Fluid velocity

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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

u = u ( x , t ) , {\displaystyle \mathbf {u} =\mathbf {u} (\mathbf {x} ,t),} {\displaystyle \mathbf {u} =\mathbf {u} (\mathbf {x} ,t),}

which gives the velocity of an element of fluid at a position x {\displaystyle \mathbf {x} \,} {\displaystyle \mathbf {x} \,} and time t . {\displaystyle t.\,} {\displaystyle t.\,}

The flow speed q is the length of the flow velocity vector[3]

q = ‖ u ‖ {\displaystyle q=\|\mathbf {u} \|} {\displaystyle 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 u {\displaystyle \mathbf {u} } {\displaystyle \mathbf {u} } does not vary with time. That is if

∂ u ∂ t = 0. {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}=0.} {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}=0.}

Incompressible flow

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

∇ ⋅ u = 0. {\displaystyle \nabla \cdot \mathbf {u} =0.} {\displaystyle \nabla \cdot \mathbf {u} =0.}

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

Irrotational flow

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

∇ × u = 0. {\displaystyle \nabla \times \mathbf {u} =0.} {\displaystyle \nabla \times \mathbf {u} =0.}

That is, if u {\displaystyle \mathbf {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 ,} {\displaystyle \Phi ,} with u = ∇ Φ . {\displaystyle \mathbf {u} =\nabla \Phi .} {\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.} {\displaystyle \Delta \Phi =0.}

Vorticity

The vorticity, ω {\displaystyle \omega } {\displaystyle \omega }, of a flow can be defined in terms of its flow velocity by

ω = ∇ × u . {\displaystyle \omega =\nabla \times \mathbf {u} .} {\displaystyle \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 ϕ {\displaystyle \phi } {\displaystyle \phi } such that

u = ∇ ϕ . {\displaystyle \mathbf {u} =\nabla \mathbf {\phi } .} {\displaystyle \mathbf {u} =\nabla \mathbf {\phi } .}

The scalar field ϕ {\displaystyle \phi } {\displaystyle \phi } is called the velocity potential for the flow. (See Irrotational vector field.)

Bulk velocity

In many engineering applications the local flow velocity u {\displaystyle \mathbf {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}}} {\displaystyle {\bar {u}}} (with the usual dimension of length per time), defined as the quotient between the volume flow rate V ˙ {\displaystyle {\dot {V}}} {\displaystyle {\dot {V}}} (with dimension of cubed length per time) and the cross sectional area A {\displaystyle A} {\displaystyle A} (with dimension of square length):

u ¯ = V ˙ A {\displaystyle {\bar {u}}={\frac {\dot {V}}{A}}} {\displaystyle {\bar {u}}={\frac {\dot {V}}{A}}}.

See also

References

  1. 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)
  2. 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)
  3. 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.