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Flow velocity

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In continuum mechanics the macroscopic velocity, also flow velocity in fluid dynamics or drift velocity in electromagnetism, is a vector field which is used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar.

Contents

Definition

The flow velocity u of a fluid is a vector field

u = u ( x , t ) ,

which gives the velocity of an element of fluid at a position x and time t .

The flow speed q is the length of the flow velocity vector

q = | | 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 u does not vary with time. That is if

u t = 0.

Incompressible flow

If a fluid is incompressible the divergence of u is zero:

u = 0.

That is, if u is a solenoidal vector field.

Irrotational flow

A flow is irrotational if the curl of u is zero:

× u = 0.

That is, if 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 Φ , with u = Φ . If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: Δ Φ = 0.

Vorticity

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

ω = × 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

u = ϕ .

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

References

Flow velocity Wikipedia


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