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

Abstract

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.

Citation

Liu, H.Y., Si, N. and He, J.-H. (2016), "A short remark on Chien’s variational principle of maximum power losses for viscous fluids", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 26 No. 3/4, pp. 694-697. https://doi.org/10.1108/HFF-09-2015-0368