field V is defined as the gradient of φ. Because the curl of a gradient is always zero, this form of V...

plane potential flows

Plane Potential Flow Patterns and Models

Plane potential flows describe fluid motion using simple flow patterns and models. They assume the fluid is irrotational and incompressible. These assumptions let the flow be written with a velocity potential function, φ, which gives the flow speed in a direction, and a stream function, ψ, which shows the flow path. Both are governed by Laplace's equation and are useful for estimating flow patterns, velocity distributions, and pressure fields around hydraulic structures. Uniform flow is the...

navier&#8211;stokes equations

Fluid Stress and Flow Equations

Fluid stress and flow equations describe how incompressible Newtonian fluids move under pressure, shear, and gravity. In these fluids, density stays constant, and stress changes linearly with the rate of deformation. The main stress types are normal stress and shear stress. Normal stress depends on pressure acting on the fluid and on deformation rate in specific directions. Shear stress acts along the surface of a fluid layer. It shows how neighboring layers slide past one another, so velocity...

steady, laminar flow between parallel plates

steady-laminar-flow-between-parallel-plates

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...

couette flow

Linear Shear Between Moving Plates

Linear shear between moving plates describes fluid motion in a simple channel with two parallel plates. One plate stays fixed, and the other moves at a constant speed. This setup helps show how viscosity and incompressible flow shape fluid motion under the Navier-Stokes equations. The model uses a steady, laminar, incompressible flow with no pressure gradient in the flow direction. The x-axis points along the moving direction of the upper plate, and the y-axis is perpendicular to both plates.

steady, laminar flow in circular tubes

Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries. Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves...

design example: flow of oil through circular pipes

design-example-flow-of-oil-through-circular-pipes

Pressure Drop in Oil Pipe Design

Pressure drop in oil pipe design depends on pipe size, oil properties, and the flow rate needed through a circular pipe. For steady, incompressible, laminar flow, Hagen-Poiseuille's law gives an exact solution from the Navier-Stokes equations. It is used to find the pressure difference needed to keep oil moving through a pipeline section. The pressure drop equation uses the oil's dynamic viscosity, the pipeline length, the desired volumetric flow rate, and the pipe's inner radius. This...

key terms and concepts

Explore how Hagen-Poiseuille flow creates a parabolic velocity profile in circular tubes, with no-slip at the wall and Poiseuille’s law for flow rate.

Parabolic Flow in Circular Tubes | JoVE K12

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.

civil engineering

Parabolic Flow in Circular Tubes

Description

Transcript

Tags