A short remark on Chien's variational principle of maximum power losses for viscous fluids

Purpose - 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. Design/methodology/approach - Restricted variables are introduced in the derivation, the first order and the second order of variation of the restricted variables are zero. Findings - An approximate variational principle of minimum power loses is established, which agrees with the Helmholtz's principle, and the paradox is solved. Research limitations/implications - 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. Practical implications - The variational principle of minimum power loses can be directly used for numerical methods and analytical analysis. Originality/value - It is proved that Chien's variational principle is a minimum principle.