Desai and Lieber 



between asymmetry and the time dependence of the flow field is shown to exist 

 from a consideration of solutions to these subsidiary equations, and a theorem 

 concerning them is proven. 



Steady motion is investigated in Part 2 of this paper. The governing sub- 

 sidiary equations for the first two iterations are solved, and the solutions are 

 obtained in the form of power series expansions of l ^c log^ r as well as 

 l/c logg (logg r+ 1), where r is the radial distance in polar coordinates and c 

 is a suitable constant scale factor. As was previously mentioned, only the 

 analysis using the second transformation is presented here; the analysis using 

 the first transformation is contained in Refs. 1 and 2. Explicit expressions for 

 drag and pressure are obtained with the help of these solutions. Streamline 

 field and separation are also discussed; expressions are obtained for the angle 

 of separation and the significance of streamline field is explained. 



Finally, in Part 3, the computed results and streamline fields are presented 

 in a series of graphs and discussed in detail, with reference to the existing body 

 of literature in the field. 



We may summarize the introduction to this paper by drawing the reader's 

 attention to the salient results and conclusions, and which may be pursued in 

 further detail by referring to Refs. 1 and 2. 



1. In this work the class of potential flows assumes a fundamental physical 

 as well as mathematical role in the construction of analytical representations of 

 steady and unsteady viscous-incompressible flow fields, which accord with the 

 Navier-Stokes equations and realistic boundary conditions. This is achieved by 

 developing an algorithm defined by an infinite sequence of iterations, successive 

 members of which are connected by linear differential relations, and by intro- 

 ducing a group of scale transformations that facilitate the numerical determina- 

 tion of the coefficients of the analytical representation of the flow by the bound- 

 ary conditions, with increasing Reynolds number. 



2. The fundamental and universal physical role, which according to the 

 present paper potential theory assumes in the development of viscous incom- 

 pressible flow fields in general and the linear differential relations connecting 

 successive steps in the iteration procedure that defines our algorithm, are un- 

 derstood here to correspond physically to fundamental aspects incurred in the 

 development of actual flows. These features free the application of the algorithm 

 presented here from the a priori restriction used in the application of small- 

 perturbation theories to the calculation of viscous incompressible flows at low 

 Reynolds number. 



3. The scale transformations introduced in the present work, to facilitate 

 the application of the analytical representation of flow fields obtained from the 

 algorithm to the numerical calculation of particular flow fields at increasing 

 Reynolds number, are shown in the present theory to comprise a group. The 

 group property of these scale transformations derives from the linear substruc- 

 ture cited above and plays a fundamental role in the present theory. Members 

 of this group of transformations may therefore, in principle, be applied succes- 

 sively to the contraction of the scale of one of the space variables of the theory, 



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