Name: stream function for incompressible flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

No fluxo incompressível bidimensional, existem componentes de velocidade horizontal e vertical.
A equação de continuidade garante a conservação de massa, exigindo que a soma das taxas de variação desses componentes seja zero, o que significa que a entrada e a saída líquidas em qualquer região são equilibradas.
Para satisfazer essa condição, a função de fluxo é introduzida.
A velocidade horizontal é a derivada parcial da função de fluxo em relação à direção vertical, enquanto a velocidade vertical é a derivada parcial negativa em relação à direção horizontal. Isso garante que a conservação em massa seja mantida automaticamente.
Uma característica fundamental da função de fluxo é que ela permanece constante ao longo das linhas de fluxo, que representam os caminhos das partículas de fluido e são tangentes às velocidades no campo de fluxo.
As linhas de fluxo, muitas vezes descritas como linhas de contorno, permitem a interpretação intuitiva do comportamento do fluido.
A função de fluxo simplifica os cálculos de movimento do fluido. A diferença nos valores da função de fluxo entre duas linhas de fluxo representa a taxa de fluxo de volume entre elas.
Isso elimina a necessidade de resolver os componentes de velocidade separadamente, tornando-o uma ferramenta eficiente para analisar o fluxo bidimensional.

função de corrente

Stream Function

Em um fluxo de fluido incompressível bidimensional, a equação de continuidade é essencial para garantir a conservação da massa, o que significa que qualquer mudança no fluido que entrando ou saindo de uma região é equilibrada por uma mudança correspondente em outro lugar. Para fluxo incompressível, onde a densidade permanece constante, esse requisito se simplifica à condição de que a divergência do campo de velocidade deve ser zero. Matematicamente, isso é expresso como,

<img alt="Equation 1" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;">

onde u e v representam, respectivamente, as componentes de velocidade horizontal e vertical.

Para satisfazer inerentemente esta equação, introduzimos a função de corrente ψ, que permite definir as componentes de velocidade em termos de ψ como:

<img alt="Equation 2" src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;">

Estas definições satisfazem automaticamente a equação de continuidade, uma vez que as derivadas parciais mistas de ψ se anulam.

A função de corrente ψ é constante ao longo das linhas de corrente, que são os caminhos que as partículas de fluido seguem dentro do fluxo. As linhas de corrente podem, portanto, ser visualizadas como linhas de contorno de ψ, proporcionando uma representação clara do movimento do fluido. Cada linha de corrente é tangente ao vetor de velocidade em qualquer ponto ao longo de seu caminho, ilustrando a direção do fluxo sem a necessidade de calcular as velocidades em cada ponto individualmente.

Além disso, a diferença nos valores de ψ\psiψ entre duas linhas de corrente representa a taxa de fluxo volumétrico por unidade de profundidade entre elas, permitindo o cálculo direto das taxas de fluxo. Essa propriedade simplifica significativamente a análise de fluxos bidimensionais incompressíveis ao eliminar a necessidade de resolver equações separadas para u e v. Como resultado, a função de corrente ψ é uma ferramenta poderosa para modelar e interpretar o comportamento dos fluidos em aplicações que envolvem fluxo incompressível e estacionário.

Função de corrente

No fluxo incompressível bidimensional, existem componentes de velocidade horizontal e vertical.
A equação de continuidade garante a conservação de massa por ...

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JoVE Core

Civil Engineering

Engenharia Civil

Unit 1

Unidade 1

Differential Analysis of Fluid Flow

Análise Diferencial do Escoamento de Fluidos

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

stream function for incompressible flow

The stream function helps describe two-dimensional incompressible flow. In this type of flow, the continuity equation keeps mass conserved. That means fluid entering a region must be balanced by fluid leaving it.

For incompressible flow, the density stays constant. The continuity requirement then becomes a zero-divergence condition for the velocity field. In two dimensions, the horizontal and vertical velocity components are usually written as u and v.

A stream function, written as ψ, is used to satisfy this equation automatically. The velocity components can be defined in terms of ψ. When they are written this way, the mixed partial derivatives of ψ cancel, so the continuity equation is already met.

The stream function is also linked to streamlines. A streamline is the path a fluid particle follows in the flow, and ψ stays constant along each one. Because of this, streamlines can be viewed as contour lines of ψ. Each streamline is tangent to the velocity vector at every point, so it shows flow direction without finding the velocity at every location.

Differences in ψ between two streamlines also have meaning. They represent the volumetric flow rate per unit depth between those lines. This makes it easier to analyze steady, two-dimensional incompressible flows. In many cases, the stream function reduces the need to solve separate equations for u and v, so it is a useful tool for modeling and interpreting fluid motion.

Stream Function for Incompressible Flow

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stream function

The stream function helps describe two-dimensional incompressible flow. In this type of flow, the continuity equation keeps mass conserved. That means fluid entering a region must be balanced by fluid leaving it. For incompressible flow, the density stays constant. The continuity requirement then becomes a zero-divergence condition for the velocity field. In two dimensions, the horizontal and vertical velocity components are usually written as u and v. A stream function, written as ψ, is used...

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key terms and concepts

Learn how the stream function describes two-dimensional incompressible flow, keeps continuity satisfied, and shows streamlines and flow rate in fluid motion.

Stream Function for Incompressible Flow | 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

Stream Function for Incompressible Flow

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