Flow velocity

In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of the fluid.

Definition
The flow velocity of a fluid is a vector field


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

which gives the velocity of an element of fluid at a point $$\mathbf{x}$$ at a time $$ t $$.

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
A fluid is incompressible if 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 fluid 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.

Vorticity
The vorticity, $$\omega$$, of a fluid 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
If an irrotational fluid occupies a simply-connected 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 fluid. (See Irrotational vector field.)