A short remark on Chien’s variational principle of maximum power losses for viscous fluids
International Journal of Numerical Methods for Heat & Fluid Flow
ISSN: 0961-5539
Article publication date: 3 May 2016
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.
Keywords
Acknowledgements
The work is supported by Shanghai Education Foundation for Young Scientists (98QN47) and National Key Basic Research Special Fund of China (No. G1998020318).
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
Publisher
:Emerald Group Publishing Limited
Copyright © 2016, Emerald Group Publishing Limited