L/ieber 



flows, and to which we refer as a linear substructure of flow fields. This idea, 

 which led us to conjecture that all flow fields are essentially endowed with 

 fundamental linear restrictions, was then reinforced by observing that in the 

 class of potential flows, a linear substructure is preeminently distinguished by 

 the fact that the velocity field is completely governed there, by a linear partial- 

 differential equation. Moreover, this remains the case even though the force 

 fields of potential flows remain conditioned in the absence of viscous forces, by 

 nonlinear partial-differential equations. 



During 1961, Lieber worked with Shrikant Desai to resolve the indetermi- 

 nacy in a coefficient appearing in the analytical representation of flow fields ob- 

 tained from the linear restrictions implied by the hydrodynamical variational 

 principles, used in conjunction with the complete Navier-Stokes equations. The 

 difficulties incurred encouraged us to conceive and develop a mathematical al- 

 gorithm which has been effectively used in the construction of analytical repre- 

 sentations of flow fields, based on the complete Navier-Stokes equations and 

 realistic boundary conditions. In so doing, Desai gives particular emphasis in 

 his Ph.D. dissertation to the idea that potential flows are fundamental in the 

 development of actual viscous flows, and he incorporated this important idea 

 in the algorithm cited above, thereby putting it to very practical use. The algo- 

 rithm consists of an iteration procedure consisting of an infinite sequence of 

 iterations applied to the complete Navier-Stokes equations, the successive steps 

 of which are joined by linear differential relations. These relations are evi- 

 dently an aspect of the fundamental linear substructure of actual flow fields, 

 discussed earlier in this paper. 



The application of this algorithm has produced analytical representations of 

 steady flow fields around a circular cylinder for a range of Reynolds numbers 

 extended from .015 to 20. These representations have been used to calculate 

 flow fields which correspond to eighteen distinct values assigned to the Reynolds 

 number. The calculations reveal, for the first time, fine detail and features of 

 the structure of a real vortex formed behind a cylinder, and in particular that 

 the outer boundary of such a vortex is like a membrane at which vorticity and 

 dissipation are concentrated with relatively high intensity. These and other re- 

 sults which are presented in detail in a joint paper with Desai have been com- 

 pared with experiments and generally supported by them. 



When, however, we endeavor to construct analytical representations of time- 

 dependent flow fields which naturally develop at higher Reynolds number, we find 

 that it is evidently necessary in such cases to incorporate in the calculations, 

 where the iteration procedure is actually applied, a model of the historical de- 

 velopment of steady- state flows. Unless we do so, we cannot in principle pro- 

 ceed to calculate by the application of our algorithm space-time flows. This 

 again supports the ideas and conjectures set forth in another related paper, 

 which hold that hydrodynamical fields in general display, and are in general de- 

 termined by, physical aspects of a universal process of evolution that follows a 

 new and fundamental law. 



Comparatively recently, Lieber formulated with L. Teuscher a new hydro- 

 dynamical variational principle based on the principle of maximum uniformity 

 that is specifically designed for application to the calculation of steady- stratified 



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