In fluid mechanics, Kelvin's minimum energy theorem (named after William Thomson, 1st Baron Kelvin who published it in 1849[1]) states that the steady irrotational motion of an incompressible fluid occupying a simply connected region has less kinetic energy than any other motion with the same normal component of velocity at the boundary (and, if the domain extends to infinity, with zero value values there).[2] [3] [4] [5]
Let
u
u1 |
u ⋅ n=u1 ⋅ n |
n
u ⋅ n=u1 ⋅ n=0 |
T1-T=
1 | |
2 |
\rho
2-u | |
\int(u | |
1 |
2) dV
can be rearranged to give
T1-T=
1 | |
2 |
\rho
2 dV | |
\int(u | |
1-u) |
+\rho\int(u1-u) ⋅ u dV.
Since
u
u=\nabla\phi
\int(u1-u) ⋅ \nabla\phi dV=\int\nabla ⋅ [(u1-u)\phi] dV-\int\phi\nabla ⋅ (u1-u) dV.
The second integral is identically zero for steady incompressible fluid, i.e.,
\nabla ⋅ u=\nabla ⋅ u1=0
\int(u1-u) ⋅ \nabla\phi dV=\int\phi(u1-u) ⋅ n dA
where the surface integral is zero since normal component of velocities are equal there. Thus, one concludes
T | ||||
|
\rho
2 dV | |
\int(u | |
1-u) |
\geq0
or in other words,
T1\geqT
u1=u