A velocity potential is a
scalar potential
In mathematical physics, scalar potential describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one p ...
used in
potential flow
In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
theory. It was introduced by
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles.
Continuum mec ...](_blank)
, when a continuum occupies a
simply-connected region and is
irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not chan ...
. In such a case,
where denotes the
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity 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 f ...
. As a result, can be represented as the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
of a
scalar function :
is known as a velocity potential for .
A velocity potential is not unique. If is a velocity potential, then is also a velocity potential for , where is a scalar function of time and can be constant. Velocity potentials are unique up to a constant, or a function solely of the temporal variable.
The
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
of a velocity potential is equal to the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of the corresponding flow. Hence if a velocity potential satisfies
Laplace equation, the
flow is
incompressible.
Unlike a
stream function
In fluid dynamics, two types of stream function (or streamfunction) are defined:
* The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, is defined for incompressible flow, incompressible (divergence-free ...
, a velocity potential can exist in three-dimensional flow.
Usage in acoustics
In theoretical
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
,
it is often desirable to work with the
acoustic wave equation of the velocity potential instead of pressure and/or
particle velocity
Particle velocity (denoted or ) is the velocity of a particle (real or imagined) in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with ...
.
Solving the wave equation for either field or field does not necessarily provide a simple answer for the other field. On the other hand, when is solved for, not only is found as given above, but is also easily found—from the (linearised)
Bernoulli equation for
irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not chan ...
and
unsteady flow—as
See also
*
Vorticity
In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point an ...
*
Hamiltonian fluid mechanics Hamiltonian fluid mechanics is the application of Hamiltonian mechanics, Hamiltonian methods to fluid mechanics. Note that this formalism only applies to non-dissipative fluids.
Irrotational barotropic flow
Take the simple example of a barotropic, ...
*
Potential flow
In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
*
Potential flow around a circular cylinder
Notes
External links
Joukowski Transform Interactive WebApp
Continuum mechanics
Physical quantities
{{Fluiddynamics-stub
Potentials