Studies on the Motion of Viscous Flows— III 



of this apparatus may require primitive notions yet to emerge. It is quite con- 

 ceivable that such primitive ideas may, as they have in the past, emerge from 

 extramathematical sources, and we believe that some of the results presented 

 here may be of this nature. 



The work presented here is part of a comprehensive exploration and study 

 in classical mechanics and hydrodynamics which was started in 1947 by Lieber 

 [3] and has been sustained and brought into sharper focus since then. This study 

 was initially motivated by a search for ways of extracting useful information 

 from the Navier-Stokes equations, which information until now has remained in- 

 accessible in terms of strictly mathematical procedures. In the works of Lieber 

 and Wan [4-7] can be found some attempts to materialize this desire by the in- 

 troduction of several significant ideas. 



In the course of the comprehensive study to which the work presented here 

 belongs, it was seen by Lieber (class notes in Relativity), that there exist in na- 

 ture distinct yet related categories of information. This came from the reali- 

 zation that the different known formulations of the principles of classical me- 

 chanics are only conditionally equivalent, and that questions concerning their 

 equivalence cannot be meaningfully considered without invoking the idea that 

 these categories exist in nature. It then became apparent that the task of ex- 

 tracting useful and testable information from the principles of classical me- 

 chanics is not strictly a mathematical endeavor, and that the feeding of explicit 

 information obtained from one formulation of the principles of mechanics into 

 another formulation can produce additional explicit information in analytical 

 terms. These observations and ideas led Lieber to the formulation of a funda- 

 mental theorem on the global distribution of internal forces which was obtained 

 on the basis of Gauss' principle of mechanics [8]. 



With the realization that this information should be implicitly contained in 

 the Navier-Stokes equations, attempts were made by Lieber and Wan to formu- 

 late statements in terms of the parameters and functions appearing in the 

 Navier-Stokes equations, in order to give this information formal representa- 

 tion in the framework of classical hydrodynamics. The dissipation mechanism 

 of Ref. 3 was used to construct a theoretical bridge between the internal forces 

 and the Rayleigh dissipation function, as it shows that dissipation is proportional 

 to the internal forces for a comparatively large class of initial conditions. In 

 this way a connection was established between the information obtained on the 

 global minimization of internal forces and a statement of the minimum dissipa- 

 tion of energy in real fluids. The physical content of this statement is restricted 

 by its mathematical formulation as given in Refs. 5 and 6, where a linear struc- 

 ture emerges for the governing equations. 



A step to give it a less restricted formulation has been made by Lieber and 

 Wan in Refs. 9 and 10 by postulating the existence of a fluid interface joining the 

 rotational and irrotational flow regimes, into which the flow field is divided 

 a priori. With the knowledge that the principle of minimum dissipation gives 

 only approximate representation to the information obtained from the theorem 

 on internal forces, they sought to give this information more complete represen- 

 tation by formulating another variational principle which maximizes a global 



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