Studies on the Motion of Viscous Flows— III 



of minimum dissipation as a restriction on a development of viscous flows. As 

 noted, we recognized from the beginning however that the principle of minimum 

 dissipation does not give full expression in hydrodynamical terms even to the 

 restricted version of the principle of maximum uniformity as presented in Ref . 

 3. For this reason we formulated in 1957 in hydrodynamical terms [7] a more 

 comprehensive statement of the principle of maximum uniformity which in fact 

 rendered the principle of minimum dissipation as a theorem for a more re- 

 stricted class of viscous flows. In both of these formulations the integrands 

 are expressed in terms of quantities which have the physical dimension of 

 energy. 



The extended version of the principle of maximum uniformity as discussed 

 in the paper appearing in the present studies with the title "Aspects of the Prin- 

 ciple of the Maximum Uniformity: a New and Fundamental Principle of Me- 

 chanics," gives increasing emphasis to the idea that the most fundamental as- 

 pects of nonuniformity in nature are directly manifested by forces rather than 

 by energy. This subsequent theoretical development and an indeterminancy in- 

 curred in the application of the hydrodynamical variational principle presented 

 in [7] accounts, in part, for the procedure which led to the algorithm presented 

 in these studies for constructing analytical representations of viscous flows by 

 using the complete Navier- Stokes equations — a procedure which gives tacit ex- 

 pression to the principle of maximum uniformity, i.e., without explicitly refer- 

 ring to a global force measure of nonuniformity. For the above reasons we have 

 tried to formulate the extended version of the principle of maximum uniformity 

 in mathematical terms for Newtonian fluids by constructing an appropriate 

 global force measure. Concurrently, we have also endeavored to carry out the 

 same program in developing a kinetic theory of gases with internal degrees of 

 freedom in which the principle of maximum uniformity is formulated as a con- 

 dition of realization of actual states. We have in both cases achieved some suc- 

 cess. In the hydrodynamical case, we have formulated new hydrodynamical 

 principles which may be effectively used for the numerical calculation of steady, 

 inviscid, stratified flow fields. This is noted in the paper of the present studies 

 entitled "Comparative Studies of Hydrodynamical Principles, Based on the Prin- 

 ciple of Maximum Uniformity." These formulations are being extended, but with 

 difficulty, to include viscous forces and time-dependent forces as well. We are 

 doing this in two steps. First, the inclusion of time-dependent forces without 

 viscous forces. This we have been able to carry out for a significant class of 

 inviscid time-dependent flow fields by using a global force in the statement of 

 the appropriate variational principles. We have considerable difficulty, how- 

 ever, in carrying out the second step, which will include viscous as well as 

 time-dependent forces in the force functional of the variational principle which 

 expresses hydrodynamically the condition of realization of actual flow fields as 

 invoked by the principle of maximum uniformity. 



Concerning the embodiment of the principle of maximum uniformity in a 

 kinetic theory of gases and its application to specific situations, we have al- 

 ready obtained some encouraging results. In so doing, we find linear substruc- 

 tures underlying solutions to the nonlinear equations obtained from the applica- 

 tion of the principle to the kinetic theory gases, that are analogous to the linear 

 substructure of the complete Navier-Stokes equations on which the algorithm 



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