Flow velocity on arvau

Wikipedia / Flow velocity

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

which gives the velocity of an element of fluid

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

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 does not vary with time. That is if

Incompressible flow

If a fluid is incompressible the divergence is zero:

is a solenoidal vector field.

Irrotational flow

A flow is irrotational if the curl is zero:

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 If the flow is both irrotational and incompressible, the Laplacian

Vorticity

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

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 such that

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

Bulk velocity

vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity (with the usual dimension of length per time), defined as the quotient between the volume flow rate (with dimension of square length):