Name: parabolic flow in circular tubes
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 23 s
Channel: JoVE

Considere o fluxo constante e incompressível através de um tubo cilíndrico reto de raio constante R, conhecido como fluxo de Hagen-Poiseuille.
A geometria cilíndrica permite o uso das coordenadas r, θ e z para medições radiais, angulares e axiais.
O fluido que flui paralelamente ao eixo do tubo tem um componente de velocidade puramente axial e varia apenas com a distância radial.
Essas condições simplificam as equações de Navier-Stokes, revelando um perfil de velocidade parabólica mostrando que a velocidade é mais alta no centro e diminui em direção às paredes, chegando a zero no limite.
Isso demonstra o fluxo laminar, onde as partículas de fluido se movem suavemente em camadas organizadas.
A integração desse perfil sobre a seção transversal do tubo fornece a lei de Poiseuille, que expressa a vazão Q como dependente da quarta potência do raio do tubo, da viscosidade do fluido e do gradiente de pressão.
A velocidade média V, calculada a partir de Q, é metade da velocidade máxima observada no centro do tubo.
Essa distribuição de velocidade confirma que o fluxo é mais alto no centro e diminui parabolicamente para zero na parede.

fluxo laminar estacionário em tubos circulares

Steady, Laminar Flow in Circular Tubes

O fluxo de Hagen-Poiseuille descreve o fluxo estável e incompressível de um fluido viscoso por meio de um tubo cilíndrico com um raio R constante. Este perfil de fluxo é frequentemente aplicado para entender o transporte de fluidos em canais estreitos, como capilares. Ele serve como um exemplo fundamental de fluxo laminar. Neste modelo, coordenadas cilíndricas (r,θ,z) são usadas para descrever as dimensões radial (r), angular (θ) e axial (z) no tubo. Para o fluxo de Hagen-Poiseuille, o perfil de velocidade é puramente axial, o que significa que o vetor de velocidade aponta apenas ao longo do eixo z e varia apenas com a distância radial r do centro do tubo.

As equações que regem o movimento do fluido, as equações de Navier-Stokes, simplificam sob essas condições. Sem componentes de velocidade angular ou radial, a velocidade axial v_z(r) surge como uma função de r sozinho. Resolver as equações reduzidas de Navier-Stokes produz um perfil de velocidade parabólico:

<img alt="Equation 1" src="/files/ftp_upload/18065/18065_Equation 1.svg" style="text-align: center; height: 27px;">


Este perfil demonstra que o arrasto viscoso diminui a velocidade do fluido do máximo no centro para zero na parede do tubo. Esta velocidade zero no limite, conhecida como condição de não deslizamento, resulta do atrito entre o fluido e a parede do tubo. Consequentemente, o fluxo é laminar, com partículas de fluido se movendo em camadas paralelas sem mistura lateral.

A integração do perfil de velocidade sobre a área da seção transversal do tubo produz a vazão volumétrica Q, conhecida como lei de Poiseuille:

<img alt="Equation 2" src="/files/ftp_upload/18065/18065_Equation 2.svg" style="text-align: center; height: 27px;">


Onde ΔP é a diferença de pressão ao longo do comprimento L do tubo, e μ é a viscosidade dinâmica do fluido. Esta relação mostra que Q é altamente sensível ao raio do tubo, aumentando com a quarta potência de R. Conclui-se que mesmo um ligeiro aumento no raio aumenta significativamente a vazão, um princípio crítico em aplicações de transporte de fluidos.

Da lei de Poiseuille, a velocidade média V do fluido pode ser expressa como:

<img alt="Equation 3" src="/files/ftp_upload/18065/18065_Equation 3.svg" style="text-align: center; height: 27px;">

Esta velocidade média é precisamente metade da velocidade máxima no centro do tubo, confirmando a distribuição de velocidade parabólica e demonstrando a natureza previsível e em camadas do fluxo laminar em um tubo cilíndrico.

Fluxo Laminar Estacionário em Tubos Circulares

Considere o fluxo constante e incompressível através de um tubo cilíndrico reto de raio constante R, conhecido como fluxo de Hagen-Poiseuille.
O ...

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parabolic flow in circular tubes

Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries.

Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves only along the z-axis and changes with distance r from the center of the tube.

Under these conditions, the Navier-Stokes equations become simpler. With no angular or radial velocity components, the axial velocity vz(r) depends only on r. Solving the reduced equations gives a parabolic velocity profile. The fluid moves fastest at the center and slows toward the wall because viscous drag reduces the speed.

At the tube wall, the velocity is zero. This is the no-slip condition, which comes from friction between the fluid and the wall. The result is laminar flow, where fluid particles travel in parallel layers with no lateral mixing.

Integrating the velocity profile over the tube cross section gives the volumetric flow rate Q. This is Poiseuille’s law. In this relation, ΔP is the pressure difference across tube length L, and μ is the fluid’s dynamic viscosity. The flow rate depends strongly on the tube radius and increases with the fourth power of R, so even a small increase in radius can greatly raise Q.

Poiseuille’s law also gives the mean velocity V of the fluid. The average velocity is exactly half of the maximum velocity at the tube center. This matches the parabolic profile and shows the predictable, layered nature of flow in a circular tube.

Parabolic Flow in Circular Tubes

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Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries. Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves...

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key terms and concepts

Explore how Hagen-Poiseuille flow creates a parabolic velocity profile in circular tubes, with no-slip at the wall and Poiseuille’s law for flow rate.

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Parabolic Flow in Circular Tubes

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