Flow velocity

Definition

The flow velocity u of a fluid is a vector field

: \mathbf{u}=\mathbf{u}(\mathbf{x},t),

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

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

:q = | \mathbf{u} |

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

Main article: Steady flow

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

: \frac{\partial \mathbf{u}}{\partial t}=0.

Incompressible flow

Main article: Incompressible flow

If a fluid is incompressible the divergence of \mathbf{u} is zero:

: \nabla\cdot\mathbf{u}=0.

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

Irrotational flow

Main article: Irrotational flow

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

: \nabla\times\mathbf{u}=0.

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

Vorticity

Main article: Vorticity

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

: \omega=\nabla\times\mathbf{u}.

If the vorticity is zero, the flow is irrotational.

The velocity potential

Main article: Potential flow

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

: \mathbf{u}=\nabla\mathbf{\phi}.

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

Bulk velocity

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

:\bar{u}=\frac{\dot{V}}{A}.

References

References

1. (1979). "Transport theory".
2. Freidberg, Jeffrey P.. (2008). "Plasma Physics and Fusion Energy".
3. (1999). "Supersonic Flow and Shock Waves". Springer-Verlag New York Inc.