Studies on the Motion of Viscous Flows— III 



Since disturbances may become, as they generally do, insignificant at 

 mathematically finite distances, a physically infinite distance may be finite 

 mathematically. Since a mathematically infinite distance embodies the idea of 

 indefinitely large distance, it would then follow that a physically infinite dis- 

 tance will always be less than, or at the most equal to, a mathematically infinite 

 distance, and hence that at a mathematically infinite distance also the physical 

 disturbances should be insignificant in the sense of being immeasurable within a 

 certain degree of accuracy. 



In contrast to the idea of physically infinite distance, a mathematically in- 

 finite distance depends on none of the four points on which the physically infinite 

 distance depends. Moreover, the mathematical idea of the point at infinity is 

 based on the idea of limit and hence a material point cannot be uniquely associ- 

 ated with such a point at infinity. This is the essential difference between the 

 abstract and the real. Herein lies both the strength and weakness of the mathe- 

 matically infinite distance. Its strength lies in the fact that in most instances it 

 leads to the possibility of a mathematical formulation which does not require 

 the information demanded under the above-mentioned four points and yet can 

 represent the corresponding physical process in essence. Its weakness is that 

 in its reinterpretation in an actual situation, where the domain is always finite 

 and hence the condition at infinity must be considered to apply at finite dis- 

 tances, one must necessarily resort to the idea of physically infinite distances. 

 Since the choice of the idea of mathematical infinity is governed by the criteria 

 of mathematical convenience, the idea of physically infinite distance may be used 

 if it is found to be more convenient than the other. It will then have the dual ad- 

 vantage of being both convenient and realistic. 



The idea of a physically infinite body of fluid involves two notions. One is 

 that of the physically infinite distance and the other is that of an inexhaustible 

 supply of fluid. Sources and sinks are devices by which the latter is accounted 

 for in some cases. For the corresponding mathematical idea, the indefinite ex- 

 tension implied by the point at infinity is sufficient. 



It might seem that before any use can be made of the idea of physically in- 

 finite distances one must necessarily possess the information about the nature 

 of the fluid, the intensity of the disturbance, degree of accuracy of the instru- 

 ments, etc. The nature of the fluid is adequately described by the knowledge of 

 physical constants like density p and viscosity m, as these are needed for any 

 mathematical formulation. That the other information need not be known a priori 

 for a mathematical formulation which makes use of the idea of physically infinite 

 distances can be shown by considering the following examples. 



The flow of an inviscid incompressible fluid around a circular cylinder of 

 radius 'a' must satisfy the harmonic equation v^/g = 0, where /q = ./'g(r,§) 

 is the stream function. 



The solution of the above equation, which renders the flow at infinity uniform 

 is given by . ... _, 



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