– The purpose of this paper is to point out a paradox in variational theory for viscous flows. Chien (1984) claimed that a variational principle of maximum power loses for viscous fluids was established, however, it violated the well-known Helmholtz’s principle.
– Restricted variables are introduced in the derivation, the first order and the second order of variation of the restricted variables are zero.
– An approximate variational principle of minimum power loses is established, which agrees with the Helmholtz’s principle, and the paradox is solved.
– This paper focusses on incompressible viscose flows, and the theory can be extended to compressible one and other viscose flows. It is still difficult to obtain a variational formulation for Navier-Stokes equations.
– The variational principle of minimum power loses can be directly used for numerical methods and analytical analysis.
– It is proved that Chien’s variational principle is a minimum principle.
