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

A equação de Bernoulli descreve o comportamento dos fluidos em movimento, relacionando pressão, velocidade e elevação ao longo de uma linha de fluxo.
Ele se baseia no princípio da conservação de energia, assumindo fluxo constante, fluidos incompressíveis e sem viscosidade, o que significa que não há forças de atrito presentes.
O termo de pressão representa a força que o fluido exerce em um ponto específico.
O termo de velocidade representa a energia cinética do fluido, indicando que a pressão diminui à medida que a velocidade do fluido aumenta.
O termo de elevação reflete a energia potencial do fluido com base em sua posição vertical em relação a um ponto de referência.
Este princípio se aplica a situações do mundo real, como sistemas de distribuição de água, vertedouros de barragens e medidores de Venturi.
Nos sistemas de distribuição de água, a equação de Bernoulli prevê como a pressão cai à medida que a água flui através de tubos de diâmetros variados, garantindo o abastecimento eficiente nas redes urbanas.
A equação de Bernoulli explica como a velocidade da água aumenta e a pressão diminui à medida que flui para baixo nos vertedouros das barragens, ajudando os engenheiros a projetar estruturas que podem gerenciar com segurança as diferentes taxas de fluxo.
Os medidores Venturi, comumente usados em sistemas de esgoto, medem as vazões detectando diferenças de pressão entre as seções dos tubos, garantindo um gerenciamento eficaz das águas residuais.

equação de bernoulli para fluxo ao longo de uma linha de corrente

Bernoulli's Equation for Flow Along a Streamline

A equação de Bernoulli relaciona a conservação de energia em um fluido que se move ao longo de uma linha de corrente. A equação se aplica a fluidos incompressíveis e invíscidos sob fluxo constante. Para tal fluxo, a segunda lei de Newton é aplicada a um pequeno elemento fluido, que experimenta forças devido a diferenças de pressão, gravidade e variações de velocidade. O equilíbrio de forças leva à seguinte forma da equação de Bernoulli:

<img alt="Equation 1" src="/files/ftp_upload/18033/18033_Equation_1.svg" style="text-align: center; height: 26px;">


Aqui, P é a pressão, ρ é a densidade do fluido, v é a velocidade, g denota a aceleração devido à gravidade e a elevação é denotada por h. Cada termo nesta equação representa energia por unidade de volume do fluido.

Considere a água fluindo por meio de um tubo horizontal com duas seções de diâmetros diferentes. No ponto 1, o tubo tem um diâmetro grande, enquanto no ponto 2, o tubo estreita. A equação de Bernoulli nos diz que se a velocidade da água aumenta na seção mais estreita (ponto 2), a pressão deve diminuir para conservar energia ao longo da linha de corrente. A aplicação da equação de Bernoulli às seções de tubulação fornece:

<img alt="Equation 2" src="/files/ftp_upload/18033/18033_Equation_2.svg" style="text-align: center; height: 26px;">

O princípio de Bernoulli é crucial para entender e projetar sistemas de fluxo de fluidos. Por exemplo, em redes de distribuição de água, variações nos diâmetros de tubulação causam mudanças de pressão que afetam a eficiência do fluxo. Conforme a equação determina, uma tubulação mais estreita aumenta a velocidade enquanto reduz a pressão, garantindo um fornecimento consistente em todos os sistemas urbanos. Em vertedouros de barragens, a energia potencial do fluido devido à elevação é convertida em energia cinética conforme a água desce. Isso resulta em um aumento na velocidade e uma queda correspondente na pressão, um fenômeno usado para projetar vertedouros capazes de gerenciar com segurança taxas de descarga variáveis.

Equação de Bernoulli para fluxo ao longo de uma linha de corrente

A equação de Bernoulli descreve o comportamento dos fluidos em movimento, relacionando pressão, velocidade e elevação ao longo de uma linha de fluxo.
Ele se baseia no ...

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

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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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