Studies on the Motion of Viscous Flows— IV 



flows under the simplifying assumption that viscous forces are negligible. In 

 the application of the principle of maximum uniformity to stratified flows, the 

 integral of the hydrodynamical variational principle is expressed in a positive, 

 definite, scalar measure of all the prevailing forces. This places it in direct 

 correspondence with the scalar force measure used in the statement of the 

 fundamental principle of maximum uniformity. We find that generalizing this 

 new hydrodynamical variational principle for application to time-dependent 

 stratified flows, again necessarily brings under consideration the idea that 

 time-dependent flows and therefore turbulent flows in particular are neces- 

 sarily strongly conditioned by the process of their historical development. 



CONCLUDING REMARKS - 



In this paper I have tried to demonstrate the crucial and unifying role of 

 the principle of maximum uniformity in revealing certain new and fundamental 

 features of hydrodynamical fields, such as (1) an underlying linear substruc- 

 ture, (2) a hydrodynamical principle of minimum dissipation, (3) fundamental 

 as well as necessary connections between spatial symmetry properties of actual 

 flow fields and time-dependent motion, and (4) the concept and discovery that 

 the space-time structures of hydrodynamical fields are in principle determined 

 by their historical development, and furthermore that the evolutionary aspects 

 of such fields are not in principle implied and therefore not mathematically de- 

 termined by the Navier-Stokes equations. This puts in perspective the signifi- 

 cance and role of various hydrodynamical variational principles which we have 

 formulated in order to give at least partial representation to the principle of 

 maximum uniformity in the context of classical hydrodynamics. 



The theoretical ground of the algorithm and the linear relations that connect 

 successive steps of an interaction process by which it is defined, are evidently 

 also contained in the principle of maximum uniformity, which does indeed ex- 

 plain why potential flows are distinguished and fundamental in the development 

 of viscous flows. The various formulations of hydrodynamical variational prin- 

 ciples cited in this paper, with the exception of the first which was conceived 

 with the object of rendering the Rayleigh-Ritz methods available to hydrody- 

 namical theory, are particular and restricted aspects of the principle of maxi- 

 mum uniformity, formulated in the context of hydrodynamical theory. The hy- 

 drodynamical variational principle, by which we recently formulated the 

 principle of maximum uniformity for stratified inviscid flows, bridges the two 

 principle objectives which directed our original work concerned with the formu- 

 lation of hydrodynamical variational principles. This new hydrodynamical vari- 

 ational principle achieves in part the two objectives simultaneously, because it 

 does afford a viable instrument for using effectively and economically the 

 Rayleigh-Ritz and Galerkin methods, for calculating with good approximation 

 steady- stratified flow fields, and because this variational principle is based on 

 a functional which represents a positive, definite, scalar measure of all the forces 

 acting in the field. We find that the generalization of this variational principle to 

 include viscous forces will necessarily require that we consider the flows as 

 time- as well as space-dependent, and the historical aspects of their development. 



679 



