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

Si consideri il flusso costante e incomprimibile attraverso un tubo cilindrico rettilineo di raggio costante R, noto come flusso di Hagen-Poiseuille.
La geometria cilindrica consente l'uso delle coordinate r, θ e z per misure radiali, angolari e assiali.
Il fluido che scorre parallelamente all'asse del tubo ha una componente di velocità puramente assiale e varia solo con la distanza radiale.
Queste condizioni semplificano le equazioni di Navier-Stokes, rivelando un profilo di velocità parabolica che mostra che la velocità è più alta al centro e diminuisce verso le pareti, raggiungendo lo zero al contorno.
Questo dimostra il flusso laminare, in cui le particelle fluide si muovono dolcemente in strati organizzati.
L'integrazione di questo profilo sulla sezione trasversale del tubo fornisce la legge di Poiseuille, che esprime la portata Q come dipendente dalla quarta potenza del raggio del tubo, dalla viscosità del fluido e dal gradiente di pressione.
La velocità media V, calcolata da Q, è la metà della velocità massima osservata al centro del tubo.
Questa distribuzione della velocità conferma che il flusso è più alto al centro e diminuisce parabolicamente a zero alla parete.

flusso laminare costante in tubi circolari

Steady, Laminar Flow in Circular Tubes

Il flusso di Hagen-Poiseuille descrive il flusso stazionario e incomprimibile di un fluido viscoso attraverso un tubo cilindrico con raggio costante R. Questo profilo di flusso viene spesso applicato per comprendere il trasporto di fluidi in canali stretti, come i capillari. Serve come esempio fondamentale di flusso laminare. In questo modello, le coordinate cilindriche (r,θ,z), vengono utilizzate per descrivere le dimensioni radiali (r), angolari (θ) e assiali (z) all'interno del tubo. Per il flusso di Hagen-Poiseuille, il profilo di velocità è puramente assiale, ovvero il vettore di velocità punta solo lungo l'asse z e varia esclusivamente con la distanza radiale r dal centro del tubo.

Le equazioni che governano il moto dei fluidi, le equazioni di Navier-Stokes, in queste condizioni si semplificano. Senza componenti di velocità angolare o radiale, la velocità assiale v_z(r) emerge come funzione della sola r. Risolvendo le equazioni ridotte di Navier-Stokes, si ottiene un profilo di velocità parabolico:

<img alt="Equation 1" src="/files/ftp_upload/18065/18065_Equation 1.svg" style="text-align: center; height: 27px;">


Questo profilo dimostra che la resistenza viscosa diminuisce la velocità del fluido dal massimo della zona centrale, a zero sulla parete del tubo. Questa velocità zero al confine, nota come condizione di non slittamento, deriva dall'attrito tra il fluido e la parete del tubo. Di conseguenza, il flusso è laminare, con particelle di fluido che si muovono in strati paralleli senza miscelazione laterale.

L'integrazione del profilo di velocità sull'area della sezione trasversale del tubo, produce la portata volumetrica Q, nota come legge di Poiseuille:

<img alt="Equation 2" src="/files/ftp_upload/18065/18065_Equation 2.svg" style="text-align: center; height: 27px;">


Dove ΔP è la differenza di pressione lungo la lunghezza L del tubo e μ è la viscosità dinamica del fluido. Questa relazione mostra che Q è altamente sensibile al raggio del tubo, aumentando con la quarta potenza di R. Ne consegue che anche un leggero aumento del raggio, provoca un aumento significativo della portata, un principio fondamentale nelle applicazioni di trasporto di fluidi.

Dalla legge di Poiseuille, la velocità media V del fluido, può essere espressa come:

<img alt="Equation 3" src="/files/ftp_upload/18065/18065_Equation 3.svg" style="text-align: center; height: 27px;">

Questa velocità media è esattamente la metà della velocità massima al centro del tubo, confermando la distribuzione parabolica della velocità e dimostrando la natura prevedibile e stratificata del flusso laminare in un tubo cilindrico.

Flusso Laminare Costante in Tubi Circolari

Si consideri il flusso costante e incomprimibile attraverso un tubo cilindrico rettilineo di raggio costante R, noto come flusso di Hagen-Poiseuille.
Il motore cilindrico ...

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Differential Analysis of Fluid Flow

Analisi differenziale del flusso di fluidi

KEY TERMS AND CONCEPTS

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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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