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# Flow velocity

In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of the fluid.

## Definition

The flow velocity of a fluid is a vector field

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

which gives the velocity of an element of fluid at a point $\mathbf{x}$ at a time t.

## 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:

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

A fluid is incompressible if the divergence of $\mathbf{u}$ is zero:

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

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

### 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.

### Vorticity

Main article: Vorticity

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

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

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 φ such that

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

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