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

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

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

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

q = ‖ u ‖ {\displaystyle 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

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

∂ u ∂ t = 0. {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}=0.}

Incompressible flow

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

∇ ⋅ u = 0. {\displaystyle \nabla \cdot \mathbf {u} =0.}

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

Irrotational flow

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

∇ × u = 0. {\displaystyle \nabla \times \mathbf {u} =0.}

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

Vorticity

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

ω = ∇ × u . {\displaystyle \omega =\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 ϕ {\displaystyle \phi } such that

u = ∇ ϕ . {\displaystyle \mathbf {u} =\nabla \mathbf {\phi } .}

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

Bulk velocity

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

u ¯ = V ˙ A {\displaystyle {\bar {u}}={\frac {\dot {V}}{A}}}.

See also