- Source: Velocity potential
A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.
It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case,
∇
×
u
=
0
,
{\displaystyle \nabla \times \mathbf {u} =0\,,}
where u denotes the flow velocity. As a result, u can be represented as the gradient of a scalar function ϕ:
u
=
∇
φ
=
∂
φ
∂
x
i
+
∂
φ
∂
y
j
+
∂
φ
∂
z
k
.
{\displaystyle \mathbf {u} =\nabla \varphi \ ={\frac {\partial \varphi }{\partial x}}\mathbf {i} +{\frac {\partial \varphi }{\partial y}}\mathbf {j} +{\frac {\partial \varphi }{\partial z}}\mathbf {k} \,.}
ϕ is known as a velocity potential for u.
A velocity potential is not unique. If ϕ is a velocity potential, then ϕ + f(t) is also a velocity potential for u, where f(t) 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 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 instead of pressure p and/or particle velocity u.
∇
2
φ
−
1
c
2
∂
2
φ
∂
t
2
=
0
{\displaystyle \nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\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 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 \varphi }{\partial t}}\,.}
See also
Vorticity
Hamiltonian fluid mechanics
Potential flow
Potential flow around a circular cylinder
Notes
External links
Joukowski Transform Interactive WebApp
Kata Kunci Pencarian:
- Energi mekanis
- Manusia
- HMX
- Keanekaragaman hayati
- Xenon
- Fosforilasi oksidatif
- Pencatatan sumur
- Prof. Mohamad Ali Fulazzaky, Ph.D
- Daftar film Amerika tahun 1997
- HD 40307 g
- Velocity potential
- Potential flow
- Flow velocity
- Potential flow around a circular cylinder
- Escape velocity
- Velocity
- Potential energy
- Bernoulli's principle
- Potential gradient
- Membrane potential
