{{Short description|Scalar potential used in fluid dynamics}} {{more citations needed|date=May 2014}} Within the applied mathematical study of fluid dynamics and continuum mechanics, a '''velocity potential''' is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.<ref>{{cite book|last=Anderson|first=John|title=A History of Aerodynamics|year=1998|publisher=Cambridge University Press|isbn=978-0521669559}}{{page needed|date=December 2017}}</ref>
Suppose a smooth vector field <math>\mathbf{u}</math> in a simple connected region represents the flow velocity of a fluid at each point. This flow field is said to be irrotational when <math display="block">\nabla \times \mathbf{u} =0 \,.</math> If the flow field is irrotational, then it can be also be represented as the gradient of a scalar function <math>\phi</math>: <math display="block"> \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} \,.</math>
<math>\phi</math> is known as a '''velocity potential''' for {{math|'''u'''}}. Velocity potentials are unique up to a constant and a function solely of the temporal variable. So if <math>\phi(x,y,z)</math> is a velocity potential, then <math>\phi(x,y,z)+f(t) + C</math> generates the same flow field as <math>\phi(x,y,z)</math>.
The Laplacian of a velocity potential is equal to the divergence of the corresponding flow. Hence if a velocity potential satisfies Laplace equation, the flow is incompressible.
Unlike a stream function, a velocity potential can exist in three-dimensional flow.
==Usage in acoustics==
In theoretical acoustics,<ref>{{cite book|last=Pierce |first=A. D.|title=Acoustics: An Introduction to Its Physical Principles and Applications|year=1994|publisher=Acoustical Society of America|isbn=978-0883186121}}{{page needed|date=December 2017}}</ref> it is often desirable to work with the acoustic wave equation of the velocity potential <math>\phi</math> instead of pressure {{mvar|p}} and/or particle velocity {{math|'''u'''}}. <math display="block"> \nabla ^2 \phi - \frac{1}{c^2} \frac{ \partial^2 \phi }{ \partial t ^2 } = 0 </math> Solving the wave equation for either {{mvar|p}} field or {{math|'''u'''}} field does not necessarily provide a simple answer for the other field. On the other hand, when <math>\phi</math> is solved for, not only is {{math|'''u'''}} found as given above, but {{mvar|p}} is also easily found—from the (linearised) Bernoulli equation for irrotational and unsteady flow—as <math display="block"> p = -\rho \frac{\partial\phi}{\partial t} \,.</math>
==See also== *Vorticity *Hamiltonian fluid mechanics *Potential flow *Potential flow around a circular cylinder
==Notes== {{Reflist}}
{{DEFAULTSORT:Velocity Potential}} Category:Continuum mechanics Category:Physical quantities
{{Fluiddynamics-stub}} Category:Potentials