Studies on the Motion of Viscous Flows— IV 



force laws is necessary to obtain a physical theory which is determinate even in 

 the strictly mathematical sense. We know from experience, of course, that actual 

 materials cannot in general be mechanically characterized by a fixed constitutive 

 relation that remains constantly appropriate for all physical environments of the 

 material. It is therefore significant to recognize that in general the restricted 

 force laws used to depict appropriately the macromechanical properties of actual 

 materials, may change significantly as their physical environment changes. We 

 are therefore given to consider the concept of the evolution of force fields, i.e., 

 the idea that different kinds of force fields individually depicted by particular 

 constitutive relations, may evolve as the environment of the material changes. . 

 We have familiar examples of marked changes in the macromechanical proper- 

 ties of solids when they yield plastically, and of fluids when turbulence develops 

 in them. 



The second type of variational principle which Wan and Lieber formulated 

 in 1956 was motivated by an attempt to express and formulate in terms of the 

 parameters of classical hydrodynamics, explicit and general information on the 

 global distribution of internal forces in a many-body classical mechanical sys- 

 tem, which Lieber obtained by using Gauss's and Hertz's formulation of the 

 principles of mechanics. This was done with a fundamental modification, which 

 reintroduces and underlines in their formulations the irreducible fundamental 

 nature of force and its nonreducibility to purely abstract geometry [3]. 



It was the emergence of this fundamental theorem on the distribution of in- 

 ternal forces within the edifice of Gaussian-Hertzian mechanics, and the reali- 

 zation that this information is not rendered explicit without integration, by 

 Newtonian mechanics, that led to the concept "Categories of Information" and to 

 the realization that the various categories of information which were identified 

 are aspects of nature that assume a fundamental role in scientific inquiry and 

 in the development of mathematics [4]. Accordingly, it was realized that ques- 

 tions concerning the equivalence between various formulations of natural laws 

 which pertain to a particular domain of experience and experimentation cannot 

 be meaningfully considered and resolved without taking cognizance of the vari- 

 ous categories of information which were identified. 



These considerations become particularly relevant when there emerged in 

 our work the question of augmenting the Navier Stokes equations by formal 

 statements of the condition on flow fields purported to comply with the restric- 

 tions implied by the Navier-Stokes equations. More specifically, questions con- 

 cerning consistency and redundancy arise, when we introduce statements of in- 

 formation augmenting the Navier-Stokes equations, as in the case when we 

 formulate variational principles that indirectly express the global information 

 on the distribution of internal forces which we have obtained as a theorem by 

 using Gauss's and Hertz's formulation of the principles of classical mechanics. 

 The question concerning the equivalence of the various known formulations of 

 the principles of mechanics is particularly relevant at this point, because the 

 Navier-Stokes equations are based on Newtonian mechanics, from which general 

 information concerning the distribution of internal forces evidently cannot be 

 derived in the category of explicit information. 



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