Studies on the Motion of Viscous Flows— III 



except the no- slip boundary condition is the most appropriate base flow for the 

 process of iteration. The solutions S^, S^, . . . , S^ define a linear substructure 

 to the solution S, and the equations L^^ri + ^n = Oj " - 1 define a system of 

 linear substructure equations underlying the Navier-Stokes equations. The equa- 

 tions for the first three iterations as given by Eqs. (1.47) to (1.58) inclusive are 

 the explicit expressions of these substructure equations for n = 1,2, and 3. We 

 emphasize that in this theory no idea of small perturbations about a given solu- 

 tion is involved and that there is no limitation imposed on the characteristic 

 parameter, the Reynolds number, of the flow field. Consequently, this theory 

 is not a small-perturbation theory. 



The potential flow solution around a circular cylinder was used by Wilson 

 (1904) [32], Boussinesq (1905) [33], Russel (1910) [34], and King (1914) [35] as 

 a means of convecting away heat from the cylinder. Later Burgers (1921) [36], 

 Zeilon (1926) [37], Southwell and Squire (1933) [24], Meksyn (1937) [38], and 

 recently, Pillow (1964) [39] have used it in a spirit of refinement over Ossen's 

 work. The conviction, at least tacitly shared by these authors, is that their 

 work is inherently restricted to flows that deviate only slightly from potential 

 flows. An immediate consequence of this convection is that these authors do 

 not consider their work as applicable close to the cylinder or in the wake. There 

 is, therefore, a conspicuous absence of the recognition of the crucial importance 

 which we have assigned to the higher iterations, among which the second itera- 

 tion equations appear to be of particular significance, as explained earlier. 

 Further, only in Southwell and Squire's work is there a clear recognition of the 

 validity of their equation for all Reynolds numbers. Burgers and Zeilon have 

 considered the case of ^ ^ o, i.e., the case of large Reynolds numbers, and 

 moreover, Zeilon has permitted convection by separated potential flows. Lewis 

 [40], in his paper, states: "Of course, it is not at all obvious which irrotational 

 motion is the one best suited in each particular case." Since the potential flow 

 solution (// = -(r- i/r) sin f approaches the potential uniform flow solution ^i = 

 -r sin d far away from a circular cylinder, there, if a real flow is viewed as a 

 slight deviation from the uniform flow, it just as well can be viewed as a slight 

 deviation from the potential flow given by t/' = -(r - 1/r) sin (9. Thus the di- 

 lemma stated by Lewis is natural when particular cases, in which some parts 

 of the complete flow field can be viewed as deviating slightly from some irro- 

 tational flow field, are considered in a technical spirit. In the present work 

 where an evolutionary point of view is taken, the potential flow = -(r - 

 1/r) sin plays a fundamental role, valid in the whole domain, as the base flow 

 from which deviations, not necessarily small in any sense, take place to accom- 

 modate the no- slip boundary condition. In this sense we have ascribed a kind of 

 reality to the potential flow solution under all flow conditions. Hence, even 

 though all the studies just cited indeed lead to equations and conditions which 

 are equivalent to our base flow and the first iteration equations and conditions, 

 the conceptual basis, motivation, and justification of our work are entirely dif- 

 ferent from those other studies. 



551 



