Name: fluid stress and flow equations
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 28 s
Channel: JoVE

Dans le mouvement des fluides dans des conditions visqueuses, la contrainte de cisaillement est directement proportionnelle à la déformation du fluide. Dans les fluides newtoniens incompressibles, cette contrainte varie linéairement avec le taux de déformation.
Les contraintes normales dépendent de la pression et des taux de déformation dans des directions spécifiques, définissant le comportement d’écoulement du fluide sous différentes pressions.
Les contraintes de cisaillement agissent de manière tangentielle, décrivant comment les différentes couches de fluide glissent les unes sur les autres, reliant les contraintes aux changements de vitesse entre les couches.
La substitution de ces relations de contrainte dans des équations différentielles de mouvements forme les équations de Navier-Stokes, qui équilibrent des forces telles que l’inertie, la pression et la gravité dans un fluide visqueux.
Le terme d’inertie capture l’accélération du fluide, montrant comment un fluide en mouvement résiste aux changements soudains de vitesse ou de direction.
Les gradients de pression entraînent le mouvement du fluide des zones de haute pression vers les zones de basse pression, tandis que les termes visqueux représentent le frottement interne dans le fluide.
Chaque équation directionnelle capture les forces internes, telles que la viscosité, et les forces externes, telles que la gravité, prédisant la réponse du fluide dans diverses conditions.
Les équations de Navier-Stokes peuvent être simplifiées pour les scénarios d’écoulement stationnaire ou laminaire, ce qui permet une analyse plus simple dans des scénarios contrôlés tels que les écoulements de couche limite et de Couette.

Équations de navier-stokes

Navier&#8211;Stokes Equations

Pour les fluides newtoniens incompressibles, où la densité reste constante, les contraintes présentent une relation linéaire avec le taux de déformation, défini par les contraintes normales et de cisaillement. Les contraintes normales dépendent de la pression exercée sur le fluide et du taux de déformation dans des directions spécifiques, ce qui détermine la manière dont le fluide s'écoule sous des pressions variables. Les contraintes de cisaillement, en revanche, agissent de manière tangentielle sur les couches de fluide. Elles expliquent comment les couches de fluide adjacentes glissent les unes par rapport aux autres, reliant la contrainte tangentielle aux différences de vitesse entre les couches. Les contraintes normales et de cisaillement définissent les forces internes au sein d'un fluide qui influencent son mouvement.

Les équations de Navier-Stokes sont dérivées en incorporant ces relations de contrainte dans des équations différentielles de mouvement.

<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;">

Ces équations équilibrent les forces d'inertie, de pression et de gravitation dans un fluide visqueux et sont fondamentales pour prédire le comportement du fluide dans diverses conditions. Le terme d'inertie dans les équations de Navier-Stokes représente l'accélération du fluide, représentant la résistance du fluide aux changements de vitesse. Ce terme est essentiel pour capturer la conservation de l'impulsion d'un fluide en mouvement, car il démontre comment les fluides résistent aux changements soudains de vitesse ou de direction.

Le terme de gradient de pression dans les équations de Navier-Stokes entraîne le mouvement du fluide en créant une force nette des zones de haute pression vers les zones de basse pression. Pendant ce temps, les termes visqueux représentent la friction interne résultant des interactions moléculaires au sein du fluide. En fonction des gradients de vitesse présents, ces forces visqueuses agissent pour ralentir le mouvement du fluide.

Chaque équation directionnelle de Navier-Stokes capture les forces internes telles que la contrainte visqueuse et les forces externes telles que la gravité. Ces équations permettent de prédire le mouvement des fluides dans diverses conditions, des écoulements simples aux écoulements complexes. Pour les scénarios d'écoulement stable ou laminaire, où l'écoulement est régulier et ordonné, les équations deviennent plus faciles à gérer, permettant une analyse plus simple. Cette simplification est utile dans des scénarios contrôlés comme l'écoulement de couche limite, où l'écoulement près d'une surface a une couche structurée, et dans l'écoulement de Couette, décrivant un fluide entre deux surfaces parallèles se déplaçant à des vitesses différentes.

Équations de Navier-Stokes

Dans le mouvement des fluides dans des conditions visqueuses, la contrainte de cisaillement est directement proportionnelle à la déformation du fluide. Dans les fluides newtoniens incompressibles, cette contrainte ...

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Civil Engineering

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Unit 1

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Differential Analysis of Fluid Flow

Analyse différentielle de l'écoulement des fluides

KEY TERMS AND CONCEPTS

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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 differences between layers matter.

These normal and shear stresses create the internal forces that control fluid motion. The Navier-Stokes equations are formed by putting these stress relations into differential equations of motion. They balance inertial, pressure, and gravitational forces in a viscous fluid and help predict how the fluid will behave.

The inertia term describes fluid acceleration and its resistance to changes in velocity. It is important for momentum conservation in moving fluids. The pressure gradient term pushes fluid from regions of high pressure to regions of low pressure, while viscous terms come from internal friction between molecules and slow the flow when velocity changes across the fluid.

Each directional Navier-Stokes equation includes both internal forces, such as viscous stress, and external forces, such as gravity. This makes the equations useful for many flow conditions, from simple to complex. For steady or laminar flow, where motion is smooth and orderly, the equations are easier to work with.

That simplification is helpful in boundary layer flow near a surface, where the fluid forms a structured layer. It is also useful for Couette flow, which describes fluid between two parallel surfaces moving at different velocities. These cases show how the same equations can describe controlled fluid motion in practical settings.

Fluid Stress and Flow Equations

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...

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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...

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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 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

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

Learn how normal stress, shear stress, and the Navier-Stokes equations explain fluid motion, pressure forces, viscosity, boundary layers, and Couette flow.

Fluid Stress and Flow Equations | 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

Fluid Stress and Flow Equations

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