Английская Википедия:Flow velocity
Шаблон:Short description In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity[1][2] in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is scalar, the flow speed. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall).
Definition
The flow velocity u of a fluid is a vector field
- <math> \mathbf{u}=\mathbf{u}(\mathbf{x},t),</math>
which gives the velocity of an element of fluid at a position <math>\mathbf{x}\,</math> and time <math> t.\,</math>
The flow speed q is the length of the flow velocity vector[3]
- <math>q = \| \mathbf{u} \|</math>
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 <math> \mathbf{u}</math> does not vary with time. That is if
- <math> \frac{\partial \mathbf{u}}{\partial t}=0.</math>
Incompressible flow
If a fluid is incompressible the divergence of <math>\mathbf{u}</math> is zero:
- <math> \nabla\cdot\mathbf{u}=0.</math>
That is, if <math>\mathbf{u}</math> is a solenoidal vector field.
Irrotational flow
A flow is irrotational if the curl of <math>\mathbf{u}</math> is zero:
- <math> \nabla\times\mathbf{u}=0. </math>
That is, if <math>\mathbf{u}</math> 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 <math>\Phi,</math> with <math>\mathbf{u}=\nabla\Phi.</math> If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: <math>\Delta\Phi=0.</math>
Vorticity
The vorticity, <math>\omega</math>, of a flow can be defined in terms of its flow velocity by
- <math> \omega=\nabla\times\mathbf{u}.</math>
If the vorticity is zero, the flow is irrotational.
The velocity potential
Шаблон:Main article If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field <math> \phi </math> such that
- <math> \mathbf{u}=\nabla\mathbf{\phi}. </math>
The scalar field <math>\phi</math> is called the velocity potential for the flow. (See Irrotational vector field.)
Bulk velocity
In many engineering applications the local flow velocity <math> \mathbf{u}</math> vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity <math>\bar{u}</math> (with the usual dimension of length per time), defined as the quotient between the volume flow rate <math>\dot{V}</math> (with dimension of cubed length per time) and the cross sectional area <math>A</math> (with dimension of square length):
- <math>\bar{u}=\frac{\dot{V}}{A}</math>.
See also
- Displacement field (mechanics)
- Drift velocity
- Enstrophy
- Group velocity
- Particle velocity
- Pressure gradient
- Strain rate
- Strain-rate tensor
- Stream function
- Velocity potential
- Vorticity
- Wind velocity
References
