Name: pressure and speed in bernoulli's equation
Uploaded: 2025-02-13T15:43:14.000Z
Duration: 1 min 30 s
Channel: JoVE

L'équation de Bernoulli décrit le comportement des fluides en mouvement en reliant la pression, la vitesse et l'élévation le long d'une ligne de courant.
Il repose sur le principe de la conservation de l’énergie, en supposant un écoulement régulier, des fluides incompressibles et aucune viscosité, ce qui signifie qu’aucune force de frottement n’est présente.
Le terme de pression représente la force que le fluide exerce en un point spécifique.
Le terme de vitesse représente l'énergie cinétique du fluide, indiquant que la pression diminue à mesure que la vitesse du fluide augmente.
Le terme d'élévation reflète l'énergie potentielle du fluide en fonction de sa position verticale par rapport à un point de référence.
Ce principe s’applique à des situations réelles telles que les systèmes de distribution d’eau, les déversoirs de barrage et les compteurs Venturi.
Dans les systèmes de distribution d'eau, l'équation de Bernoulli prédit comment la pression diminue lorsque l'eau s'écoule dans des tuyaux de différents diamètres, assurant ainsi un approvisionnement efficace sur les réseaux urbains.
L'équation de Bernoulli explique comment la vitesse de l'eau augmente et la pression diminue à mesure qu'elle s'écoule vers le bas dans les déversoirs des barrages, aidant les ingénieurs à concevoir des structures capables de gérer en toute sécurité des débits variables.
Les compteurs Venturi, couramment utilisés dans les systèmes d’égouts, mesurent les débits en détectant les différences de pression entre les sections de conduite, assurant ainsi une gestion efficace des eaux usées.

Équation de bernoulli pour l'écoulement le long d'une ligne de courant

Bernoulli's Equation for Flow Along a Streamline

L'équation de Bernoulli concerne la conservation de l'énergie dans un fluide se déplaçant le long d'une ligne de courant. L'équation s'applique aux fluides incompressibles et non visqueux sous écoulement constant. Pour un tel écoulement, la deuxième loi de Newton s'applique à un petit élément fluide, qui subit des forces dues aux différences de pression, à la gravité et aux variations de vitesse. L'équilibre des forces conduit à la forme suivante de l'équation de Bernoulli:

<img alt="Equation 1" src="/files/ftp_upload/18033/18033_Equation_1.svg" style="text-align: center; height: 26px;">

Ici, P est la pression, ρ est la densité du fluide, v est la vitesse, g désigne l'accélération due à la gravité et l'élévation est désignée par h. Chaque terme de cette équation représente l'énergie par unité de volume du fluide.

Considérons l'eau qui s'écoule dans un tuyau horizontal avec deux sections de diamètres différents. Au point 1, le tuyau a un grand diamètre, tandis qu'au point 2, le tuyau se rétrécit. L'équation de Bernoulli nous dit que si la vitesse de l'eau augmente dans la section la plus étroite (point 2), la pression doit diminuer pour conserver l'énergie le long de la ligne de courant. En appliquant l'équation de Bernoulli aux sections de conduites, on obtient:

<img alt="Equation 2" src="/files/ftp_upload/18033/18033_Equation_2.svg" style="text-align: center; height: 26px;">

Le principe de Bernoulli est essentiel pour comprendre et concevoir des systèmes d'écoulement des fluides. Par exemple, dans les réseaux de distribution d'eau, les variations de diamètre des conduites entraînent des changements de pression qui affectent l'efficacité de l'écoulement. Comme le montre l'équation, un tuyau plus étroit augmente la vitesse tout en réduisant la pression, ce qui garantit un approvisionnement constant dans les systèmes urbains. Dans les déversoirs de barrage, l'énergie potentielle du fluide due à l'élévation est convertie en énergie cinétique lorsque l'eau descend. Il en résulte une augmentation de la vitesse et une baisse correspondante de la pression, un phénomène utilisé pour concevoir des déversoirs capables de gérer en toute sécurité des débits de décharge variables.

Équation de Bernoulli pour l'écoulement le long d'une ligne de courant

L'équation de Bernoulli décrit le comportement des fluides en mouvement en reliant la pression, la vitesse et l'élévation le long d'une ligne de courant.
Il s’appuie sur le ...

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

Education

Enseignement

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Civil Engineering

Génie civil

Unit 1

Unité 1

Fluid Dynamics

Dynamique des fluides

KEY TERMS AND CONCEPTS

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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 elevation. Each term represents energy per unit volume of the fluid. This makes Bernoulli's equation a useful way to track how energy changes as fluid moves.

A simple pipe example shows the main effect. Water flows through a horizontal pipe with two sections. Point 1 has a large diameter, and point 2 has a narrower diameter. When the water speeds up at the narrow section, the pressure must drop to keep energy conserved along the streamline.

Bernoulli's principle also helps explain real flow systems. In water distribution networks, changes in pipe diameter create pressure changes that affect flow efficiency. A narrower pipe raises velocity and lowers pressure, which helps support a steady supply across urban systems. In dam spillways, water loses potential energy as it falls. That energy turns into kinetic energy, so velocity increases and pressure decreases. Engineers use this effect to design spillways that can handle changing discharge rates safely.

Pressure and Speed in Bernoulli's Equation

pressure-and-speed-in-bernoullis-equation

bernoulli's equation for flow along a streamline

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

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 Bernoulli's equation links pressure, speed, and height in flowing fluids, with pipe and dam spillway examples for high school science.

Pressure and Speed in Bernoulli's Equation | 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

Pressure and Speed in Bernoulli's Equation

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