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

Stellen Sie sich eine stetige, inkompressible Strömung durch ein gerades zylindrisches Rohr mit konstantem Radius R vor, die als Hagen-Poiseuille-Strömung bekannt ist.
Die zylindrische Geometrie ermöglicht die Verwendung der Koordinaten r, θ und z für radiale, winklige und axiale Messungen.
Das parallel zur Rohrachse strömende Fluid hat eine rein axiale Geschwindigkeitskomponente und variiert nur mit dem radialen Abstand.
Diese Bedingungen vereinfachen die Navier-Stokes-Gleichungen und zeigen ein parabolisches Geschwindigkeitsprofil, das zeigt, dass die Geschwindigkeit im Zentrum am höchsten ist und zu den Wänden hin abnimmt und an der Grenze Null erreicht.
Dies zeigt die laminare Strömung, bei der sich Flüssigkeitspartikel gleichmäßig in organisierten Schichten bewegen.
Durch die Integration dieses Profils über den Querschnitt des Rohrs ergibt sich das Poiseuillesche Gesetz, das die Durchflussrate Q in Abhängigkeit von der vierten Potenz des Rohrradius, der Viskosität des Fluids und dem Druckgradienten ausdrückt.
Die mittlere Geschwindigkeit V, berechnet aus Q, ist die Hälfte der maximalen Geschwindigkeit, die im Zentrum der Röhre beobachtet wird.
Diese Geschwindigkeitsverteilung bestätigt, dass die Strömung in der Mitte am höchsten ist und an der Wand parabolisch auf Null abnimmt.

gleichmäßige, laminare strömung in runden rohren

Steady, Laminar Flow in Circular Tubes

Die Hagen-Poiseuille-Strömung beschreibt die gleichmäßige, inkompressible Strömung einer viskosen Flüssigkeit durch ein zylindrisches Rohr mit einem konstanten Radius R. Dieses Strömungsprofil wird häufig angewendet, um den Flüssigkeitstransport in engen Kanälen wie Kapillaren zu verstehen. Es dient als grundlegendes Beispiel für eine laminare Strömung. In diesem Modell werden Zylinderkoordinaten (r, θ, z) verwendet, um die radialen (r), Winkel- (θ) und axialen (z) Abmessungen innerhalb des Rohrs zu beschreiben. Bei der Hagen-Poiseuille-Strömung ist das Geschwindigkeitsprofil rein axial, d. h. der Geschwindigkeitsvektor zeigt nur entlang der z-Achse und variiert nur mit dem radialen Abstand r vom Zentrum des Rohrs.

Die maßgeblichen Gleichungen für die Flüssigkeitsbewegung, die Navier-Stokes-Gleichungen, vereinfachen sich unter diesen Bedingungen. Ohne Winkel- oder Radialgeschwindigkeitskomponenten ergibt sich die axiale Geschwindigkeit v_z(r) als Funktion von r allein. Das Lösen der reduzierten Navier-Stokes-Gleichungen ergibt ein parabolisches Geschwindigkeitsprofil:

<img alt="Equation 1" src="/files/ftp_upload/18065/18065_Equation 1.svg" style="text-align: center; height: 27px;">


Dieses Profil zeigt, dass der viskose Widerstand die Flüssigkeitsgeschwindigkeit vom Maximum in der Mitte auf Null an der Rohrwand verringert. Diese Nullgeschwindigkeit an der Grenze, bekannt als Haftbedingung, resultiert aus der Reibung zwischen der Flüssigkeit und der Rohrwand. Folglich ist die Strömung laminar, wobei sich die Flüssigkeitspartikel in parallelen Schichten bewegen, ohne sich seitlich zu vermischen.

Die Integration des Geschwindigkeitsprofils über die Querschnittsfläche des Rohrs ergibt die volumetrische Durchflussrate Q, bekannt als Poiseuille-Gesetz:

<img alt="Equation 2" src="/files/ftp_upload/18065/18065_Equation 2.svg" style="text-align: center; height: 27px;">


Wobei ΔP die Druckdifferenz über die Länge L des Rohrs und μ die dynamische Viskosität der Flüssigkeit ist. Diese Beziehung zeigt, dass Q sehr empfindlich auf den Radius des Rohrs reagiert und mit der vierten Potenz von R zunimmt. Daraus folgt, dass selbst eine geringfügige Erhöhung des Radius die Durchflussrate erheblich steigert, ein Prinzip, das bei Flüssigkeitstransportanwendungen von entscheidender Bedeutung ist.

Nach dem Poiseuille-Gesetz kann die mittlere Geschwindigkeit V der Flüssigkeit wie folgt ausgedrückt werden:

<img alt="Equation 3" src="/files/ftp_upload/18065/18065_Equation 3.svg" style="text-align: center; height: 27px;">

Diese Durchschnittsgeschwindigkeit beträgt genau die Hälfte der maximalen Geschwindigkeit in der Mitte des Rohrs, was die parabolische Geschwindigkeitsverteilung bestätigt und die vorhersehbare, geschichtete Natur der laminaren Strömung in einem zylindrischen Rohr demonstriert.

Gleichmäßige, laminare Strömung in runden Rohren

Stellen Sie sich eine stetige, inkompressible Strömung durch ein gerades zylindrisches Rohr mit konstantem Radius R vor, die als Hagen-Poiseuille-Strömung bekannt ist.
Die zylindrische ...

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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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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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