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

Considere el agua que fluye constantemente a través de una tubería larga y recta con una sección transversal uniforme.
En las regiones centrales de la tubería, donde el flujo es irrotacional, un potencial de velocidad define el flujo, y la velocidad se expresa como el gradiente de este potencial.
La sustitución de los componentes de velocidad verifica directamente que el flujo satisface esta condición de irrotación.
El potencial de velocidad se aplica a flujos tridimensionales, como este flujo de tubería, mientras que la función de corriente es específica de flujos bidimensionales, como resultado de la conservación de la masa.
En el flujo incompresible, la ecuación de continuidad requiere que la divergencia del vector velocidad sea cero.
Al sustituir el potencial de velocidad, esta condición se convierte en la ecuación de Laplace, una ecuación diferencial clave que gobierna los flujos invisibles, incompresibles e irrotacionales.
En coordenadas cilíndricas, la ecuación de Laplace se ajusta a las variaciones radiales, angulares y axiales. La resolución de esta ecuación con condiciones de contorno en las paredes de la tubería define el potencial de velocidad para el flujo.
A partir de esto, se determina el campo de velocidad completo tomando derivadas espaciales, revelando el patrón de flujo dentro de la tubería.

potencial de velocidad

Velocity Potential

En un flujo constante e incompresible a través de una tubería larga y recta con una sección transversal uniforme, el flujo en la región central (lejos de las paredes de la tubería) es irrotacional. Esta naturaleza irrotacional significa que las partículas del fluido no rotan alrededor de sus ejes, y se puede utilizar una función escalar llamada potencial de velocidad, representada por ϕ, para describir su movimiento. En flujos irrotacionales, el campo de velocidad V se define como el gradiente del potencial de velocidad: <img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;"> Dado que tomar el rizo de un gradiente siempre es cero, definir la velocidad en términos de ϕ garantiza que el flujo no tenga vorticidad, lo que lo hace irrotacional. Para flujos incompresibles, la ecuación de continuidad establece que la divergencia del campo de velocidad debe ser cero: <img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;"> Sustituyendo V en esta ecuación, la ecuación se convierte en: <img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;"> Esto se conoce como ecuación de Laplace. En regiones donde el flujo es irrotacional e incompresible, el potencial de velocidad ϕ debe satisfacer la ecuación de Laplace. En el caso del flujo en tuberías, las coordenadas cilíndricas (r, θ, z) son una opción natural para describir el flujo, donde r es la distancia radial desde la línea central de la tubería, θ es la posición angular y z es la distancia axial a lo largo de la tubería. La ecuación de Laplace en coordenadas cilíndricas, que tiene en cuenta las variaciones radiales, angulares y axiales, se expresa como: <img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;"> Encontrar la forma exacta de ϕ en la tubería requiere condiciones de contorno. Por ejemplo, en las paredes de la tubería, la condición de no deslizamiento implicaría una velocidad tangencial cero, mientras que en la región irrotacional central, la función potencial seguiría la simetría de la tubería. Resolver la ecuación de Laplace con estas condiciones define el potencial de velocidad a través del campo de flujo. Una vez que se determina ϕ, los componentes de velocidad en coordenadas cilíndricas se pueden derivar tomando derivadas parciales de ϕ. Estos componentes son: Velocidad radial: <img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;"> Velocidad angular <img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;"> Velocidad axial <img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;"> Estas expresiones para los componentes de velocidad confirman que el flujo satisface la condición irrotacional, ya que surgen de un gradiente.

Potencial de velocidad

Considere el agua que fluye constantemente a través de una tubería larga y recta con una sección transversal uniforme.
En las regiones centrales de la tubería, donde el flujo es ...

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JoVE Core

Civil Engineering

Ingeniería Civil

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Differential Analysis of Fluid Flow

Análisis Diferencial del Flujo de Fluidos

KEY TERMS AND CONCEPTS

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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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civil engineering

Velocity Potential in Pipe Flow

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