19.2
Função de corrente
Em um fluxo de fluido incompressível bidimensional, a equação de continuidade é essencial para garantir a conservação da massa, o que significa que qu…
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.
View the full transcript and gain access to JoVE Core videos
Q1: What is the stream function and why is it used in fluid flow analysis?
The stream function is a mathematical tool that simplifies two-dimensional incompressible flow analysis by automatically satisfying the continuity equation for mass conservation. Instead of solving for horizontal and vertical velocity components separately, the stream function allows velocities to be defined as partial derivatives, eliminating tedious calculations and making fluid motion analysis more efficient.
Q2: How do velocity components relate to the stream function?
The horizontal velocity equals the partial derivative of the stream function with respect to the vertical direction, while the vertical velocity equals the negative partial derivative with respect to the horizontal direction. These definitions automatically satisfy the continuity equation because the mixed partial derivatives of the stream function cancel out.
Q3: What do streamlines represent in a flow field?
Streamlines are the paths that fluid particles follow within a flow field and are tangent to velocity vectors at every point. They can be visualized as contour lines of constant stream function values, providing an intuitive representation of fluid behavior without requiring individual velocity calculations at each location.
Q4: How does the stream function help calculate flow rates between streamlines?
The difference in stream function values between two streamlines directly represents the volumetric flow rate per unit depth flowing between them. This property eliminates the need to integrate velocity components separately, making flow rate calculations straightforward and efficient for two-dimensional incompressible flows.
Q5: Why must the continuity equation be satisfied in incompressible flow?
The continuity equation ensures mass conservation by requiring that the net inflow and outflow in any region remain balanced. For incompressible flow where density is constant, this means the divergence of the velocity field must equal zero, guaranteeing that fluid mass is neither created nor destroyed within the flow domain.
Q6: How does the stream function automatically satisfy mass conservation?
The stream function satisfies mass conservation because velocity components are defined as specific partial derivatives of the stream function. When these derivatives are substituted into the continuity equation, the mixed partial derivatives cancel out mathematically, automatically ensuring the divergence condition equals zero without additional constraints.
Q7: What advantage does using stream function provide over solving velocity components directly?
Using the stream function eliminates the need to solve separate equations for horizontal and vertical velocity components in steady laminar flow between parallel plates or other two-dimensional incompressible flows. This reduces computational complexity while providing direct access to flow rates and streamline patterns through a single scalar function.