Studies on the Motion of Viscous Flows— III 



number. We think that these views are not well justified. It is true that Kap- 

 lun's expression for drag represents a higher approximation than Lamb's ex- 

 pression to which, surely, the expressions of Bairstow et al. and Tomotika 

 and Aoi reduce for the range of Reynolds number in which both Lamb's and 

 Kaplun's expressions are meaningful. However the expressions due to Bair- 

 stow et al. and Tomotika and Aoi are asymptotic in nature with increasing 

 Reynolds number and show no unboundedness, at least within the range they 

 have investigated, whereas those due to Lamb and Kaplun are otherwise and 

 become unbounded for Re = 3.73. The results that do not become unbounded are 

 more of value, even if they are quantitatively somewhat different from the ex- 

 act results produced by those expressions that do. This shows that expansions 

 in terms of Re as a perturbation parameter, as obtained by Lamb and Kaplun, 

 are not mathematically equivalent to the solutions obtained by Bairstow et al., 

 Tomotika and Aoi, and others, except in a narrow range, for otherwise, they 

 all should show unboundedness at Re = 3.73. The range < Re < 12 which is 

 the one investigated by Tomotika and Aoi as well as Bairstow and others is a 

 range in which the assumptions of the boundary layer theory are invalid. Con- 

 sequently, the thickness of the boundary layer argument cannot be applied to 

 this range. For this range, then, these works cannot be invalidated totally on 

 this count. As for Yamada's work [64], his results as shown in Fig. 3 of Ref. 

 64 do not seem to be correct. The reasons are as follows. 



The experimental work of Thom shows that for R = 3.5 the maximum 

 stagnation pressure in front of the cylinder is p - Po/(/ov2/2) = 2.3. Figure 6 

 in his work shows that for higher values of R, this must be decreasing. Conse- 

 quently for R = 4, the value must be less than 2.3. Since the pressure drag is 

 somewhat directly related to, and has a value in excess of, that of the maximum 

 stagnation pressure for this value of Reynolds number, we can also estimate 

 what value this maximum pressure may possibly have. From Tritton's work 

 [59] the value of total drag at R = 4 is about 4.85. According to Oseen's theory 

 the pressure drag is half the total drag. This would give the pressure drag 

 2.425. On the other hand, if we take the pressure drag as 0.65 times the total 

 drag as found by Kawaguti [65] and as can be roughly estimated from Fig. 8 in 

 Thom's work [66], it turns out to be 3.152. In any case, then, the maximum 

 pressure cannot be larger than 3.152. However, Fig. 3 in Yamada's work [64] 

 shows this value obtained by considering the exact Navier-Stokes equations to 

 be 5.2, which is much too large in comparison to the maximum estimated value 

 3.152 and the possible experimental value which may be less than 2.3. Hence 

 we cannot but conclude that Yamada's results are in error. Consequently, Van 

 Dyke's statement, which is based on Yamada's work, has to be discounted. 



The method of inner and outer expansions has been utilized by Blair et al. 

 [43], Brenner [67], Brenner and Cox [68], Caswell and Schwartz [69], Chester 

 [70], and many other authors. But the limitations of the small-perturbation 

 theories are too severe to help us understand the evolution of a flow field for 

 the complete range of a characteristic parameter. Lagerstrom [60] rightly 

 considers that the main importance of their work lies in the analysis of basic 

 problems in asymptotic expansions. 



585 



