Desai and Lieber 



usually regarded as less than 1, of the Reynolds number. Two ideas generated 

 by S. Kaplun have made it possible to extend methods in this field of asymptotic 

 expansions. Lagerstrom [60] has stated them as: "(1) The discovery of suit- 

 able inner and outer limits; (2) An extension of the technique of matching be- 

 tween various expansions." The works of Lagerstrom and Cole [61], Kaplun 

 and Lagerstrom [62], and Kaplun [22,23] in this field are quite significant. 

 Kaplun has given a higher approximation solution than Lamb's for a circular 

 cylinder that is valid for small Reynolds numbers. We have plotted Lamb's as 

 well as Kaplun' s results for drag in Fig. IE. Lamb's expression for drag is as 

 follows: 



D 



477 / 1 \ 



/ \, Re = Reynolds number based on radius 



Re Re 



\ 0. 5 - y - log — / 



\ ^ 4 / 7 = 0.5772157, Euler's constant . 



Kaplun' s expression for drag includes one more term than Lamb's. It is 



" Re ^ ^ 



where 



Re 



0.5 - 7 - log — 

 4 



It is easy to see that both of these expressions become unbounded for Re = 

 3.73. For this value Lamb's expression becomes +oo and Kaplun' s becomes -oo, 

 though the latter obviously tends to -oo faster than does Lamb's, which tends to 

 + 00 as Re -^3.73. As noted by Van Dyke [42], all such higher approximations 

 will have expressions for drag which are expansions in powers of Aj. Hence 

 they cannot be meaningful beyond Re = 3.73, although their actual range of 

 validity is successively increased within these limits < Re < 3.73. 



By using ideas of inner and outer expansions and matching procedures, 

 Proudman and Pearson [63] have also obtained higher approximations to the 

 flow past a sphere and a circular cylinder than those represented by the solu- 

 tions of Stokes and Oseen. The results are essentially same as Kaplun' s for a 

 circular cylinder. Proudman and Pearson as well as Van Dyke [42] regard the 

 works of Bairstow et al. [25], Goldstein [56], and Tomotika and Aoi [58] as of 

 limited value. Proudman and Pearson [63] remark that "there is no point in 

 solving the linear equation (2.12) to a greater degree of approximation than that 

 of the inertial terms neglected by substituting the Oseen equations for Navier- 

 Stokes equations, and so the simple solution given by Lamb (1911) is as good 

 an approximation as it is possible to obtain from the linearized equation." The 

 equation (2.12) to which the remark refers is (V^.^ - r a/3x) v^2(^ - q^ van Dyke 

 [42] states that the approximation is qualitatively as well as quantitatively in- 

 valid at high Reynolds number and, to support his view, gives the reason that 

 Oseen' s approximation gives boundary layers whose thickness is of the order 

 R'l rather than W'^'"^, as in Prendtl's correct theory. Moreover he points to 

 Yamada's work [64] to invalidate qualitative results even at low Reynolds 



584 



