Studies on the Motion of Viscous Flows — III 



difference is reflected in the fact that the continuation of the iteration procedure 

 was not recognized as an instrument for constructing analytical representations 

 of actual flows at higher Reynolds numbers, particularly if the higher order it- 

 erations are used in conjunction with higher harmonics as used in our represen- 

 tation of the stream function. Our algorithm consists of an analytical procedure, 

 i.e., of a set of mathematically specified rules for constructing analytical rep- 

 resentations of viscous incompressible flow fields which are based on the com- 

 plete Navier-Stokes equations and on realistic boundary conditions which define 

 solid bodies fixed in the flow field. This algorithm is free of the a priori re- 

 strictions used in the application of the small-perturbation theory for the con- 

 struction of approximate solutions to the Navier-Stokes equations, and which, 

 consistent with the reasoning underlying small-perturbation theories, have been 

 severely limited by their authors (Oseen, Kaplun, Van Dyke, Lager strom, etc.) 

 to a very restricted range of Reynolds numbers. Among these cases, in those 

 in which solutions based on the potential flow theory are used as the basis for 

 the application of the perturbations, it is either implicitly assumed or explicitly 

 stated, that actual flows in general deviate strongly, i.e., significantly and thus 

 finitely from potential flows, and that only in such cases where these deviations 

 are very small, is it justifiable to use potential flow solutions as a basis for 

 constructing analytical representations of viscous flows by a method of succes- 

 sive approximations. The theoretical basis of our work has freed us from these 

 ad hoc restrictions by rendering a new interpretation and understanding of the 

 nature of potential flows, which ascribes to them a fundamental and distinguished 

 position in the development of actual flow fields and therefore necessarily en- 

 dows them with a dissipation process. The insight which revealed in our work 

 that potential flows are essentially and universally connected with the detailed 

 development of actual flow fields at all locations in the field, was inspired by 

 phenomenologieal considerations similar to those which reinforced the concep- 

 tual grounds on which we originally conceived the principle of minimum dissi- 

 pation as a fundamental restriction on the development of viscous flow fields — 

 a restriction not reported or implied by the Navier-Stokes equations. It is 

 indeed the absence of the interpretation given here to potential flows and of an 

 appreciation of their fundamental and universal role in the development of actual 

 viscous-incompressible flow fields, which accounts for the fact that it was pre- 

 viously assumed that potential flows can only be used to calculate viscous flows 

 which deviate from them only slightly, thus justifying the application of the 

 methods of small perturbation. The algorithm presented here is defined by an 

 infinite sequence of iterations, successive steps of which are connected by linear 

 differential relations. These linear differential relations are understood to rep- 

 resent actual linear restrictions which constrain the development of actual flow 

 fields. It is with this understanding that we assert that actual flow fields are 

 essentially endowed with a linear substructure, and that this linear substructure 

 frees the iteration procedure from restrictions adopted in the application of 

 small-perturbation methods. These differences are further brought out in Part 1 

 of the paper, where our equations are formulated, and in Part 3, where the re- 

 sults of our work are discussed. 



Subsidiary equations governing the coefficients of the Fourier expansions of 

 the stream functions of the first two iterations are obtained in Part 2. Expres- 

 sions for drag and pressure are obtained here in terms of these coefficients. 

 An intimate relation, which has been discussed by Lieber and Wan in their work, 



497 



