Flow velocity

Short description: Vector field which is used to mathematically describe the motion of a continuum

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

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

$q=\Vert \mathbf{𝐮}\Vert$

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 $\mathbf{𝐮}$ does not vary with time. That is if

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

Incompressible flow

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

$\mathrm{\nabla }\cdot \mathbf{𝐮}=0.$

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

Irrotational flow

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

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

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

Vorticity

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

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

If the vorticity is zero, the flow is irrotational.

The velocity potential

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

$\mathbf{𝐮}=\mathrm{\nabla }\mathit{\varphi }.$

The scalar field $\varphi$ 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 $\overline{u}$ (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):

$\overline{u}=\frac{\dot{V}}{A}$.

See also

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