Studies on the Motion of Viscous Flows— II 



The uniform connections are represented by what corresponds to a locally iso- 

 tropic stress field, whereas the nonuniform connections may be represented 

 symbolically by the stress deviator tensor. 



The constitutive properties by which specific materials are conventionally 

 identified, as well as their heat baths and thermal fluctuations, are all relevant 

 in producing at each instant a manifold of actual possibilities, available for the 

 selection at each instant, of a particular and preferred stress field as required 

 and selected by the principle of maximum uniformity. Strictly speaking, ac- 

 cording to this conception of the operation in nature of the principle of maximum 

 uniformity, the so-called constitutive properties are not strictly constant, but 

 may, according to this principle and the mental picture drawn above of its oper- 

 ation, undergo change, which is tantamount to a change of state or of mechanical 

 phase. In actual and familiar cases, what I am describing here is manifested in 

 the plastic yield of solids and in the turbulence in fluids. 



With this background, we can present a statement of the principle of maxi- 

 mum uniformity as it pertains specifically to the domain of classical mechanical 

 experience. An amplified statement of this principle will be given in a separate 

 volume, in the broader context of a unifying evolutionary principle which may 

 pertain to all aspects of nature and consequently to hydrodynamical and biologi- 

 cal phenomena in particular. The principle of maximum uniformity asserts — 

 that among the manifold of actual-possible stress fields which are immediately 

 and instantaneously available for selection in a continuously extended and 

 bounded material domain, and which accord with the following conditions and 

 aspects of the domain: (a) the instantaneous constitutive properties of the do- 

 main; (b) the temperature field and its fluctuations; (c) the forces impressed 

 and sustained at the boundaries and within the domain; (d) the established laws 

 of classical mechanics; (e) the principle of conservation of energy; and (f) the 

 appropriate equation of state —that the stress field which actually evolves, min- 

 imizes an integral of a positive measure of the shear stresses extending over 

 the whole domain. 



This is equivalent to maximizing a global measure of uniformity of the do- 

 main, since according to the ideas of the present paper the shear stress of a 

 differential element of the material is the direct manifestation in experience of 

 its nonuniform connections with the universe. 



This statement of the principle of maximum uniformity, as applied to con- 

 tinuously extended rheological materials, neglects nonuniformities in the inertia 

 forces as manifested by the nonhomogeneity in their spatial distribution. When 

 these are significant, they must, of course, be included in the total measure of 

 global nonuniformity. Indeed, in an application of the principle of maximum 

 tiniformity to stratified flows presented in a later part of these studies, the 

 global measure of nonuniformity includes only pressures and inertia forces, as 

 viscous forces may be neglected in comparison. 



As this statement of the principle depends incisively on the idea of a mani- 

 fold of actual-possible states of stress which are available for the selection of a 

 particular member, it is necessary to consider this concept in some detail. For 

 this purpose, we first introduce the concepts microstress and macrostress 



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