Name: bernoulli’s equation in streamlined flow
Uploaded: 2025-05-22T09:06:09.000Z
Duration: 1 min 16 s
Channel: JoVE

Suponha um fluido laminar incompressível em estado estacionário passando por um tubo de seção transversal não uniforme.
Como o fluido é incompressível, o volume do elemento fluido e sua massa permanecem constantes.
A velocidade do elemento fluido que entra e sai do tubo é diferente, resultando em mudança líquida de energia cinética.
A força que atua no elemento fluido é devida à pressão circundante do fluido.
Assim, o trabalho líquido feito no elemento fluido pelo fluido circundante durante o deslocamento é a diferença entre o trabalho realizado ao entrar e sair do tubo.
Quando o elemento fluido atravessa a região elevada, ele ganha energia potencial gravitacional e a mudança na energia potencial gravitacional é determinada.
De acordo com o teorema da energia cinética de trabalho, o trabalho líquido realizado no elemento fluido é igual à mudança na energia cinética.
Dividindo por um valor comum, uma equação é obtida.
Ao longo de uma linha de fluxo, a energia total por unidade de volume de um fluido que flui é uma constante, denominada equação de Bernoulli.

<ol>
	<li>Young, H.D. e Freedman, R.A. (2012). Física Universitária com Física Moderna. São Francisco, CA: Pearson: seção 12.5; páginas 385–386.</li>
	<li>OpenStax. (2019). Física Universitária Vol. 1. [Versão Web]. Obtido de <a href="https://openstax.org/books/university-physics-volume-1/pages/1-introduction">https://openstax.org/books/university-physics-volume-1/pages/1-introduction</a>: seção 14.6; páginas 716–718.</li>
</ol>

conservação de energia e a equação de bernoulli

Aplicar o princípio da conservação de energia ou o teorema trabalho-energia a um fluido incompressível, invíscido, em fluxo lamelar, estável e irrotacional leva à Equação de Bernoulli. Esta equação afirma que a soma da pressão do fluido, da energia potencial e da energia cinética por unidade de volume é constante ao longo de uma linha de corrente.

Todos os termos na equação têm a dimensão de energia por unidade de volume. A energia cinética por unidade de volume é chamada de densidade de energia cinética, e a energia potencial por unidade de volume é chamada de densidade de energia potencial.

É importante observar que a densidade do líquido não deve mudar ao longo do fluxo, ou seja, ele deve ser incompressível. O fluxo também deve ser lamelar e não turbulento. A equação de Bernoulli é aplicável a gases que têm efeitos de compressibilidade negligíveis. Para esses gases, assume-se que a densidade é constante e eles são tratados como um fluido incompressível. Uma vez que os gases são, geralmente, compressíveis, a equação não se aplica a eles.

Apesar de ser uma simples reformulação do princípio da conservação de energia com algumas suposições cruciais, a equação facilita o cálculo da pressão em diferentes pontos, se as velocidades forem conhecidas.

Conservação de Energia e a Equação de Bernoulli

Aplicar o princípio da conservação de energia ou o teorema da energia de trabalho a um fluido incompressível e invíscido em chumbos de fluxo laminar, constante e irrotacional ...

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energy-conservation-and-bernoulli-s-equation

Education

JoVE Core

Physics

Física

Unit 1

Unidade 1

Fluid Mechanics

Mecânica dos Fluidos

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

Energy Conservation and Bernoulli's Equation

bernoulli’s equation in streamlined flow

Bernoulli’s equation describes energy in a moving fluid. It comes from the conservation of energy principle, or the work-energy theorem, applied to an incompressible, inviscid fluid in laminar, steady, irrotational flow.

The equation says that pressure energy, potential energy, and kinetic energy per unit volume stay constant along a streamline. Each term has the units of energy per unit volume. The kinetic energy per unit volume is called kinetic energy density, and the potential energy per unit volume is called potential energy density.

The fluid must keep the same density as it flows, so it must be incompressible. The flow also needs to be laminar, not turbulent. Bernoulli’s equation can also apply to gases when compressibility effects are negligible. In that case, the gas density is treated as constant and the gas is handled like an incompressible fluid.

Because gases are usually compressible, the equation does not apply to them in many situations. Even with these limits, Bernoulli’s equation is useful for finding pressure at different points when the speed is known. It gives a simple way to connect pressure and motion in fluid flow.

Bernoulli’s Equation in Streamlined Flow

characteristics of fluids

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Fluid behavior under force differs from the behavior of solids. When a force acts parallel to the top surface of a solid, the solid resists it because of internal friction between its layers. This is called shearing resistance. If the force is removed, the shearing forces help the solid return to its original shape. Other deformation forces can also change the shape of a solid for a short time, as long as the force stays below a threshold magnitude. Solids usually keep their shape, which makes...

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density

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Density helps explain why objects sink or float in a fluid. It is also useful for identifying a substance and for learning about its phase and substructure. The SI unit of density is kg/m3, and the cgs unit is g/cm3. Liquids and solids usually have densities that are roughly comparable. That matches the fact that their atoms are in close contact. Gases have much lower densities than liquids and solids because their atoms are separated by large amounts of empty space. Density does not always...

pressure of fluids

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pascal's law

Pascal’s Principle in Static Fluids

Pascal’s principle explains how pressure changes move through a fluid that is not flowing. A fluid at rest is in static equilibrium. If the fluid is water, it is in hydrostatic equilibrium. In this state, the net force on any part of the fluid must be zero, or the fluid will begin to flow. Blaise Pascal described this idea in 1653 in Treatise on the Equilibrium of Liquids. He showed that when pressure changes in an enclosed fluid, that change is transmitted without loss through the fluid and...

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archimedes' principle

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accelerating fluids

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Surface tension sets the pressure difference in small drops and bubbles. For a tiny liquid drop, gravity can be neglected, so the drop is treated as spherical. It can also be viewed as two equal hemispherical drops joined together by surface tension. The forces on the drop come from three sources. The liquid inside pushes outward, the air outside pushes inward, and surface tension acts along the diameter between the two hemispheres. Since surface tension is force per unit length, it creates a...

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buoyancy

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contact angle

Wetting and Surface Curvature

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rise of liquid in a capillary tube

Capillary Action in Thin Tubes

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laminar and turbulent flow

Fluid Motion Patterns in Air and Water

Fluid dynamics studies fluids in motion. Scientists often use velocity vectors to show fluid motion in places like meteorology. For example, wind in the atmosphere can be mapped with vectors that show the speed and direction of the wind at each point. Another way to show fluid motion is with a streamline. A streamline shows the path of a small volume of fluid as it flows. When the flow changes with time, the streamlines do not match the flow lines. The fluid velocity is always tangential to...

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bernoulli's equation

Fluid Pressure and Speed in Motion

Bernoulli's equation connects fluid pressure with fluid speed. It helps explain why moving fluids can create pressure changes in everyday situations. One example is when two trains pass each other quickly and seem to pull inward. The same effect can happen when two cars pass each other at high speed. As the air between the trains or cars speeds up, its pressure drops. The air on the outside stays at atmospheric pressure. That pressure difference creates a net inward force on the vehicles.

bernoulli's principle

Fluid Pressure and Speed Changes

Bernoulli's equation links fluid pressure and fluid speed in a static, incompressible fluid. For horizontal flow, where gravitational energy density stays constant, the equation is especially useful. In that setting, the result is known as Bernoulli's principle. Bernoulli's principle says that fluid pressure drops when speed increases, and pressure rises when speed decreases. This pressure-speed relationship helps explain how moving fluids behave in simple flow situations. It also provides a...

bernoulli's principle: applications

Bernoulli's Principle in Fluid Devices

Bernoulli's principle helps explain how many fluid devices work when the fluid moves at a constant height. It is useful for devices that use fast flow to create low pressure and for devices that measure fluid speed. Some devices use this pressure change to pull one fluid into another. These are called entrainment devices. A Bunsen burner uses an adjustable gas nozzle to entrain air for proper combustion. An atomizer uses a squeeze bulb to make a jet of air that pulls perfume drops into the...

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energy conservation and bernoulli's equation

Bernoulli’s equation describes energy in a moving fluid. It comes from the conservation of energy principle, or the work-energy theorem, applied to an incompressible, inviscid fluid in laminar, steady, irrotational flow. The equation says that pressure energy, potential energy, and kinetic energy per unit volume stay constant along a streamline. Each term has the units of energy per unit volume. The kinetic energy per unit volume is called kinetic energy density, and the potential energy per...

viscosity

Viscosity and How Fluids Flow

Viscosity describes how easily a fluid flows. Water moves quickly, while honey and maple syrup flow slowly and stick to a surface. That difference comes from fluid friction between liquid layers and between a fluid and the material around it. In this example, water has a lower viscosity than honey and maple syrup. Viscosity is measured in newton-second per meter squared (N·s/m2) or pascal-second (Pa·s). It also changes with temperature and can differ from one fluid to another by several orders...

poiseuille's law and reynolds number

Fluid Resistance, Viscosity, and Flow Regime

Fluid resistance, viscosity, and flow regime shape how liquid moves through a horizontal tube. Fluid flows from high pressure to low pressure. The flow rate, Q, is the ratio of the pressure difference to the resistance in the tube. A larger pressure difference gives a higher flow rate. Poiseuille's law describes laminar flow of an incompressible fluid in a tube. In this relation, the factors that affect flow rate except pressure are grouped into resistance. Resistance depends on the tube size...

stokes' law

Terminal Velocity in a Viscous Fluid

Viscous forces act like friction in a liquid. They resist the relative motion of molecules and pull back on a solid body moving through the fluid. The size of this force depends on the body’s shape and size, its speed, and the liquid’s viscosity, density, and temperature. For a spherical solid object, the force in a fluid is described by Stokes’ law. This law applies only at low Reynolds number, when the flow is laminar and not turbulent. Experiments show that the force is proportional to the...

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key terms and concepts

Learn how Bernoulli’s equation links pressure, speed, and energy in laminar fluid flow, and why incompressible conditions matter.

Bernoulli’s Equation in Streamlined Flow | JoVE K12

physics

JoVE Core - Fisica apresenta a teoria e a aplicação de tópicos fundamentais da física, incluindo cinemática, dinâmica, mecânica dos fluidos, estática, ondas e acústica, com vídeos animados concisos e de fácil compreensão. Além disso, os vídeos de cientistas em ação demonstram a aplicação de conceitos relacionados em experimentos de pesquisa clássica e original realizados nos laboratórios de todo o mundo hoje em dia.

Bernoulli’s Equation in Streamlined Flow

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