Name: laminar flow profile between parallel plates
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 17 s
Channel: JoVE

Die stetige, laminare Strömung zwischen zwei parallelen Platten beschreibt die Bewegung des Fluids in einem engen, begrenzten Kanal, der häufig zur Modellierung einfacher Fluidsysteme verwendet wird.
Für die Strömung zwischen zwei festen Platten vereinfachen sich die Navier-Stokes-Gleichungen, da die Geschwindigkeit nur senkrecht zu den Platten variiert, ohne dass eine Beschleunigung entlang der Strömungsrichtung erfolgt.
Dies führt zu einem parabolischen Geschwindigkeitsprofil, bei dem sich die Flüssigkeit in der Mitte am schnellsten bewegt und an den Platten aufgrund von Reibung auf Null abbremst, was als rutschfeste Bedingung bezeichnet wird.
Die Durchflussrate zwischen den Platten hängt vom Druckgradienten, der Viskosität des Fluids und der Plattentrennung ab; eine Vergrößerung des Spalts oder eine Erhöhung des Druckgradienten erhöht die Strömung.
Die mittlere Strömungsgeschwindigkeit hängt auch mit dem Druckabfall zusammen, wobei die Spitzengeschwindigkeit in der Mitte des Kanals das 1,5-fache der mittleren Geschwindigkeit beträgt.
Diese Analyse bleibt nur für niedrige Reynoldszahlen gültig, was auf laminare Bedingungen hinweist, und schlägt bei turbulenten Strömungen mit hoher Reynoldszahl fehl.
Anwendungen dieser Analyse sind Bewässerungskanäle, Grundwasserströmungs- und Sedimentationsbecken in Wasseraufbereitungsanlagen, in denen laminare Bedingungen herrschen.

gleichmäßige, laminare strömung zwischen parallelen platten

Steady, Laminar Flow Between Parallel Plates

Das Verständnis der gleichmäßigen, laminaren Strömung zwischen parallelen Platten ist für die Analyse und Gestaltung der Strömung in schmalen rechteckigen Kanälen, wie sie häufig in verschiedenen Wassertransport- und Entwässerungssystemen vorkommen, von wesentlicher Bedeutung. Die Navier-Stokes-Gleichungen bestimmen die Flüssigkeitsbewegung und sind aufgrund ihrer Nichtlinearität im Allgemeinen schwierig zu lösen. In bestimmten Fällen sind jedoch Vereinfachungen möglich, beispielsweise bei der gleichmäßigen, laminaren Strömung zwischen parallelen Platten. Für dieses Szenario gehen wir von einer gleichmäßigen, inkompressiblen, laminaren Strömung aus.

Wenn wir zwei horizontale Platten betrachten, die entlang der x-Achse ausgerichtet und in einem Abstand von „2h“ voneinander angeordnet sind, wobei die y-Achse senkrecht zu den Platten steht, wird angenommen, dass die Flüssigkeitsbewegung parallel ist. Hier zeigt die Kontinuitätsgleichung keine Geschwindigkeitsänderung entlang der x-Achse an, und die Navier-Stokes-Gleichungen reduzieren sich wie folgt:

<img alt="Equation 1" src="/files/ftp_upload/18063/18063_Equation_1.svg" style="text-align: center; height: 30px;">


Die Gleichungen in y- und z-Richtung zeigen keinen Druckgradienten entlang dieser Achsen, was darauf hinweist, dass der Druck in y-Richtung hydrostatisch variiert, sodass die Integration zu folgendem Ergebnis führt:

<img alt="Equation 2" src="/files/ftp_upload/18063/18063_Equation_2.svg" style="text-align: center; height: 12px;">


Für parallele Platten mit einem Druckgradienten entlang der x-Achse vereinfacht sich die Gleichung:

<img alt="Equation 3" src="/files/ftp_upload/18063/18063_Equation_3.svg" style="text-align: center; height: 30px;">


Durch zweimalige Integration und Anwendung von Randbedingungen (u = 0 bei y = +h und y = -h) erhalten wir:

<img alt="Equation 4" src="/files/ftp_upload/18063/18063_Equation_4.svg" style="text-align: center; height: 30px;">

Dieser Ausdruck zeigt, dass das Geschwindigkeitsprofil über den Spalt zwischen den Platten parabolisch ist, mit maximaler Geschwindigkeit bei y = 0, auf halbem Weg zwischen den Platten. Die volumetrische Strömungsrate durch eine Einheitsbreite der Platten wird durch Integration des Geschwindigkeitsprofils über den Spalt ermittelt.

Das Verständnis der laminaren Strömung zwischen parallelen Platten hat praktische Anwendungen in der Grundwasserströmung, Abwasserbehandlung und Schmierung. Beispielsweise kann dieses Modell die Strömung in Bodenschichten abbilden, wo sich Wasser zwischen Bodenpartikeln bewegt, wie die Strömung zwischen Platten. Die Kenntnis der Strömungsrate und der Geschwindigkeitsverteilung hilft Ingenieuren, effiziente Entwässerungssysteme zu entwerfen und die Bewegung des Grundwassers vorherzusagen.

Gleichmäßige, laminare Strömung zwischen parallelen Platten

Die stetige, laminare Strömung zwischen zwei parallelen Platten beschreibt die Bewegung des Fluids in einem engen, begrenzten Kanal, der häufig zur Modellierung einfacher Fluidsysteme verwendet wird.
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JoVE Core

Civil Engineering

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Unit 1

Einheit 1

Differential Analysis of Fluid Flow

Differentielle Analyse der Fluidströmung

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

laminar flow profile between parallel plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings.

The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem becomes simpler because the flow is steady, incompressible, and laminar. Two horizontal plates are placed along the x-axis, and the gap between them is 2h. The y-axis is perpendicular to the plates, and the fluid moves parallel to the plates.

With these assumptions, the continuity equation shows that the velocity does not change along the x-axis. The Navier-Stokes equations also simplify in the y and z directions. They show no pressure gradient along those axes, so pressure changes hydrostatically in the y-direction. After integration, the x-direction equation includes the pressure gradient along the plates.

Solving that equation requires integrating twice and applying the no-slip boundary conditions, where u = 0 at y = +h and y = -h. The result is a parabolic velocity profile across the gap. The fluid moves fastest at y = 0, the midpoint between the plates. The volumetric flow rate for a unit width of the plates is then found by integrating the velocity profile across the space.

This flow model has practical uses in groundwater flow, wastewater treatment, and lubrication. It can also represent water moving through soil layers, where the flow resembles movement between plates. Engineers use the velocity distribution and flow rate to help design drainage systems and predict groundwater movement.

Laminar Flow Profile Between Parallel Plates

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Learn how steady laminar flow between parallel plates leads to a parabolic velocity profile, pressure-gradient behavior, and flow-rate calculations.

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Laminar Flow Profile Between Parallel Plates

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