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Velocity potential

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Scalar potential used in fluid dynamics
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Within the applied mathematical study offluid dynamics andcontinuum mechanics, avelocity potential is ascalar potential used inpotential flow theory. It was introduced byJoseph-Louis Lagrange in 1788.[1]

Suppose a smoothvector fieldu{\displaystyle \mathbf {u} } in a simple connected region represents theflow velocity of a fluid at each point. This flow field is said to beirrotational when∇×u=0.{\displaystyle \nabla \times \mathbf {u} =0\,.}If the flow field is irrotational, then it can be also be represented as thegradient of ascalar functionϕ{\displaystyle \phi }:u=∇ϕ =∂ϕ∂xi+∂ϕ∂yj+∂ϕ∂zk.{\displaystyle \mathbf {u} =\nabla \phi \ ={\frac {\partial \phi }{\partial x}}\mathbf {i} +{\frac {\partial \phi }{\partial y}}\mathbf {j} +{\frac {\partial \phi }{\partial z}}\mathbf {k} \,.}

ϕ{\displaystyle \phi } is known as avelocity potential foru. Velocity potentials are unique up to a constant and a function solely of the temporal variable. So ifϕ(x,y,z){\displaystyle \phi (x,y,z)} is a velocity potential, thenϕ(x,y,z)+f(t)+C{\displaystyle \phi (x,y,z)+f(t)+C} generates the same flow field asϕ(x,y,z){\displaystyle \phi (x,y,z)}.

TheLaplacian of a velocity potential is equal to thedivergence of the corresponding flow. Hence if a velocity potential satisfiesLaplace equation, theflow isincompressible.

Unlike astream function, a velocity potential can exist in three-dimensional flow.

Usage in acoustics

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In theoreticalacoustics,[2] it is often desirable to work with theacoustic wave equation of the velocity potentialϕ{\displaystyle \phi } instead of pressurep and/orparticle velocityu.∇2ϕ−1c2∂2ϕ∂t2=0{\displaystyle \nabla ^{2}\phi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}=0}Solving the wave equation for eitherp field oru field does not necessarily provide a simple answer for the other field. On the other hand, whenϕ{\displaystyle \phi } is solved for, not only isu found as given above, butp is also easily found—from the (linearised)Bernoulli equation forirrotational andunsteady flow—asp=−ρ∂ϕ∂t.{\displaystyle p=-\rho {\frac {\partial \phi }{\partial t}}\,.}

See also

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Notes

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  1. ^Anderson, John (1998).A History of Aerodynamics. Cambridge University Press.ISBN 978-0521669559.[page needed]
  2. ^Pierce, A. D. (1994).Acoustics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America.ISBN 978-0883186121.[page needed]


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