Name: velocity potential in pipe flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

Considere a água fluindo de forma constante através de um tubo longo e reto com uma seção transversal uniforme.
Nas regiões centrais do tubo, onde o fluxo é irrotacional, um potencial de velocidade define o fluxo, e a velocidade é expressa como o gradiente desse potencial.
A substituição dos componentes de velocidade verifica diretamente se o fluxo satisfaz essa condição irrotacional.
O potencial de velocidade se aplica a fluxos tridimensionais, como este fluxo de tubo, enquanto a função de fluxo é específica para fluxos bidimensionais, resultantes da conservação de massa.
No fluxo incompressível, a equação de continuidade requer que a divergência do vetor velocidade seja zero.
Ao substituir o potencial de velocidade, essa condição se torna a equação de Laplace, uma equação diferencial chave que governa fluxos invíscidos, incompressíveis e irrotacionais.
Em coordenadas cilíndricas, a equação de Laplace se ajusta para variações radiais, angulares e axiais. Resolver esta equação com condições de contorno nas paredes do tubo define o potencial de velocidade para o fluxo.
A partir disso, o campo de velocidade completo é determinado tomando derivadas espaciais, revelando o padrão de fluxo dentro do tubo.

potencial de velocidade

Velocity Potential

Em um fluxo constante e incompressível por meio de um tubo longo e reto com uma seção transversal uniforme, o fluxo na região central (longe das paredes do tubo) é irrotacional. Essa natureza irrotacional significa que as partículas do fluido não giram em torno de seus eixos, e uma função escalar chamada potencial de velocidade, representada por ϕ, pode ser usada para descrever seu movimento. Em fluxos irrotacionais, o campo de velocidade V é definido como o gradiente do potencial de velocidade:

<img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;">

Como rotação da curva de um gradiente é sempre zero, definir a velocidade em termos de ϕ garante que o fluxo não tenha vorticidade, tornando-o irrotacional.

Para fluxos incompressíveis, a equação de continuidade afirma que a divergência do campo de velocidade deve ser zero:

<img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;">

Substituindo V nesta equação, a equação se torna:

<img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;">

Isso é conhecido como equação de Laplace. Em regiões onde o fluxo é irrotacional e incompressível, o potencial de velocidade ϕ deve satisfazer a equação de Laplace.

No caso do fluxo de tubulação, as coordenadas cilíndricas (r, θ, z) são uma escolha natural para descrever o fluxo, onde r é a distância radial da linha central da tubulação, θ é a posição angular e z é a distância axial ao longo da tubulação. A equação de Laplace em coordenadas cilíndricas, considerando variações radiais, angulares e axiais, é expressa como:

<img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;">

Encontrar a forma exata de ϕ na tubulação requer condições de contorno. Por exemplo, nas paredes do tubo, a condição de não escorregamento implicaria velocidade tangencial zero, enquanto na região irrotacional central, a função potencial seguiria a simetria do tubo. Resolver a equação de Laplace com essas condições define o potencial de velocidade através do campo de fluxo.

Uma vez que ϕ é determinado, os componentes de velocidade em coordenadas cilíndricas podem ser derivados tomando derivadas parciais de ϕ. Esses componentes são:

Velocidade radial:

<img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;">

Velocidade angular

<img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;">

Velocidade axial

<img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;">

Essas expressões para componentes de velocidade confirmam que o fluxo satisfaz a condição irrotacional, pois elas derivam de um gradiente.

Potencial de Velocidade

Considere a água fluindo de forma constante através de um tubo longo e reto com uma seção transversal uniforme.
Nas regiões centrais do tubo, onde o fluxo é irrotacional, um ...

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Differential Analysis of Fluid Flow

Análise Diferencial do Escoamento de Fluidos

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velocity potential in pipe flow

Velocity potential in pipe flow describes the motion of fluid in the central region of a long, straight pipe with a uniform cross-section. In this region, the flow is steady, incompressible, and irrotational. Irrotational means the fluid particles do not spin around their own axes.

A scalar function called the velocity potential, written as φ, is used to describe this motion. The velocity field V is defined as the gradient of φ. Because the curl of a gradient is always zero, this form of V gives the flow no vorticity and keeps it irrotational.

For an incompressible fluid, the continuity equation requires the divergence of the velocity field to be zero. When V is written in terms of φ, this condition becomes Laplace's equation. In regions where the flow is both incompressible and irrotational, the velocity potential must satisfy this equation.

Pipe flow is often described with cylindrical coordinates: r, θ, and z. Here, r is the radial distance from the pipe centreline, θ is the angular position, and z is the distance along the pipe. In these coordinates, Laplace's equation includes radial, angular, and axial variation.

To find the exact form of φ in the pipe, boundary conditions are needed. At the pipe walls, the no-slip condition means the tangential velocity is zero. In the central irrotational region, the potential function must also match the symmetry of the pipe. Solving Laplace's equation with these conditions gives the velocity potential across the flow field.

Once φ is known, the velocity components can be found by taking partial derivatives. These components are the radial velocity, the angular velocity, and the axial velocity. Together, they show how the flow field comes from a gradient and therefore satisfies the irrotational condition.

Velocity Potential in Pipe Flow

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Velocity Potential in Pipe Flow

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