Name: fluid flow and area changes
Uploaded: 2026-01-30T03:08:26.000Z
Duration: 1 min 28 s
Channel: JoVE

L’équation de continuité dérive du principe de conservation de la masse, selon lequel la masse de fluide entrant et sortant d’un système reste constante.
Pour les fluides incompressibles, le produit de la surface d’écoulement et de la vitesse reste constant tout au long du trajet d’écoulement, s’ajustant aux changements de sections transversales.
À mesure que la section transversale diminue, la vitesse du fluide augmente pour maintenir un débit constant.
Cette relation est cruciale pour l’analyse de systèmes tels que les canalisations d’eau, la gestion des eaux pluviales et les canaux ouverts, garantissant la stabilité et l’efficacité dans diverses applications d’écoulement des fluides.
Dans les systèmes d’approvisionnement en eau, l’équation de continuité aide à maintenir une distribution stable en prédisant les changements de vitesse lorsque les tuyaux se rétrécissent, ce qui est essentiel pour éviter les chutes de pression.
Il aide à concevoir des bassins de rétention pour la gestion des eaux pluviales. Lorsque les eaux de ruissellement pénètrent dans le bassin pendant une tempête, le débit sortant doit correspondre au débit entrant, ajusté pour éviter les inondations en aval.
Pour les canaux ouverts comme les rivières ou les canaux, l’équation de continuité permet de comprendre comment les variations de largeur ou de profondeur affectent les niveaux d’eau et les vitesses d’écoulement. Cette compréhension est essentielle pour concevoir des systèmes efficaces de contrôle des inondations et de navigation.

Équation de continuité

L'équation de continuité affirme que le débit massique doit rester constant pour un écoulement stable d'un fluide incompressible dans un système confiné. Ce principe s'applique aux systèmes dans lesquels le fluide traverse des sections transversales variables, telles que des buses, des seringues et des tuyaux.

Le débit massique est exprimé comme suit:

<img alt="Equation 1" src="/files/ftp_upload/18038/18038_Equation_1.svg" style="text-align: center; height: 12px;">

Pour un écoulement incompressible, où la densité reste constante, l'équation de continuité se simplifie en:

<img alt="Equation 2" src="/files/ftp_upload/18038/18038_Equation_2.svg" style="text-align: center; height: 12px;">


Cette équation illustre que la vitesse augmente lorsque la section transversale diminue et vice versa, garantissant ainsi la conservation de la masse. Un exemple est l'écoulement à travers une seringue. Lorsque le piston comprime le fluide à travers une buse, l'équation de continuité montre que si la zone de sortie est la moitié de la zone d'entrée, la vitesse de sortie doit être le double de la vitesse d'entrée.

Un autre exemple est l'écoulement dans un réservoir à gravité. L'application de l'équation de Bernoulli et de l'équation de continuité à des points situés le long de l'écoulement permet de déterminer la vitesse de sortie et le débit volumétrique:

<img alt="Equation 3" src="/files/ftp_upload/18038/18038_Equation_3a.svg" style="text-align: center; height: 75px;">
<img alt="Equation 3" src="/files/ftp_upload/18038/18038_Equation_3b.svg" style="text-align: center; height: 20px;">

où h est la hauteur du fluide, D le diamètre du réservoir et d le diamètre de sortie. Cette relation montre que le débit converge vers une valeur stable lorsque le diamètre de sortie devient petit par rapport au diamètre du réservoir. Ces dérivations illustrent l'utilité de l'équation de continuité pour prédire le comportement des fluides sous diverses contraintes, garantissant ainsi la conservation de la masse dans les systèmes dynamiques.

L’équation de continuité dérive du principe de conservation de la masse, selon lequel la masse de fluide entrant et sortant d’un système reste constante.
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Education

Enseignement

JoVE Core

Civil Engineering

Génie civil

Unit 1

Unité 1

Fluid Dynamics

Dynamique des fluides

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Continuity Equation

fluid flow and area changes

Fluid flow and area changes are linked by the continuity equation. For a steady flow of an incompressible fluid in a confined system, the mass flow rate stays constant. This idea is useful in nozzles, syringes, and pipes where the cross-sectional area changes.

The mass flow rate is written as an equation that connects density, area, and velocity. For incompressible flow, the density stays the same, so the equation becomes simpler. When the area gets smaller, the fluid speed must increase. When the area gets larger, the speed must decrease. This keeps mass conserved as the fluid moves.

A syringe shows this relationship clearly. When the plunger pushes fluid through a nozzle, the outlet speed rises if the outlet area is smaller than the inlet area. If the outlet area is half the inlet area, the outlet velocity is twice the inlet velocity.

A gravity-driven tank gives another example. Here, Bernoulli's equation and the continuity equation can be used together at points along the flow. They help find the outlet velocity and the volumetric flow rate. In this setup, h is the fluid height, D is the tank diameter, and d is the outlet diameter. The flow rate approaches a stable value as the outlet diameter becomes small compared with the tank diameter.

Fluid Flow and Area Changes

bernoulli's equation for flow along a streamline

bernoulli-s-equation-for-flow-along-a-streamline

Pressure and Speed in Bernoulli's Equation

Bernoulli's equation links pressure, speed, and height in a moving fluid along a streamline. It is used for incompressible, inviscid fluids under steady flow. The idea comes from applying Newton's second law to a small fluid element. That element feels forces from pressure differences, gravity, and changes in velocity. The resulting equation shows how fluid energy is shared. In the equation, P is pressure, ρ is fluid density, v is velocity, g is the acceleration due to gravity, and h is...

bernoulli's equation for flow normal to a streamline

bernoulli-s-equation-for-flow-normal-to-a-streamline

Pressure Changes Across Curved Streamlines

Bernoulli's equation for flow normal to a streamline describes how pressure changes across curved streamlines. When a fluid follows a curved path, the outer side of the curve experiences centrifugal force, which pushes the fluid outward. The pressure is therefore higher on the inner side, near the center of curvature, and lower farther out. The size of this pressure difference depends on the fluid's velocity and the radius of curvature. In flows with nearly straight streamlines, the pressure...

bernoulli's equation: problem solving

Venturi Meter Flow Rate from Pressure

A Venturi meter measures fluid flow rate in a pipeline by using pressure changes and Bernoulli's equation. In a sewage system, it can be used to determine wastewater flow rate by reading the pressure difference across the meter. The narrowed throat makes the fluid speed up, and that change in speed is linked to pressure drop. The first step is to find the cross-sectional areas of the pipe and the Venturi throat. These areas are needed to interpret the pressure difference shown on the pressure...

static, stagnation, dynamic and total pressure

static-stagnation-dynamic-and-total-pressure

Pitot-Static Measurement in Bernoulli Flow

Static pressure, dynamic pressure, stagnation pressure, and total pressure are key ideas in fluid dynamics. They are often explained with Bernoulli's equation. In that equation, p is static pressure, 1/2ρV^2 is dynamic pressure, γz is hydrostatic pressure or potential energy pressure, and pT is total pressure. For incompressible flow, the total pressure stays constant along a streamline. Static pressure is the pressure a fluid exerts when it is at rest, or when flowing fluid is brought to rest...

free jet

Liquid Speed from Reservoir Outlets

Free jets show how liquid moves after leaving a reservoir through an opening and flowing into the air without resistance. The jet speed is found with Bernoulli’s principle, and it depends on the height of liquid above the opening. This relationship assumes atmospheric pressure at both the reservoir surface and the jet exit. The liquid column’s height controls the speed at the nozzle because the fluid’s potential energy changes into kinetic energy during flow. As the liquid leaves the...

continuity equation

Fluid flow and area changes are linked by the continuity equation. For a steady flow of an incompressible fluid in a confined system, the mass flow rate stays constant. This idea is useful in nozzles, syringes, and pipes where the cross-sectional area changes. The mass flow rate is written as an equation that connects density, area, and velocity. For incompressible flow, the density stays the same, so the equation becomes simpler. When the area gets smaller, the fluid speed must increase. When...

energy line and hydraulic gradient line

Fluid Head Profiles in Pipes and Channels

Fluid head profiles in pipes and channels show how pressure, speed, and height change in a flowing fluid. These profiles come from Bernoulli's equation for steady, incompressible, inviscid flow. In that equation, total head includes pressure head, velocity head, and elevation head. The energy line (EL) shows the total head available in the flow. It sits above the datum and includes all three energy parts. Under ideal conditions, the EL stays horizontal because the total energy does not change...

design example: designing water slide

Water Slide Speed and Bernoulli's Principle

Water slide speed is controlled by Bernoulli's principle and the height of the slide. Engineers use these ideas to keep the ride exciting while making sure the water stays safe for riders. As water moves downhill, gravity makes it speed up. The water at the bottom depends on the height it starts from. A taller slide gives the water more potential energy at the top. That energy changes into kinetic energy as the water descends, so the flow becomes faster. Bernoulli's principle helps describe...

key terms and concepts

Learn how the continuity equation links area, velocity, and mass conservation in steady fluid flow, with examples from syringes and gravity-driven tanks.

Fluid Flow and Area Changes | 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 Flow and Area Changes

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