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In continuum mechanics the macroscopic velocity, also flow velocity in fluid dynamics or drift velocity in electromagnetism, 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

$q=\left|\right|\mathbf{u}\left|\right|$

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

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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

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If a fluid is incompressible the divergence of $\mathbf{u}$ is zero:

$\mathrm{\nabla }\cdot \mathbf{u}=0.$

That is, if $\mathbf{u}$ is a solenoidal vector field.

Irrotational flow

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A flow is irrotational if the curl of $\mathbf{u}$ is zero:

$\mathrm{\nabla }\times \mathbf{u}=0.$

That is, if $\mathbf{u}$ 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 $\mathrm{\Phi },$ with $\mathbf{u}=\mathrm{\nabla }\mathrm{\Phi }.$ If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: $\mathrm{\Delta }\mathrm{\Phi }=0.$

Vorticity

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

$\omega =\mathrm{\nabla }\times \mathbf{u}.$

Thus in irrotational flow the vorticity is zero.

The velocity potential

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field $\varphi$ such that

$\mathbf{u}=\mathrm{\nabla }\mathit{\varphi }$

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

Notes and references

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Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534