Studies on the Motion of Viscous Flows— III 



h = H This is defined either by the transformation 

 h = H = 1 /c log^ h|, or by the transformation 

 h - H - 1/c' log^(log^ h*+ 1) ,^ ...,.:,.,,,,.. 



a Angle of separation 



a J Angle of separation obtained without considering 

 iterations higher than the first 



INTRODUCTION AND THEORETICAL BACKGROUND 



It is well known that there is a vast gap in our theoretical understanding of 

 the flows around solid obstacles. At the extreme values, viz., Re -» and Re-> o, 

 of the characteristic flow parameter Re, the Reynolds number, we have some 

 insight, but in the vast intermediate range our lack of knowledge is disquieting. 

 The usual remarks about the difficulty of solving the highly nonlinear Navier- 

 Stokes equations are, accordingly, quite challenging. The small perturbation 

 theories which deal with flows with Re^ o set up a priori limitations to what is 

 possible to investigate and then use a mathematical apparatus which is consist- 

 ent with and circumscribed by these self-imposed limitations. Consequently, 

 the possibility of being able to obtain a coherent and complete description is 

 there abandoned at the beginning. For Re^*, where the boundary layer theory 

 has prevailed, no satisfying description and explanation of the evolution and 

 structure of the wake region can be found, because the assumptions of that theory 

 make it invalid for these regions. It is evident that without this knowledge and 

 understanding of these regions, the mechanism of turbulence still remains an 

 open question, as does its theoretical foundation. 



During a period of more than a century, a large body of experimental work 

 has accumulated. Thus information about the flows as they exist in nature is 

 not lacking. Theoretical work and, wherever possible, its experimental confir- 

 mation on all aspects of fluid mechanics so far seem to indicate that the Navier- 

 Stokes equations do contain implicitly all the essential information for the flows 

 of a large class of fluids. It appears, therefore, that if mathematical knowledge 

 about the theory of nonlinear partial differential equations, in particular the 

 Navier-Stokes equations, were sufficiently advanced, we could have all the in- 

 formation about flow structures etc. in explicit and detailed form. The question 

 then is, since we do not have this mathematical knowledge, how can we obtain it 

 in the first place. 



It is our belief that to make explicit any information from the Navier-Stokes 

 equations, which are presumed to contain it implicitly, we must provide some 

 prior information to initiate a process which leads to the information desired in 

 explicit form. Evidently, the prior information to be provided must be consist- 

 ent with the implicit information embodied in the Navier-Stokes equations. 

 Therefore, an important question needs to be resolved first; i.e., what informa- 

 tion should be provided and how can one be assured about its consistency with 

 the implicit information embodied in the Navier-Stokes equations? Since we 

 feel reasonably certain that the Navier-Stokes equations do represent the physi- 

 cal laws governing the behavior of a class of fluids and since we have a large 



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