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


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

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
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:


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

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.

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


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

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 $$ \phi $$ such that


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

The scalar field $$\phi$$ 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 $$\bar{u}$$ (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):


 * $$\bar{u}=\frac{\dot{V}}{A}$$.