flow velocity

{{Short description|Vector field which is used to mathematically describe the motion of a continuum}}

In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity{{cite book |author1=Duderstadt, James J. |author2=Martin, William R. | title= Transport theory | editor=Wiley-Interscience Publications | location= New York| year= 1979 | ISBN=978-0471044925|chapter=Chapter 4:The derivation of continuum description from transport equations|page=218}}{{cite book | author=Freidberg, Jeffrey P.|title=Plasma Physics and Fusion Energy|edition=1|editor=Cambridge University Press|location=Cambridge|year=2008| ISBN=978-0521733175|chapter=Chapter 10:A self-consistent two-fluid model|page=225}} 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

: $\mathbf{u}=\mathbf{u}(\mathbf{x},t),$

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{{cite book | first1=R. | last1=Courant | author1-link=Richard Courant | first2=K.O. | last2=Friedrichs | author2-link=Kurt Otto Friedrichs | edition=5th | orig-year=unabridged republication of the original edition of 1948 | isbn=0387902325 | pages=[https://archive.org/details/supersonicflowsh0000cour/page/24 24] | title=Supersonic Flow and Shock Waves | oclc=44071435 | publisher=Springer-Verlag New York Inc | year=1999 | series=Applied mathematical sciences | url=https://archive.org/details/supersonicflowsh0000cour/page/24 }}

: $q = \| \mathbf{u} \|$

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 $\mathbf{u}$ does not vary with time. That is if

: $\frac{\partial \mathbf{u}}{\partial t}=0.$

=Incompressible flow=

If a fluid is incompressible 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.

A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential $\Phi,$ with $\mathbf{u}=\nabla\Phi.$ If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: $\Delta\Phi=0.$

=Vorticity=

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

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

If the vorticity is zero, the flow is irrotational.

The velocity potential

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

: $\mathbf{u}=\nabla\mathbf{\phi}.$

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

Bulk velocity

In many engineering applications the local flow velocity $\mathbf{u}$ vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity $\bar{u}$ (with the usual dimension of length per time), defined as the quotient between the volume flow rate $\dot{V}$ (with dimension of cubed length per time) and the cross sectional area $A$ (with dimension of square length):

: $\bar{u}=\frac{\dot{V}}{A}$ .