## Frequent Links

# Flow velocity

In continuum mechanics the **macroscopic velocity**,^{[1]}^{[2]} also **flow velocity** in fluid dynamics or **drift velocity** in electromagnetism, is a vector field which is used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the **flow speed** and is a scalar.

## Contents

## Definition

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

- <math> \mathbf{u}=\mathbf{u}(\mathbf{x},t)</math>

which gives the velocity of an *element of fluid* at a position <math>\mathbf{x}\,</math> and time <math> t\,</math>.

The flow speed *q* is the length of the flow velocity vector^{[3]}

- <math>q = || \mathbf{u} ||</math>

and is a scalar field.

## 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:

### Steady flow

The flow of a fluid is said to be *steady* if <math> \mathbf{u}</math> does not vary with time. That is if

- <math> \frac{\partial \mathbf{u}}{\partial t}=0.</math>

### Incompressible flow

If a fluid is incompressible the divergence of <math>\mathbf{u}</math> is zero:

- <math> \nabla\cdot\mathbf{u}=0.</math>

That is, if <math>\mathbf{u}</math> is a solenoidal vector field.

### Irrotational flow

A flow is *irrotational* if the curl of <math>\mathbf{u}</math> is zero:

- <math> \nabla\times\mathbf{u}=0. </math>

That is, if <math>\mathbf{u}</math> 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 <math>\Phi,</math> with <math>\mathbf{u}=\nabla\Phi.</math> If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: <math>\Delta\Phi=0.</math>

### Vorticity

The *vorticity*, <math>\omega</math>, of a flow can be defined in terms of its flow velocity by

- <math> \omega=\nabla\times\mathbf{u}.</math>

Thus in irrotational flow the vorticity is zero.

## The velocity potential

If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field <math> \phi </math> such that

- <math> \mathbf{u}=\nabla\mathbf{\phi} </math>

The scalar field <math>\phi</math> is called the velocity potential for the flow. (See Irrotational vector field.)

## References

**^**Duderstadt, James J., Martin, William R. (1979). "Chapter 4:The derivation of continuum description from transport equations". In Wiley-Interscience Publications.*Transport theory*. New York. p. 218. ISBN 978-0471044925.**^**Freidberg, Jeffrey P. (2008). "Chapter 10:A self-consistent two-fluid model". In Cambridge University Press.*Plasma Physics and Fusion Energy*(1 ed.). Cambridge. p. 225. ISBN 978-0521733175.**^**Courant, R.; Friedrichs, K.O. (1999) [unabridged republication of the original edition of 1948].*Supersonic Flow and Shock Waves*. Applied mathematical sciences (5th ed.). Springer-Verlag New York Inc. p. 24. ISBN 0387902325. OCLC 44071435.

## See also

- Velocity gradient
- Velocity potential
- Drift velocity
- Group velocity
- Particle velocity
- Vorticity
- Enstrophy
- Strain rate
- Stream function
- Pressure gradient