Name: pressure and velocity in inviscid flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 28 s
Channel: JoVE

Dans un fluide en mouvement, des contraintes de cisaillement se développent en raison de la viscosité. Pour l’eau, qui a une faible viscosité, ces effets sont négligeables, ce qui simplifie l’analyse du mouvement des fluides.
Lorsque les contraintes de cisaillement sont négligeables, le fluide est considéré comme invisqueux ou sans frottement. Dans les fluides visqueux, la contrainte normale est la même dans toutes les directions, ce qui signifie que la pression est indépendante de la direction de la contrainte.
La pression est le négatif de la contrainte normale, ce qui garantit que les contraintes de compression entraînent une valeur de pression positive.
Les équations générales du mouvement pour l'écoulement invisqueux, connues sous le nom d'équations d'Euler, décrivent la conservation de la quantité de mouvement. Ces équations représentent la deuxième loi de Newton appliquée à un élément fluide, reliant les changements de vitesse aux forces agissant sur le fluide.
Les équations d'Euler tiennent compte des gradients de pression, des forces corporelles comme la gravité et des accélérations, locales et convectives, qui entraînent des changements de vitesse du fluide.
Ils négligent les forces visqueuses, simplifiant l’analyse, mais restent difficiles à résoudre en raison des termes de vitesse non linéaires dans les équations aux dérivées partielles.
L'intégration des équations d'Euler le long d'une rationalisation dérive de l'équation de Bernoulli, fournissant des informations sur les variations de pression, de vitesse et d'élévation dans les écoulements invisqueux, ce qui en fait un outil clé en mécanique des fluides.

Équations d’euler du mouvement

Euler&#39;s Equations of Motion

En mécanique des fluides, les contraintes de cisaillement proviennent de la viscosité, qui représente la résistance interne d’un fluide à la déformation. Pour les fluides à faible viscosité, comme l’eau, ces contraintes sont minimales, ce qui simplifie l’analyse de l’écoulement en permettant de traiter le fluide comme non visqueux ou sans frottement. Dans un fluide non visqueux, les contraintes de cisaillement sont absentes, ne laissant que des contraintes normales, qui agissent perpendiculairement aux éléments fluides. Notamment, la pression (définie comme l’inverse de la contrainte normale) reste uniforme dans différentes directions et n’est pas affectée par l’orientation de la contrainte.

Le mouvement des fluides non visqueux est régi par les équations d’Euler, dérivées de la deuxième loi de Newton appliquée à la dynamique des fluides. Ces équations décrivent la conservation de l’impulsion en reliant le taux de changement de vitesse aux forces agissant sur un élément fluide. Pour un fluide non visqueux, les forces pertinentes comprennent les gradients de pression et les forces du corps, telles que la gravité. Les équations d'Euler expriment l'équilibre de ces forces, à l'exclusion des termes visqueux, comme suit:

<img alt="Equation 1" src="/files/ftp_upload/18057/18057_Equation_1.svg" style="text-align: center; height: 30px;">

où ρ est la densité du fluide, v est le vecteur vitesse, p est la pression et g​ représente l'accélération gravitationnelle.

Bien que les équations d'Euler ignorent la viscosité, elles restent complexes en raison de leurs termes non linéaires, qui tiennent compte de l'accélération convective des particules fluides. Lorsqu'elles sont intégrées le long d'une ligne de courant, ces équations donnent l'équation de Bernoulli, qui décrit une relation fondamentale en mécanique des fluides, reliant la pression, la vitesse et l'élévation pour les écoulements non visqueux. L'équation de Bernoulli est un outil puissant pour analyser la distribution d'énergie dans les écoulements fluides, prenant en charge les applications dans le transport des fluides, l'aérodynamique et l'hydrodynamique.

Équations d’Euler du mouvement

Dans un fluide en mouvement, des contraintes de cisaillement se développent en raison de la viscosité. Pour l’eau, qui a une faible viscosité, ces effets sont négligeables, simplifiant le mouvement des fluides...

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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

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 same in different directions and does not depend on how the stress is oriented.

Euler's equations come from Newton's second law applied to fluid motion. They describe momentum conservation by linking the change in velocity to the forces on a fluid element. For inviscid flow, those forces are pressure gradients and body forces such as gravity. The equations include density, velocity, pressure, and gravitational acceleration, but they leave out viscous terms.

Even without viscosity, Euler's equations are still nonlinear. That means they are complex because they include convective acceleration, which follows the motion of fluid particles. When the equations are integrated along a streamline, they lead to Bernoulli's equation. Bernoulli's equation connects pressure, velocity, and elevation in inviscid flows and is useful for fluid transport, aerodynamics, and hydrodynamics.

Pressure and Velocity in Inviscid Flow

euler&#39;s equations of motion

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...

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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...

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key terms and concepts

Learn how pressure, velocity, and gravity shape inviscid fluid motion through Euler's equations, and how they lead to Bernoulli's equation along a streamline.

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civil engineering

Pressure and Velocity in Inviscid Flow

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