Studies on the Motion of Viscous Flows— III 



These formulations of the condition at infinity are found in many books on 

 fluid mechanics and in technical papers. In particular, references may be made 

 to books by Milne-Thomson [11], Lamb [18], Schlichting [19], Rosenhead [20], 

 and Landau and Lifshitz [21], and to papers by Kaplun [22,23], Southwell and 

 Squire [24], Bairstow, Cave, and Lang [25], and Hollingdale [26]. 



This manner of stating this condition is not entirely appropriate because it 

 conceals a very significant point. 



If we consider the ideas underlying such a condition we see that the condi- 

 tion stems from a feeling that the changes introduced in the flow field by an ob- 

 stacle in an infinite body of fluid must be finitely extended. The principle of 

 conservation of energy would imply that an infinite domain cannot be disturbed 

 everywhere at finite amplitudes. It might be asked what we mean by an infinite 

 body of fluid. Experiments show that for steady motion of a fluid around an ob- 

 stacle the flow field significantly far away is essentially the same as when the 

 obstacle was absent. What is actually and decisively observed is that the dis- 

 turbances, the physical changes in the flow field, attenuate with distance away 

 from the obstacle. 



There are two categories of variables involved in the measurement and ob- 

 servation of a physical process. The first category is the geometric category, 

 i.e., the category to which the coordinate variables r, 6, and t belong. The 

 second category is the dynamic category to which belong the variables u, v and 

 p. There is a significant difference between the variables belonging to these 

 two categories in relation to their measurements. 



Let us consider a disturbance at some point in a body of homogeneous and 

 isotropic fluid. Suppose there is some particular law of decay of this disturb- 

 ance as it propagates outwards in the fluid from the source of disturbance. This 

 law must be the same for all directions in the fluid because the fluid is assumed 

 to be isotropic. However, the intensity of the disturbance at the source may be 

 different in different directions. Now let us consider the measurement of the 

 intensity of the propagated disturbance at some distance i away from the source. 

 Suppose the measurements are made in some suitable but definite system of 

 units. The measured intensity at I can be expressed as some multiple of the 

 unit of measurement selected or as a percentage of the magnitude of the inten- 

 sity in this direction at the source. If the law of decay is such that increasing 

 the intensity at the source increases the intensity at I linearly, the percentage 

 which expresses the ratio of these two intensities will remain unchanged thereby. 

 Hence, considering other directions in which the intensity of disturbance at the 

 source may be different in magnitudes, we come to the conclusion that with such 

 a law of decay this percentage will remain unchanged at a distance I in all di- 

 rections; i.e., the isopercentage surface will be spherical, considering the 

 source to be at the center of the sphere. We can obtain an isopercentage sur- 

 face even if the law of decay does not exhibit the property of linearty mentioned 

 above. If the fluid is nonhomogeneous and nonisotropic, even then one can obtain 

 isopercentage surfaces, though the law of decay will depend on the direction of 

 propagation of the disturbance. In these cases the surface is not spherical. 

 However, on an isopercentage surface the magnitudes of the intensity of dis- 

 turbance will, in general, vary from point to point. Just as we can obtain an 



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