Desai and Lieber 



COMMENTS ON THE DISCUSSION OF DR. SCHMIECHEN 



Dr. M. Schmiechen's encouraging written comments are very much appreci- 

 ated, as they are evidently motivated by an exceptional insight into the nature of 

 our work, which is concerned with the foundations of the theory of viscous flows, 

 and by a grasp of its significance in producing an algorithm for constructing 

 analytical representations of flow fields using the complete Navier-Stokes equa- 

 tions and realistic boundary conditions. We are especially grateful to Dr. 

 Schmiechen, as his comments are somewhat characteristic of the encouraging 

 response to this work which some colleagues kindly communicated to me in per- 

 son during the symposium. Accordingly, the present reply affords the oppor- 

 tunity to also express here in writing my thanks to Professor R. Timman, Pro- 

 fessor E. Laitone, Dr. H. H. Chen, and Dr. N. Francev in this regard. 



Dr. Schmiechen's insight into our work no doubt stems partially from his 

 researches concerned with the hydrodynamical implications of the thermody- 

 namics of irreversible processes, and in particular with the principle of mini- 

 mum entropy production, which he cites in his comments and from which he 

 derives a very interesting criterion of minimum instability pertaining to the 

 development of the Karman vortex street. The reported criterion of minimum 

 instability appears highly significant in the context of our study, as it evidently 

 bears a correspondence to and may be a particular aspect of a general stability 

 principle which has emerged from our work. This stability principle and the 

 conceptual background which lead to its identification are discussed in some de- 

 tail in the paper appearing in the present studies with the title "Aspects of the 

 Principle of Maximum Uniformity: a New and Fvindamental Principle of 

 Mechanics." 



The theoretical basis of the materials presented in the six papers included 

 in the present studies were originally and conceptionally motivated by informa- 

 tion we obtained by using Carl Gauss's [l] and Heinrich Hertz's [2] formulations 

 of the principle of classical mechanics, and by introducing and underlining 

 therein the concept force which they, in fact, endeavored to completely eliminate 

 in their formulations by formal representations of geometrical constraints. 



This information which was obtained as a theorem on the distribution of in- 

 ternal forces for a hydrodynamically significant class of mechanical systems 

 [3], and which was then generalized by hypothesis to be an aspect of all mechani- 

 cal systems, was the theoretical basis for introducing the principle of minimum 

 dissipation as a general restriction on realizable flow fields in nature [4]. This 

 restriction was thus originally introduced in hydrodynamical theory with the 

 understanding that it augments and compliments the restrictions imposed by the 

 Navier-Stokes equations which were then and are still understood in our work to 

 admit a larger class of flows than the class of realizable flows. However, when 

 we originally introduced the principle of minimum dissipation in 1957 as a gen- 

 eral condition on realizable flows, we did not realize, as we do now, that the 

 Navier-Stokes equations do not in principle afford a criterion by which realiz- 

 able flows are selected in nature from a larger class of admissible flows which 

 also satisfy the physical principle expressed by the Navier-Stokes equations, 

 but which do not in general satisfy a condition of realizability. 



666 



