Studies on the Motion of Viscous Flows ~IV 



IV— Comparative Study of 



Hydrodynamical Variational Principles, 



Based on the Principle of 



Maximum Uniformity 



Paul Lieber - 



, . University of California ., 



- Berkeley, California ■,■_■.• 



In this paper I endeavor to communicate the salient features and results of 

 a study originating in 1953, which has since then been committed to the concep- 

 tion and application of variational principles to flow fields conditioned by the 

 complete Navier- Stokes equations. 



At the beginning of this search, we formulated two distinct types of varia- 

 tional principles which were motivated by essentially different considerations 

 and objectives. In one case we sought to formulate a statement as a variational 

 principle which renders, according to a prescribed procedure for performing 

 the variations, the complete Navier- Stokes equations, as its Euler- Lagrange 

 differential equations. In so doing, the variational methods of Rayleigh-Ritz, 

 Galerkin, and related methods, may be used to obtain approximate but neverthe- 

 less useful analytical representations of viscous flow fields, in a manner which 

 is analogous to the application of these variational methods in the mechanics of 

 solids, where the variational principles to which they are applied have already 

 been known for some time. Such a variational principle was formulated and ef- 

 fectively applied by Lieber and Wan, and is presented in the Proceedings of the 

 IX International Congress of Theoretical and Applied Mechanics, published in 

 Brussels, Belgium, in 1957 [ij. Since then, it has been successfully applied by 

 Wan and others (Prigogine and Shecter), with small modifications, to obtain use- 

 ful approximate mathematical representations of viscous flow fields produced 

 in nature. These successful applications of this variational principle attest to 

 its power and practical value. 



At this point, it is convenient and important to draw attention to the funda- 

 mental distinction that must be made between a variational principle and a 

 variational method, a distinction which evidently is not understood even by indi- 

 viduals who write comprehensive papers on the search for variational princi- 

 ples [2].* A variational principle is a proposition that refers to and conditions 



*Ref. (2) overlooks this distinction by not grasping the outstanding contribution 

 of M.A. Biot. 



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