Desai and Lieber 



used in the present volume to construct analytical representations of viscous 

 incompressible flows is based. 



In concluding my response to Dr. Schmiechen's obviously inspiring dis- 

 cussion, it is relevant to present here the conjecture that the question of sta- 

 tistical stability versus the stability of individual flows as noted by Professor 

 R. Kraichnan may possibly be resolved, if we understand averaging to be an 

 aspect of uniformity and therefore to offer implicitly a condition of realization 

 for actual fields. This, in principle, is not included in the Navier-Stokes equa- 

 tions for the reasons given above, and which are given more extensively in 

 some of the papers we present in these studies. It is also relevant to draw 

 attention here to the possibility that the dramatic reduction of the friction co- 

 efficient obtained by the addition of minute quantities of certain polymers to 

 turbulent flow may also be an aspect of the principle of maximum uniformity. 

 We are trying to pinpoint the connection between them, with the guiding hy- 

 pothesis that the polymer material physically produces sites of high nonuni- 

 formity in the viscous flow fields, and therefore as a consequence of the prin- 

 ciple of maximum uniformity both assume the role of control centers that 

 dominate the evolution of actual flow fields. 



COMMENTS ON THE DISCUSSION OF DR. WIEGHARDT 



First, in Ref . 8, we presented in 1957 a formulation of a hydrodynamical 

 variational principle that gives the complete Navier-Stokes equations as the 

 Euler- Lagrange condition expressed in terms of the Eulerian description of 

 flows. As noted in the paper in the present studies entitled "Comparative 

 Study of Hydrodynamical Variational Principles, Based on the Principle of 

 Maximum Uniformity," this hydrodynamical variational principle has been suc- 

 cessively applied by us and subsequently by others for the purpose for which it 

 was invented. 



I believe that an adequate response to the important question Dr. Wieghardt 

 raises concerning the range of validity of the principle of minimum dissipation 

 is contained in my response to Dr. Schmiechen's discussion and in the materials 

 included in the six papers we present in these studies. It was this question 

 which in part motivated my extended response to Dr. Schmiechen's remarks. 



CONCLUDING REMARK 



In responding above to the comments of Dr. Schmiechen and Dr. K. Wie- 

 ghardt, a fundamental distinction is made between flow fields allowed by certain 

 established laws of physics and flow fields that are realized. This distinction 

 is grounded in the observation that the laws of classical mechanics are essen- 

 tially devoid of evolutionary content and information and that a principle of 

 realization is a principle of evolution, i.e., evolution is the process of 

 realization. 



670 



