Flow velocity


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.

Definition

The flow velocity 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, .

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:

teady 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:

: ablacdotmathbf{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:

: abla imesmathbf{u}=0.

That is, if mathbf{u} is an irrotational vector field.

Vorticity

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

: omega= abla imesmathbf{u}.

Thus in irrotational flow the vorticity is zero.

The velocity potential

If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field phi such that

: mathbf{u}= ablamathbf{phi}

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


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