Name: pressure and velocity in inviscid flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 28 s
Channel: JoVE

In einem bewegten Fluid entstehen viskositätsbedingt Scherspannungen. Bei Wasser mit niedriger Viskosität sind diese Effekte vernachlässigbar, was die Analyse der Flüssigkeitsbewegung vereinfacht.
Wenn die Scherspannungen vernachlässigbar sind, gilt das Fluid als viskos oder reibungsfrei. In viskosen Flüssigkeiten ist die Normalspannung in alle Richtungen gleich, was bedeutet, dass der Druck unabhängig von der Spannungsrichtung ist.
Der Druck ist der Nachteil der Normalspannung, wodurch sichergestellt wird, dass Druckspannungen zu einem positiven Druckwert führen.
Die allgemeinen Bewegungsgleichungen für viskose Strömungen, bekannt als Euler-Gleichungen, beschreiben die Impulserhaltung. Diese Gleichungen stellen das zweite Newtonsche Gesetz dar, das auf ein flüssiges Element angewendet wird und Geschwindigkeitsänderungen mit den auf das Fluid wirkenden Kräften in Beziehung setzt.
Die Euler-Gleichungen berücksichtigen Druckgradienten, Körperkräfte wie die Schwerkraft und Beschleunigungen – sowohl lokale als auch konvektive – die Änderungen der Fluidgeschwindigkeit antreiben.
Sie vernachlässigen viskose Kräfte, was die Analyse vereinfacht, bleiben aber aufgrund der nichtlinearen Geschwindigkeitsterme in den partiellen Differentialgleichungen schwierig zu lösen.
Durch die Integration der Euler-Gleichungen entlang einer Stromlinie wird die Bernoulli-Gleichung abgeleitet, die Einblicke in Druck-, Geschwindigkeits- und Höhenvariationen in viskosen Strömungen bietet, was sie zu einem wichtigen Werkzeug in der Strömungsmechanik macht.

eulersche bewegungsgleichungen

Euler&#39;s Equations of Motion

In der Strömungsmechanik entstehen Scherspannungen durch die Viskosität, die den inneren Widerstand einer Flüssigkeit gegen Verformung darstellt. Bei Flüssigkeiten mit niedriger Viskosität, wie Wasser, sind diese Spannungen minimal, was die Strömungsanalyse vereinfacht, da die Flüssigkeit als nichtviskos oder reibungslos behandelt werden kann. In einer nichtviskosen Flüssigkeit gibt es keine Scherspannungen, sodass nur Normalspannungen vorhanden sind, die senkrecht auf die Flüssigkeitselemente wirken. Insbesondere bleibt der Druck – definiert als das Negativ der Normalspannung – in verschiedene Richtungen gleichmäßig und wird von der Spannungsausrichtung nicht beeinflusst.

Die Bewegung nichtviskoser Flüssigkeiten wird durch die Eulerschen Gleichungen bestimmt, die aus Newtons zweitem Gesetz in Anwendung auf die Strömungsdynamik abgeleitet wurden. Diese Gleichungen beschreiben die Impulserhaltung, indem sie die Änderungsrate der Geschwindigkeit mit den auf ein Flüssigkeitselement wirkenden Kräften in Beziehung setzen. Bei einer nichtviskosen Flüssigkeit umfassen die relevanten Kräfte Druckgradienten und Körperkräfte wie die Schwerkraft. Die Euler-Gleichungen drücken das Gleichgewicht dieser Kräfte aus, wobei viskose Terme ausgeschlossen werden, und zwar wie folgt:

<img alt="Equation 1" src="/files/ftp_upload/18057/18057_Equation_1.svg" style="text-align: center; height: 30px;">

wobei ρ die Fluiddichte, v der Geschwindigkeitsvektor, p der Druck und g​ die Erdbeschleunigung ist.

Obwohl die Euler-Gleichungen die Viskosität ignorieren, bleiben sie aufgrund ihrer nichtlinearen Terme, die die konvektive Beschleunigung von Fluidpartikeln berücksichtigen, komplex. Wenn diese Gleichungen entlang einer Stromlinie integriert werden, ergeben sie die Bernoulli-Gleichung, die eine grundlegende Beziehung in der Fluidmechanik beschreibt und Druck, Geschwindigkeit und Höhe für nichtviskose Strömungen verbindet. Die Bernoulli-Gleichung ist ein leistungsstarkes Werkzeug zur Analyse der Energieverteilung innerhalb von Fluidströmen und unterstützt Anwendungen im Fluidtransport, der Aerodynamik und der Hydrodynamik.

Eulersche Bewegungsgleichungen

In einem bewegten Fluid entstehen viskositätsbedingt Scherspannungen. Bei Wasser mit niedriger Viskosität sind diese Effekte vernachlässigbar, was die Bewegung der Flüssigkeit ...

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pressure and velocity in inviscid flow

Pressure and velocity in inviscid flow are linked by Euler's equations of motion. In fluid mechanics, viscosity creates shear stress, which is a force that acts along a surface. For low-viscosity fluids like water, these shear stresses are very small. That lets the fluid be treated as inviscid, or frictionless.

In an inviscid fluid, shear stresses are absent. Only normal stresses remain, and these act perpendicular to a fluid element. Pressure is the negative of the normal stress. It stays the same in different directions and does not depend on how the stress is oriented.

Euler's equations come from Newton's second law applied to fluid motion. They describe momentum conservation by linking the change in velocity to the forces on a fluid element. For inviscid flow, those forces are pressure gradients and body forces such as gravity. The equations include density, velocity, pressure, and gravitational acceleration, but they leave out viscous terms.

Even without viscosity, Euler's equations are still nonlinear. That means they are complex because they include convective acceleration, which follows the motion of fluid particles. When the equations are integrated along a streamline, they lead to Bernoulli's equation. Bernoulli's equation connects pressure, velocity, and elevation in inviscid flows and is useful for fluid transport, aerodynamics, and hydrodynamics.

Pressure and Velocity in Inviscid Flow

euler&#39;s equations of motion

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Learn how pressure, velocity, and gravity shape inviscid fluid motion through Euler's equations, and how they lead to Bernoulli's equation along a streamline.

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Pressure and Velocity in Inviscid Flow

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