studies on the Motion of Viscous Flows— III 



6 = 77/2, the sum also displays this behavior. K more harmonics are taken into 

 account this behavior will be smoothed out. In anticipation of this and for the 

 sake of clarity, the discontinuous behavior in Fig. 22D is smoothed out by draw- 

 ing tangential lines to opposite branches of the streamlines. Figures 22G and 

 22H, 23C and 23D, and 23G and 23H are also included to show that the discon- 

 tinuous behavior takes place for all Reynolds numbers. Figure 28B and similar 

 other plots of the second harmonic for higher Reynolds numbers show the stream- 

 lines intersecting each other near the ±45° lines. This is purely due to the fact 

 that the points on the streamlines are obtained as intersections of the radial 

 lines with the streamlines, and hence when these lines are more or less parallel 

 to each other their intersections cannot be determined accurately. In short, this 

 feature is attributable to the mode of obtaining points on the streamline, and is 

 not a property of the streamlines. ,^ - 



Figure 27C shows a tiny vortex behind the cylinder for Re = 1.0. But this 

 disappears when the second iteration contribution is added, and all that is left is 

 a wake without a vortex. The same happens at Re = 2.0, except that the vortex 

 in Fig. 28C is larger. However, at Re = 2.75, the vortex appears with both iter- 

 ations; but the second iteration vortex is much smaller than the first iteration 

 vortex. This shows that the effect of the second iteration is to delay the separa- 

 tion and also affect the size and structure of the vortex. The calculations for 

 Reynolds numbers between Re = 2.0 and 2.75 show that the vortex begins to ap- 

 pear in the second iteration plots from Re = 2.3 onwards, i.e., the flow separa- 

 tion begins at Re = 2.3. The vortex structure does not show fully rounded con- 

 tours, because only two harmonics are taken into account. There are three 

 noteworthy features here. If we observe Figs. 29G and 29H we see that the 

 gradients within the vortex are smaller than the outside flow field by some or- 

 der of magnitudes. This means that in the initial stages of the development of 

 a vortex its appearance may not be noticeable by the naked eye or even by a 

 microscope, because its size as well as the movement within it are extremely 

 small in the beginning. The experiments, therefore, must give a higher value of 

 Re for separation than does the theory. Further, the velocities in a vortex such 

 as in Fig. 31H are higher near the separation streamline and the cylinder wall 

 than near the center, in contrast to the case with potential vortices. This is, of 

 course, what is observed in real vortices. The vortex is here obtained as a 

 result of the addition of two harmonics which are continuous functions of the 

 space variables r and d and hence must be represented by a continuous func- 

 tion. Therefore, a vortex need not be represented by or viewed as a singular 

 structure in the flow field. With increasing Reynolds number the point of sepa- 

 ration first moves forward on the cylinder wall, and then attains a limiting posi- 

 tion as shown by the plots of the angles of separation a^ and a in Fig. 2B. 

 This agrees with what is observed in nature. 



DISCUSSION 



The existing explicit theoretical knowledge about flows of viscous fluids is, 

 for the most part, obtained from the Navier-Stokes equations by the application 

 of small-perturbation techniques [42]. Here we have attempted to depart from 

 this thinking. Although we have used an iterative process for solving the 



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