Desai and Lieber 



measure of uniformity. The results obtained from both of these variational 

 principles share a common denominator, by showing: 



1. That the velocity fields are conditioned by linear partial differential 

 equations, thereby suggesting a linear substructure underlying solutions to 

 Navier -Stokes equations for actual flows. 



2. That there exist necessary connections between the global geometrical 

 symmetry properties of a body and the production of time -dependent flows under 

 time -independent boundary conditions. 



3. That there is the necessity to postulate a fluid interface joining the ro- 

 tational flow with a potential flow extending from the interface to infinity. 



An hypothesis has come out of our work which is strongly related to the 

 principle of minimum dissipation and to its basis as established in the theorem 

 on the global distribution of internal forces, obtained by Lieber [8]. The hypoth- 

 esis is that solutions to the Navier -Stokes equations for actual flows tend every- 

 where, as far as the actual boundary conditions permit, to approach asymptoti- 

 cally solutions for a class of ideal flows which satisfy the Navier-Stokes equa- 

 tions everywhere and a set of ideal boundary conditions at solid boundaries. 

 The ideal flows of the Navier-Stokes equations are the class of potential flows 

 which can dissipate only under very special conditions at fluid interfaces joining 

 rotational-irrotational flow regimes [9,10j. Since these flows are kinematically 

 determined by conditions expressed as linear partial differential equations, we 

 may conjecture from the above hypothesis, that solutions to the Navier-Stokes 

 equations for actual flows which are formally represented by an iterative proc- 

 ess applied to a well-defined sequence of functions, must converge asymptoti- 

 cally toward, and are thus bounded by, functions which satisfy linear equations. 

 Such is the theoretical background of our work. 



In the present work, the iterative process is assumed to have a fundamental 

 validity, and governing equations for successive iterations are obtained by as- 

 serting the fundamental role of the potential flow as a base flow that is valid in 

 the whole domain for all flow conditions to start the process of iteration. This 

 is done on the basis of physical and mathematical reasoning. A real flow is 

 viewed as a deviation, not necessarily small, from the basic potential flow. The 

 linear equations governing the iterations are called here linear substructure 

 equations. It is shown that at least two iterations are necessary and are suffi- 

 cient for the restricted range of Reynolds number to represent the flow field 

 around a circular cylinder. Boundary conditions are discussed and an idea of a 

 physically infinite distance is introduced. 



Historically, starting with Boussinesq, various authors have used the poten- 

 tial solution with a conviction that the results so obtained describe flows which 

 deviate only slightly from potential flows, thus ruling out, a priori, any consid- 

 eration of the regions close to the cylinder and in the wake. The governing 

 equation obtained by them, which has been recently called "Burger's equation" 

 by Pillow, is formally equivalent to our base flow and first iteration equations 

 taken together. The conceptual basis, motivation, and justification — mathemati- 

 cal and physical — of our work is, however, entirely different from theirs. This 



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