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

A equação de continuidade deriva do princípio da conservação de massa, afirmando que a massa de fluido que entra e sai de um sistema permanece constante.
Para fluidos incompressíveis, o produto da área de fluxo e da velocidade permanece consistente ao longo do caminho do fluxo, ajustando-se às mudanças nas seções transversais.
À medida que a área da seção transversal diminui, a velocidade do fluido aumenta para manter uma taxa de fluxo constante.
Essa relação é crucial para a análise de sistemas como tubulações de água, gerenciamento de águas pluviais e canais abertos, garantindo estabilidade e eficiência em várias aplicações de fluxo de fluidos.
Em sistemas de abastecimento de água, a equação de continuidade ajuda a manter uma distribuição estável, prevendo mudanças de velocidade quando os tubos se estreitam, o que é essencial para evitar quedas de pressão.
Ajuda a projetar bacias de detenção para gerenciamento de águas pluviais. Quando o escoamento entra na bacia durante uma tempestade, a taxa de fluxo deve corresponder à taxa de entrada, ajustada para evitar inundações a jusante.
Para canais abertos, como rios ou canais, a equação de continuidade ajuda a entender como as variações de largura ou profundidade afetam os níveis de água e as velocidades de fluxo. Esse entendimento é vital para projetar sistemas eficazes de controle de inundações e navegação.

equação de continuidade

A equação de continuidade afirma que a taxa de fluxo da massa deve permanecer constante para um fluxo estável de um fluido incompressível dentro de um sistema confinado. Este princípio se aplica a sistemas onde o fluido passa por áreas transversais variáveis, como bicos, seringas e tubulações.

A taxa de fluxo da massa é expressa como:

<img alt="Equation 1" src="/files/ftp_upload/18038/18038_Equation_1.svg" style="text-align: center; height: 12px;">

Para fluxo incompressível, onde a densidade permanece constante, a equação de continuidade é simplificada para:

<img alt="Equation 2" src="/files/ftp_upload/18038/18038_Equation_2.svg" style="text-align: center; height: 12px;">

Esta equação ilustra que a velocidade aumenta quando a área transversal diminui e vice-versa, garantindo a conservação da massa. Um exemplo é o fluxo através de uma seringa. Quando o êmbolo comprime o fluido através de um bico, a equação de continuidade mostra que, se a área de saída for metade da área de entrada, a velocidade de saída deve ser o dobro da velocidade de entrada.

Outro exemplo é o fluxo em um tanque acionado por gravidade. Aplicar tanto a equação de Bernoulli quanto a equação de continuidade aos pontos ao longo do fluxo ajuda a determinar a velocidade de saída e a taxa de fluxo volumétrico:

<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;">

onde h é a altura do fluido, D é o diâmetro do tanque e d é o diâmetro de saída. Essa relação mostra que a taxa de fluxo converge para um valor estável à medida que o diâmetro de saída se torna pequeno em relação ao diâmetro do tanque. Essas derivações exemplificam a utilidade da equação de continuidade na previsão do comportamento do fluido sob várias restrições, garantindo a conservação de massa em sistemas dinâmicos.

Equação de continuidade

A equação de continuidade deriva do princípio da conservação de massa, afirmando que a massa de fluido que entra e sai de um sistema permanece constante.
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Education

JoVE Core

Civil Engineering

Engenharia Civil

Unit 1

Unidade 1

Fluid Dynamics

Dinâmica de Fluidos

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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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