Flow velocity

In continuum mechanics the macroscopic velocity, 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.

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

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

Thus in irrotational flow the vorticity is zero.

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.)

See also

• Particle velocity
• Velocity field
• Group velocity
• Drift velocity
• Vorticity
• Enstrophy
• Strain rate
• Stream function
• Pressure gradient