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.

velocity potential

Velocity Potential

In steady, incompressible flow through a long, straight pipe with a uniform cross-section, the flow in the central region (far from the pipe walls) is irrotational. This irrotational nature means that fluid particles do not rotate around their axes, and a scalar function called the velocity potential, represented by &#981;, can be used to describe their movement. In irrotational flows, the velocity field V is defined as the gradient of the velocity potential:
<img alt="Divergence operator equation &#8711;&#8901;V=&#8711;&#966;, vector calculus concept in mathematics and physics." src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
Since taking the curl of a gradient is always zero, defining the velocity in terms of &#981; ensures the flow has no vorticity, making it irrotational.
For incompressible flows, the continuity equation states that the divergence of the velocity field must be zero:
<img alt="Divergence theorem equation (&#8711;&#183;V=0), vector field analysis, fluid dynamics concept." src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
Substituting V into this equation, the equation becomes:
<img alt="Laplace's equation &#8711;&#178;&#966;=0, mathematical expression, potential field, static equilibrium." src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;" data-altupdatedat="1762173717000" data-alt="Equation 3" />
This is known as Laplace's equation. In regions where the flow is irrotational and incompressible, the velocity potential &#981; must satisfy Laplace's equation.
In the case of pipe flow, cylindrical coordinates (r, &#952;, z) are a natural choice for describing the flow, where r is the radial distance from the pipe centreline, &#952; is the angular position, and z is the axial distance along the pipe. Laplace's equation in cylindrical coordinates, accounting for radial, angular, and axial variations, is expressed as:
<img alt="Partial differential equation &#8711;&#178;&#934;=0, Laplace's equation, mathematical analysis, formula." src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 4" />
Finding the exact form of &#981; in the pipe requires boundary conditions. For instance, at the pipe walls, the no-slip condition would imply zero tangential velocity, while in the central irrotational region, the potential function would follow the symmetry of the pipe. Solving Laplace's equation with these conditions defines the velocity potential across the flow field.
Once &#981; is determined, the velocity components in cylindrical coordinates can be derived by taking partial derivatives of &#981;. These components are:
Radial velocity:
<img alt="Velocity potential equation, Vr=&#8706;&#966;/&#8706;r, in fluid dynamics context." src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 5" />
Angular velocity
<img alt="angular velocity equation v&#952;; mathematical formula, partial derivative notation; educational use" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 6" />
Axial velocity
<img alt="Fluid dynamics equation, Vz=&#8706;&#966;/&#8706;z, diagram for velocity potential analysis." src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 7" />
These expressions for velocity components confirm that the flow satisfies the irrotational condition, as they stem from a gradient.

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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key terms and concepts

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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