Name: velocity potential in pipe flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

Consider water flowing steadily through a long, straight pipe with a uniform cross-section.
In central regions of the pipe, where flow is irrotational, a velocity potential defines the flow, and the velocity is expressed as the gradient of this potential.
Substituting the velocity components directly verifies that the flow satisfies this irrotational condition.
The velocity potential applies to three-dimensional flows, like this pipe flow, while the stream function is specific to two-dimensional flows, resulting from mass conservation.
In incompressible flow, the continuity equation requires the divergence of the velocity vector to be zero.
By substituting the velocity potential, this condition becomes Laplace&#39;s equation, a key differential equation governing inviscid, incompressible, and irrotational flows.
In cylindrical coordinates, Laplace&#39;s equation adjusts for radial, angular, and axial variations. Solving this equation with boundary conditions at the pipe walls defines the velocity potential for the flow.
From this, the complete velocity field is determined by taking spatial derivatives, revealing the flow pattern within the pipe.

snelheidspotentiaal

Velocity Potential

In een stabiele, onsamendrukbare stroming door een lange, rechte pijp met een uniforme doorsnede, is de stroming in het centrale gebied (ver van de pijpwanden) rotatievrij. Deze rotatievrije aard betekent dat vloeistofdeeltjes niet rond hun assen roteren en een scalaire functie genaamd de snelheidspotentiaal, weergegeven door ϕ, kan worden gebruikt om hun beweging te beschrijven. In rotatievrije stromingen wordt het snelheidsveld V gedefinieerd als de gradiënt van de snelheidspotentiaal: <img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;"> Aangezien de krul van een gradiënt altijd nul is, zorgt het definiëren van de snelheid in termen van ϕ ervoor dat de stroming geen werveling heeft, waardoor deze rotatievrij is. Voor onsamendrukbare stromingen stelt de continuïteitsvergelijking dat de divergentie van het snelheidsveld nul moet zijn: <img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;"> Als u V in deze vergelijking vervangt, wordt de vergelijking: <img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;"> Dit staat bekend als de vergelijking van Laplace. In gebieden waar de stroming niet-roterend en onsamendrukbaar is, moet het snelheidspotentiaal ϕ voldoen aan de vergelijking van Laplace. In het geval van pijpstroming zijn cilindrische coördinaten (r, θ, z) een natuurlijke keuze om de stroming te beschrijven, waarbij r de radiale afstand vanaf de middellijn van de pijp is, θ de hoekpositie en z de axiale afstand langs de pijp. De vergelijking van Laplace in cilindrische coördinaten, rekening houdend met radiale, hoekige en axiale variaties, wordt uitgedrukt als: <img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;"> Om de exacte vorm van ϕ in de pijp te vinden, zijn randvoorwaarden nodig. Bijvoorbeeld, bij de pijpwanden zou de no-slip-voorwaarde een tangentiële snelheid van nul impliceren, terwijl in het centrale irrotationele gebied de potentiaalfunctie de symmetrie van de pijp zou volgen. Door de vergelijking van Laplace op te lossen met deze voorwaarden, wordt het snelheidspotentiaal over het stromingsveld gedefinieerd. Zodra ϕ is bepaald, kunnen de snelheidscomponenten in cilindrische coördinaten worden afgeleid door partiële afgeleiden van ϕ te nemen. Deze componenten zijn: Radiale snelheid: <img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;"> Hoeksnelheid <img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;"> Axiale snelheid <img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;"> Deze uitdrukkingen voor snelheidscomponenten bevestigen dat de stroming voldoet aan de rotatievrije voorwaarde, aangezien ze voortkomen uit een gradiënt.

Snelheidspotentiaal

Consider water flowing steadily through a long, straight pipe with a uniform cross-section.
In central regions of the pipe, where flow is irrotational, a ...

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Differential Analysis of Fluid Flow

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velocity potential in pipe flow

Velocity potential in pipe flow describes the motion of fluid in the central region of a long, straight pipe with a uniform cross-section. In this region, the flow is steady, incompressible, and irrotational. Irrotational means the fluid particles do not spin around their own axes.

A scalar function called the velocity potential, written as φ, is used to describe this motion. The velocity field V is defined as the gradient of φ. Because the curl of a gradient is always zero, this form of V gives the flow no vorticity and keeps it irrotational.

For an incompressible fluid, the continuity equation requires the divergence of the velocity field to be zero. When V is written in terms of φ, this condition becomes Laplace's equation. In regions where the flow is both incompressible and irrotational, the velocity potential must satisfy this equation.

Pipe flow is often described with cylindrical coordinates: r, θ, and z. Here, r is the radial distance from the pipe centreline, θ is the angular position, and z is the distance along the pipe. In these coordinates, Laplace's equation includes radial, angular, and axial variation.

To find the exact form of φ in the pipe, boundary conditions are needed. At the pipe walls, the no-slip condition means the tangential velocity is zero. In the central irrotational region, the potential function must also match the symmetry of the pipe. Solving Laplace's equation with these conditions gives the velocity potential across the flow field.

Once φ is known, the velocity components can be found by taking partial derivatives. These components are the radial velocity, the angular velocity, and the axial velocity. Together, they show how the flow field comes from a gradient and therefore satisfies the irrotational condition.

Velocity Potential in Pipe Flow

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Velocity potential in pipe flow describes the motion of fluid in the central region of a long, straight pipe with a uniform cross-section. In this region, the flow is steady, incompressible, and irrotational. Irrotational means the fluid particles do not spin around their own axes. A scalar function called the velocity potential, written as φ, is used to describe this motion. The velocity field V is defined as the gradient of φ. Because the curl of a gradient is always zero, this form of V...

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Explore how velocity potential describes irrotational, incompressible flow in a straight pipe, and see how Laplace's equation and cylindrical coordinates apply.

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Velocity Potential in Pipe Flow

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