In fluid mechanics, Helmholtz minimum dissipation theorem (named after Hermann von Helmholtz who published it in 1868[1] [2]) states that the steady Stokes flow motion of an incompressible fluid has the smallest rate of dissipation than any other incompressible motion with the same velocity on the boundary.[3] [4] The theorem also has been studied by Diederik Korteweg in 1883[5] and by Lord Rayleigh in 1913.[6]
This theorem is, in fact, true for any fluid motion where the nonlinear term of the incompressible Navier-Stokes equations can be neglected or equivalently when
\nabla x \nabla x \boldsymbol{\omega}=0
\boldsymbol{\omega}
Let
u, p
E= | 1 |
2 |
(\nablau+(\nablau)T)
u', p'
E'= | 1 |
2 |
(\nablau'+(\nablau')T)
u=u'
ui
eij
Consider the following integral,
\begin{align} \int(eij'-eij)eij dV&=\int
\partial(ui'-ui) | |
\partialxj |
eij dV \end{align}
where in the above integral, only symmetrical part of the deformation tensor remains, because the contraction of symmetrical and antisymmetrical tensor is identically zero. Integration by parts gives
\int(eij'-eij)eij dV=\int(ui'-ui)eijnj dA-
1 | |
2 |
\int(ui'-ui)(\nabla2ui) dV.
The first integral is zero because velocity at the boundaries of the two fields are equal. Now, for the second integral, since
ui
\mu\nabla2ui=\nablap
\int(eij'-eij)eij dV=-
1 | |
2\mu |
\int(ui'-ui)
\partialp | |
\partialxi |
dV.
Again doing an Integration by parts gives
\int(eij'-eij)eij dV=-
1 | |
2\mu |
\intp(ui'-ui)ni dA+
1 | |
2\mu |
\intp
\partial(ui'-ui) | |
\partialxi |
dV.
\nabla ⋅ u=\nabla ⋅ u'=0
\int(eij'-eij)eij dV=0.
The total rate of viscous dissipation energy over the whole volume of the field
u'
D'=\int\Phi'dV=2\mu\inteij'eij' dV=2\mu\int[eijeij+eij'eij'-eijeij] dV
and after a rearrangement using above identity, we get
D'=2\mu\int[eijeij+(eij'-eij)(eij'-eij)] dV
If
D
u
D'=D+2\mu\int(eij'-eij)(eij'-eij) dV
The second integral is non-negative and zero only if
eij=eij'
The Poiseuille flow theorem[7] is a consequence of the Helmholtz theorem states that The steady laminar flow of an incompressible viscous fluid down a straight pipe of arbitrary cross-section is characterized by the property that its energy dissipation is least among all laminar (or spatially periodic) flows down the pipe which have the same total flux.