Studies on the Motion of Viscous Flows— III 



In any iterative procedure working inwards and based on a Burgers 

 flow as a first approximation, it becomes important to construct 

 particular solutions of the non- homogeneous form of the Burgers 

 vorticity equation in which the successive estimates of the self- 

 convection effect appear as perturbation terms on the right-hand 

 side . . . 



. . . However, it must be clearly understood that boundary layer 

 flow must inevitably dominate the inner flow region sufficiently 

 far downstream. In such a region, non-linear self -convection 

 effects become comparable with Burgers convection ... 



These statements show that the ideas of inner and outer expansions are cen- 

 tral in his work. Although the last quotation shows, in a sense, the recognition 

 of the importance of our second iteration, it is used to refute the validity of the 

 process of iteration for the whole domain consistent with the ideas of inner and 

 outer expansions, as the following statement indicates: "... In its outer region, 

 this solution merges with a suitable Burgers flow but, owing to the finite differ- 

 ence of displacement thickness, not the one one obtains by a naive application 

 of linear theory right up to the boundary of the parabola. A blind iteration from 

 such a linear solution fails . . ." This statement shows the fundamental differ- 

 ences between his and our work. 



Starting with Stokes' [16] treatment of the creeping motion of a sphere, which 

 neglected the inertia terms completely, and Oseen's work [51,52,53] which by 

 taking into account, to some measure, these inertia terms, aimed at resolving 

 Whitehead's paradox [54], a large body of work has been based on their approach 

 to the external flow problems for low values of Reynolds number. Lamb's solu- 

 tion [18], based on Oseen's equations for a circular cylinder, has provided a 

 milestone for work on cylindrical objects. Stokes' paradox [16] in case of a cir- 

 cular cylinder is resolved in a sense by the use of Oseen's equations instead of 

 Stokes' equations for creeping motion. It is also resolved, as Bairstow [55] has 

 shown, by using Stokes' equations together with a flow field which is partially 

 bounded at infinity. S. Goldstein [56] has given an exact analytical solution of 

 Oseen's equation for the case of the steady flow of an incompressible viscous 

 fluid past a sphere. For the case of a circular cylinder, Faxen [57] provided 

 the solution. The solution given by Bairstow, Cave, and Lang [25] for a circular 

 cylinder is based on an extension of Lamb's treatment, Tomotika and Aoi [58] 

 have given similar solutions for a sphere and a circular cylinder along lines 

 following Goldstein's work on a sphere. Both of these works have carried out 

 calculations of drag for a circular cylinder in the range < R < 23, where R is 

 based on the diameter of the cylinder. This is just a little more than half the 

 range which we have examined in detail. As noted by Tritton [59], the results of 

 Bairstow et al. and Tamotika and Aoi are essentially the same as far as the drag 

 is concerned. Hence only the results of Bairstow et al. are plotted for compari- 

 son in Fig. IB. 



The methods of small perturbation in fluid mechanics are discussed in de- 

 tail by Van Dyke [42], Stokes' and Oseen's solutions are shown to be asymptotic 

 expansions of the solutions of the Navier- Stokes equations for small values, 



583 



