# Kelvin's minimum-energy theorem

## Kelvin's minimum-energy theorem

A theorem in fluid dynamics that pertains to the kinetic energy of an ideal fluid (that is, inviscid, incompressible, and irrotational) and provides uniqueness statements concerning the solution of potential-flow problems. The theorem states that the irrotational motion of a liquid occupying a simply connected region has less kinetic energy than any other motion consistent with the same normal motion of the boundary S. See Potentials

The implications of the minimum-energy of irrotational motion are: (1) irrotational motion is impossible in a simply connected region bounded by fixed walls since in this case the normal derivative of the velocity potential vanishes at all points on the boundary, and therefore, according to the energy theorem, the kinetic energy is zero, or the system is at rest; (2) irrotational motion is impossible in a fluid in which the velocity at infinity vanishes if the internal boundaries are also at rest; (3) if the velocity at infinity vanishes, then the irrotational motion due to prescribed motion of an internal boundary is unique; and (4) if a fluid is in motion with uniform velocity at infinity, then the irrotational motion due to prescribed motion of an internal boundary is unique. See Laplace's irrotational motion

## Kelvin's minimum-energy theorem

[′kel·vənz ¦min·ə·məm ′en·ər·jē ‚thir·əm]
(fluid mechanics)
The theorem that the irrotational motion of an incompressible, inviscid fluid occupying a simply connected region has less kinetic energy than any other fluid motion consistent with the boundary condition of zero relative velocity normal to the boundaries of the region.
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