Helmholtz minimum dissipation theorem

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 ${\displaystyle \nabla \times \nabla \times {\boldsymbol {\omega }}=0}$, where ${\displaystyle {\boldsymbol {\omega }}}$ is the vorticity vector. For example, the theorem also applies to unidirectional flows such as Couette flow and Hagen–Poiseuille flow, where nonlinear terms disappear automatically.