Studies on the Motion of Viscous Flows — IV 



Both the principle of minimum dissipation as we originally formulated it in 

 Ref. 5 and the hydrodynamical variational principle we subsequently formulated 

 in Ref. 6, produced some fundamental information and implications for hydro- 

 dynamical fields. These were obtained by requiring that the restrictions these 

 variational principles impose upon admissible flow fields be compatible with the 

 restrictions imposed by the conservation laws, including, of course, the Navier- 

 Stokes equations. It was in this way that we originally conceived the idea that 

 actual hydrodynamical fields are subject to linear differential restrictions, 

 which we now realize are not implied by the Navier- Stokes equations and which 

 must be explicitly expressed by statements such as variational principles, or 

 implicitly expressed by an analytical algorithm, which augment the Navier- 

 Stokes equations. We refer to these linear differential restrictions as the linear 

 substructure of actual hydrodynamical fields. Another result originally re- 

 vealed by the application of the variational principles of Refs. 5 and 6, concerns 

 necessary and evidently fundamental connections between spatial symmetry 

 features of flow fields and their dependence upon time. The necessary connec- 

 tion between the time-dependent features of hydrodynamical fields and the sym- 

 metry properties of their space-dependent structures, was first revealed by the 

 compatibility conditions obtained by formally requiring that the hydrodynamical 

 variational principles cited above be compatible with the restrictions imposed 

 by the Navier- Stokes equations. 



We then endeavored to construct analytical representations of actual flow 

 fields by jointly applying the linear differential restrictions on flow fields im- 

 plied by the variational principles, in conjunction with the nonlinear compatibility 

 equation which insures the compatibility of these linear restrictions with the 

 nonlinear restrictions implied by the Navier-Stokes equations. In so doing, we 

 restricted our attention and objective to the class of fully developed steady- state 

 flows, which are maintained by boundai^y conditions that are fixed in time. Thus, 

 by construction, we removed from our consideration the historical development 

 of these fields and of the boundary conditions by which they are maintained, an 

 aspect which we have since discovered to be fundamental and evidently essential 

 for the production in nature of space-time-dependent flow fields. This ad hoc 

 restriction which we used in trying to obtain actual steady- state flow fields from 

 the hydrodynamical variational principles noted in conjunction with the Navier- 

 Stokes equations, may explain the fact that in every case we were left with an 

 arbitrary coefficient not determined by the formal statement of the problem — 

 a statement which omits the historical development of steady- state flows, as 

 well as the historical development of the boundary conditions which maintain 

 them. If this explanation is valid, then the fact that we were consistently left 

 with an undetermined coefficient when certain essential historical aspects were 

 excluded by the formal statement of the problem, is then indeed a positive and 

 possibly important result. This conclusion obtains, because if the space-time 

 structure of steady- state flows depend in fact upon their historical development, 

 then obtaining an analytical representation of them without giving representa- 

 tion in the analysis to their historical development would be untenable. 



When we originally conceived and applied the hydrodynamical variational 

 principles used to augment the Navier-Stokes equations, we interpreted the 

 linear differential restrictions on the actual flow field implied by them as cor- 

 responding to real linear restrictions which are understood to exist in actual 



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