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. Suppose a smooth vector field u {\displaystyle \mathbf {u} } in a simple connected region represents the flow velocity of a fluid at each point. This flow field is said to be irrotational when
∇ × u = 0 . {\displaystyle \nabla \times \mathbf {u} =0\,.}
If the flow field is irrotational, then it can be also be represented as the gradient of a scalar function ϕ {\displaystyle \phi } :
u = ∇ ϕ = ∂ ϕ ∂ x i + ∂ ϕ ∂ y j + ∂ ϕ ∂ z k . {\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 a velocity potential for u. Velocity potentials are unique up to a constant and a function solely of the temporal variable. So if ϕ ( x , y , z ) {\displaystyle \phi (x,y,z)} is a velocity potential, then ϕ ( x , y , z ) + f ( t ) + C {\displaystyle \phi (x,y,z)+f(t)+C} generates the same flow field as ϕ ( x , y , z ) {\displaystyle \phi (x,y,z)} . 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, it is often desirable to work with the acoustic wave equation of the velocity potential ϕ {\displaystyle \phi } instead of pressure p and/or particle velocity u.
∇ 2 ϕ − 1 c 2 ∂ 2 ϕ ∂ t 2 = 0 {\displaystyle \nabla ^{2}\phi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}=0}
Solving the wave equation for either p field or u field does not necessarily provide a simple answer for the other field. On the other hand, when ϕ {\displaystyle \phi } is solved for, not only is u found as given above, but p is also easily found—from the (linearised) Bernoulli equation for irrotational and unsteady flow—as
p = − ρ ∂ ϕ ∂ t . {\displaystyle p=-\rho {\frac {\partial \phi }{\partial t}}\,.}
See also Vorticity Hamiltonian fluid mechanics Potential flow Potential flow around a circular cylinder
Notes