Studies on the Motion of Viscous Flows— II 



Hydrodynamical Aspects of Evolution 



The Navier-Stokes equations have long been considered and are still con- 

 sidered to contain all of the essential information relevant to the development of 

 actual flow fields in materials well-modelled by the Newtonian fluid. In mathe- 

 matical terms, this is equivalent to saying that the Navier-Stokes equations are 

 supposed to imply this information mathematically, and consequently that it is, 

 in principle, obtainable v/ithout requiring the explicit statement of additional in- 

 formation. The reason the fundamental problem of theoretical hydrodynamics 

 has remained open for about a hundred years is that the Navier-Stokes equations 

 do not contain, in principle, all the information which is necessary to obtain 

 mathematically the kind of information which we so long expected of them; i.e., 

 analytical representations of actual flow fields. This section of the paper en- 

 deavors to explain on the basis of ideas and conclusions developed in the pre- 

 ceding sections, why, in principle, this is so. We shall point out that it is nec- 

 essary, in principle, to augment the information reported by the Navier-Stokes 

 theory as general restrictions on actual flow fields, by statements which directly 

 or indirectly pertain to the evolutionary development of such fields. The algo- 

 rithm presented in the joint paper with S. M. Desai, included in the proceedings 

 of this symposium, is not implied by the Navier-Stokes equations. It endeavors 

 to give tacit operational expression to some information concerning actual flow 

 fields contained in the principle of maximum uniformity — information which 

 evidently is not included in the Navier-Stokes equations. 



The Navier-Stokes equations formulate for a Newtonian fluid the law of 

 classical mechanics, which invokes the equilibrium of forces as a general con- 

 dition that applies to all elements of the fluid, and which is constantly maintained 

 at all locations and for all time. The information content of the Navier-Stokes 

 equations is therefore equivalent to the information content of this law of me- 

 chanics, which as explained earlier in this paper does not constitute a definition 

 of force, and therefore does not, in principle, determine the space-time devel- 

 opment of the forces between which this law expresses a simple relation — a 

 relation which is in fact independent of space and time, and which is consequently 

 devoid of historical information. The historical content and evolutionary aspect 

 with which, by the present thesis, all forces are essentially endowed is not 

 therefore implicitly or explicitly expressed by this law and consequently not by 

 the Navier-Stokes equations. If the space-time structure of actual flow fields 

 sustained by boundary conditions that are maintained constant with time depend 

 on the historical process by which these boundary conditions are actually pro- 

 duced in nature, then it follows that their dependence on the historical develop- 

 ment of the boundary conditions, i.e., on their evolution, is, in principle, not 

 implied and therefore not predictable by the Navier-Stokes equations. If this is 

 generally the case, as our work indicates, then it follows that actual flow fields 

 cannot, in principle, be predicted by the complete Navier-Stokes equations with- 

 out augmenting them with general and fundamental information about these fields 

 which they do not imply, and which specifically concerns their evolution. This 

 information is, I believe, contained in the principle of maximum uniformity. It 

 was given only an approximate representation in the original formulation of the 

 Principle of Minimum Dissipation (6), where it was introduced as a fundamental 

 restriction on viscous flow fields, which augments at all Reynolds numbers the 

 restrictions implied by the complete Navier-Stokes equations. Only in the 



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