Studies on the Motion of Viscous Flows — III 



Since the wall has the surface as shown in Fig. B, if we interpret 'the cur- 

 vature of the wall' to mean the curvature of the surface of the wall at different 

 points on the surface then this curvature must be different at its different points. 

 In that case, according to the first assumption, the thickness must be the same 

 everywhere, while according to the second it cannot be so. This is a contradic- 

 tion. However, if we interpret 'the curvature of the wall' to mean the curvature 

 of the surface at various points after the irregularities are neglected and a 

 curve such as the one in Fig, C is considered, the contradiction appears in an- 

 other way. The curvature changes from point to point on the curve in Fig. C. 

 Consequently, the thickness of the layer of fluid adhering to it must change from 

 point to point. The resulting shape of the outer surface of the layer, therefore, 

 will be different from that of the wall when the irregularities are neglected. If 

 we now superimpose the original irregularities on this shape of the outer sur- 

 face of the layer, the final shape presented to the current will have irregulari- 

 ties which are oriented somewhat differently than before. Hence they cannot be 

 regarded as the same irregularities as those presented by the wall itself. Thus 

 we again reach a contradiction. , • ; ,' 



Looking at the two assumptions from another point of view, a deeper ques- 

 tion arises. The thickness of the layer may vary from point to point and ac- 

 cordingly be a local attribute of the layer. Then how can it be influenced by the 

 curvature of an imaginary surface obtained by neglecting the irregularities of 

 the surface of the wall and not by the curvature of the actual surface of the wall 

 which has these irregularities? Moreover, whether or not a surface variation 

 can be regarded as a surface irregularity should also depend on the actual 

 thickness of the layer as compared to the depth A. This would involve a serious 

 investigation into the idea of the 'relative scales' of different aspects of an ac- 

 tual physical process. 



Besides these questions there are other objections which Lighthill [14] has 

 clearly presented as follows: "Supporters of this view do not seem to have dis- 

 cussed the dynamics of such a layer, or thought about the necessarily continuous 

 variation of velocity across it which results from viscous action, or about the 

 effect on the fluid in this layer of the pressure gradients to which it is sub- 

 jected." It is then not difficult to see why this hypothesis has fallen into disfavor. 



About the first hypothesis, Lighthill says: ' -■ - ' ' 



Stokes (1851), in his great paper "On Effect of the Internal Fric- 

 tion of Fluids on the Motion of Pendulums," had shown that the 

 condition of zero relative velocity of the fluid at a solid surface 

 both was the most physically tenable boundary condition for the 

 equations of motion of a viscous fluid, and led to remarkable 

 agreement with a wide range of experiments in that problem, as 

 it had also in the capillary -tube resistance experiments of Poi- 

 seuille (1840) and Hagen (1839). 



Three questions arise in relation to this statement. First, how can one decide 

 that this condition is the most physically tenable condition? Second, in what way 

 is it the most physically tenable condition /or the equations of motion of a vis- 

 cous fluid, i.e., can it be deduced from or can it be shown to be the only condition 



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