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 a fluid. The length of the flow velocity vector is the flow speed.

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

Main article: 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

Main article: 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

Main article: 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.

Vorticity

Main article: Vorticity

The vorticity, \(\omega\), of a flow 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

Main article: Potential flow

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.)

Notes and references

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