Helmholtz minimum dissipation theorem

In fluid mechanics, Helmholtz minimum dissipation theorem (named after Hermann von Helmholtz who published it in 1868) 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. The theorem also has been studied by Diederik Korteweg in 1883 and by Lord Rayleigh in 1913.

Let ${\displaystyle \mathbf {u} ,\ p}$ and ${\displaystyle E={\frac {1}{2}}(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{T})}$ be the velocity, pressure and strain rate tensor of the Stokes flow and ${\displaystyle \mathbf {u} ',\ p'}$ and ${\displaystyle E'={\frac {1}{2}}(\nabla \mathbf {u} '+(\nabla \mathbf {u} ')^{T})}$ be the velocity, pressure and strain rate tensor of any other incompressible motion with ${\displaystyle \mathbf {u} =\mathbf {u} '}$ on the boundary. Let ${\displaystyle u_{i}}$ and ${\displaystyle e_{ij}}$ be the representation of velocity and strain tensor in index notation, where the index runs from one to three.

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