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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.

Contents

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

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

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, ω, 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

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

\mathbf{u}=\nabla\mathbf{\phi}

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

 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Flow_velocity". A list of authors is available in Wikipedia.
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