Flow velocity

 ( Fluid Dynamics ) 

In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity

Freidberg, Jeffrey P. (2025). 9780521733175 ISBN 9780521733175

in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is scalar, the flow speed. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall).

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Definition The flow velocity u of a fluid is a vector field

$\backslash mathbf\{u\}=\backslash mathbf\{u\}(\backslash mathbf\{x\},t),$

which gives the velocity of an fluid parcel at a position $\backslash mathbf\{x\}\backslash ,$ and time $t.\backslash ,$

The flow speed q is the length of the flow velocity vector

$q\; =\; \backslash |\; \backslash mathbf\{u\}\; \backslash |$

and is a scalar field.

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Uses The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow:

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Steady flow The flow of a fluid is said to be steady if $\backslash mathbf\{u\}$ does not vary with time. That is if

$\backslash frac\{\backslash partial\; \backslash mathbf\{u\}\}\{\backslash partial\; t\}=0.$

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Incompressible flow If a fluid is incompressible the divergence of $\backslash mathbf\{u\}$ is zero:

$\backslash nabla\backslash cdot\backslash mathbf\{u\}=0.$

That is, if $\backslash mathbf\{u\}$ is a solenoidal vector field.

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Irrotational flow A flow is irrotational if the curl of $\backslash mathbf\{u\}$ is zero:

$\backslash nabla\backslash times\backslash mathbf\{u\}=0.$

That is, if $\backslash mathbf\{u\}$ is an irrotational vector field.

A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential $\backslash Phi,$ with $\backslash mathbf\{u\}=\backslash nabla\backslash Phi.$ If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: $\backslash Delta\backslash Phi=0.$

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Vorticity The vorticity, $\backslash omega$, of a flow can be defined in terms of its flow velocity by

$\backslash omega=\backslash nabla\backslash times\backslash mathbf\{u\}.$

If the vorticity is zero, the flow is irrotational.

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The velocity potential If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field $\backslash phi$ such that

$\backslash mathbf\{u\}=\backslash nabla\backslash mathbf\{\backslash phi\}.$

The scalar field $\backslash phi$ is called the velocity potential for the flow. (See Irrotational vector field.)

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Bulk velocity In many engineering applications the local flow velocity $\backslash mathbf\{u\}$ vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity $\backslash bar\{u\}$ (with the usual dimension of length per time), defined as the quotient between the volume flow rate $\backslash dot\{V\}$ (with dimension of cubed length per time) and the cross sectional area $A$ (with dimension of square length):

$\backslash bar\{u\}=\backslash frac\{\backslash dot\{V\}\}\{A\}$.

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See also

• Displacement field (mechanics)
• Drift velocity
• Enstrophy
• Group velocity
• Particle velocity
• Pitot tube
• Pressure gradient
• Strain rate
• Strain-rate tensor
• Stream function
• Velocity potential
• Vorticity
• Wind velocity

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