The flow velocity u of a fluid is a vector field
which gives the velocity of an element of fluid at a position x {\displaystyle \mathbf {x} \,} and time t . {\displaystyle t.\,}
The flow speed q is the length of the flow velocity vector3
and is a scalar field.
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:
Main article: Steady flow
The flow of a fluid is said to be steady if u {\displaystyle \mathbf {u} } does not vary with time. That is if
Main article: Incompressible flow
If a fluid is incompressible the divergence of u {\displaystyle \mathbf {u} } is zero:
That is, if u {\displaystyle \mathbf {u} } is a solenoidal vector field.
Main article: Irrotational flow
A flow is irrotational if the curl of u {\displaystyle \mathbf {u} } is zero:
That is, if u {\displaystyle \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 Φ , {\displaystyle \Phi ,} with u = ∇ Φ . {\displaystyle \mathbf {u} =\nabla \Phi .} If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: Δ Φ = 0. {\displaystyle \Delta \Phi =0.}
Main article: Vorticity
The vorticity, ω {\displaystyle \omega } , of a flow can be defined in terms of its flow velocity by
If the vorticity is zero, the flow is irrotational.
Main article: Potential flow
If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field ϕ {\displaystyle \phi } such that
The scalar field ϕ {\displaystyle \phi } is called the velocity potential for the flow. (See Irrotational vector field.)
In many engineering applications the local flow velocity u {\displaystyle \mathbf {u} } vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity u ¯ {\displaystyle {\bar {u}}} (with the usual dimension of length per time), defined as the quotient between the volume flow rate V ˙ {\displaystyle {\dot {V}}} (with dimension of cubed length per time) and the cross sectional area A {\displaystyle A} (with dimension of square length):
Duderstadt, James J.; Martin, William R. (1979). "Chapter 4:The derivation of continuum description from transport equations". In Wiley-Interscience Publications (ed.). Transport theory. New York. p. 218. ISBN 978-0471044925.{{cite book}}: CS1 maint: location missing publisher (link) 978-0471044925 ↩
Freidberg, Jeffrey P. (2008). "Chapter 10:A self-consistent two-fluid model". In Cambridge University Press (ed.). Plasma Physics and Fusion Energy (1 ed.). Cambridge. p. 225. ISBN 978-0521733175.{{cite book}}: CS1 maint: location missing publisher (link) 978-0521733175 ↩
Courant, R.; Friedrichs, K.O. (1999) [unabridged republication of the original edition of 1948]. Supersonic Flow and Shock Waves. Applied mathematical sciences (5th ed.). Springer-Verlag New York Inc. pp. 24. ISBN 0387902325. OCLC 44071435. 0387902325 ↩