Name: velocity potential in pipe flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

Considérez que l’eau s’écoule régulièrement dans un long tuyau droit de section transversale uniforme.
Dans les régions centrales du tuyau, où l’écoulement est irrotationnel, un potentiel de vitesse définit l’écoulement et la vitesse est exprimée comme le gradient de ce potentiel.
La substitution des composantes de vitesse permet de vérifier directement que l’écoulement satisfait à cette condition d’irrotation.
Le potentiel de vitesse s’applique aux écoulements tridimensionnels, comme cet écoulement de tuyau, tandis que la fonction de flux est spécifique aux écoulements bidimensionnels, résultant de la conservation de la masse.
Dans un écoulement incompressible, l’équation de continuité exige que la divergence du vecteur vitesse soit nulle.
En substituant le potentiel de vitesse, cette condition devient l'équation de Laplace, une équation différentielle clé régissant les écoulements visqueux, incompressibles et irrotationnels.
En coordonnées cylindriques, l'équation de Laplace s'ajuste pour les variations radiales, angulaires et axiales. La résolution de cette équation avec des conditions aux limites au niveau des parois du tuyau définit le potentiel de vitesse de l’écoulement.
À partir de là, le champ de vitesse complet est déterminé en prenant des dérivées spatiales, révélant le modèle d’écoulement à l’intérieur du tuyau.

potentiel de vitesse

Velocity Potential

Dans un écoulement stable et incompressible à travers un long tuyau droit de section uniforme, l'écoulement dans la région centrale (loin des parois du tuyau) est irrotationnel. Cette nature irrotationnelle signifie que les particules de fluide ne tournent pas autour de leurs axes, et une fonction scalaire appelée potentiel de vitesse, représentée par ϕ, peut être utilisée pour décrire leur mouvement. Dans les écoulements irrotationnels, le champ de vitesse V est défini comme le gradient du potentiel de vitesse:

<img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;">

Étant donné que la courbe d'un gradient est toujours nulle, la définition de la vitesse en termes de ϕ garantit que l'écoulement n'a pas de vorticité, ce qui le rend irrotationnel.

Pour les écoulements incompressibles, l'équation de continuité stipule que la divergence du champ de vitesse doit être nulle:

<img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;">


En remplaçant V dans cette équation, l'équation devient:

<img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;">


C'est ce qu'on appelle l'équation de Laplace. Dans les régions où l'écoulement est irrotationnel et incompressible, le potentiel de vitesse ϕ doit satisfaire l'équation de Laplace.

Dans le cas d'un écoulement dans un tuyau, les coordonnées cylindriques (r, θ, z) sont un choix naturel pour décrire l'écoulement, où r est la distance radiale par rapport à l'axe central du tuyau, θ est la position angulaire et z est la distance axiale le long du tuyau. L'équation de Laplace en coordonnées cylindriques, tenant compte des variations radiales, angulaires et axiales, s'exprime comme suit:

<img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;">


Trouver la forme exacte de ϕ dans le tuyau nécessite des conditions aux limites. Par exemple, au niveau des parois du tuyau, la condition de non-glissement impliquerait une vitesse tangentielle nulle, tandis que dans la région irrotationnelle centrale, la fonction de potentiel suivrait la symétrie du tuyau. La résolution de l'équation de Laplace avec ces conditions définit le potentiel de vitesse à travers le champ d'écoulement.

Une fois ϕ déterminé, les composantes de vitesse dans les coordonnées cylindriques peuvent être dérivées en prenant des dérivées partielles de ϕ. Ces composantes sont:

Vitesse radiale:

<img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;">


Vitesse angulaire

<img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;">


Vitesse axiale

<img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;">

Ces expressions pour les composantes de vitesse confirment que l'écoulement satisfait la condition irrotationnelle, car elles proviennent d'un gradient.

Potentiel de vitesse

Considérez que l’eau s’écoule régulièrement dans un long tuyau droit de section transversale uniforme.
Dans les régions centrales du tuyau, où l’écoulement est irrotationnel, un ...

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Differential Analysis of Fluid Flow

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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 gives the flow no vorticity and keeps it irrotational.

For an incompressible fluid, the continuity equation requires the divergence of the velocity field to be zero. When V is written in terms of φ, this condition becomes Laplace's equation. In regions where the flow is both incompressible and irrotational, the velocity potential must satisfy this equation.

Pipe flow is often described with cylindrical coordinates: r, θ, and z. Here, r is the radial distance from the pipe centreline, θ is the angular position, and z is the distance along the pipe. In these coordinates, Laplace's equation includes radial, angular, and axial variation.

To find the exact form of φ in the pipe, boundary conditions are needed. At the pipe walls, the no-slip condition means the tangential velocity is zero. In the central irrotational region, the potential function must also match the symmetry of the pipe. Solving Laplace's equation with these conditions gives the velocity potential across the flow field.

Once φ is known, the velocity components can be found by taking partial derivatives. These components are the radial velocity, the angular velocity, and the axial velocity. Together, they show how the flow field comes from a gradient and therefore satisfies the irrotational condition.

Velocity Potential in Pipe Flow

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Velocity Potential in Pipe Flow

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