Studies on the Motion of Viscous Flows— III 



program based on the same transformation r = e*^^. It is still further de- 

 ferred to r > 340 by the use of a double-precision program based on the trans- 

 formation r = e^'^^'"!. This shows two things. First, the numerical breakdown 

 can be deferred so that it occurs at some later stage by increasing precision as 

 well as by using a transformation belonging to the group of transformations dis- 

 cussed in Part 2 of this paper. Second, the behavior of the solutions when the 

 boundary conditions are applied at points for which r is less than 7 is not due 

 to any numerical errors, but indicates that significant viscous effects cannot be 

 restricted to a domain bound by r < 7. Consequently, the imposition of the 

 boundary conditions in a domain bound by r < 7 is physically unrealistic and 

 hence unacceptable mathematically. In other words, the domain bound by r < 7 

 is the smallest domain for Re = 0.05 in which the viscous effects must be con- 

 sidered extremely significant, and therefore the physically infinite distance h| 

 cannot be smaller than r = 7. We define the lower bound of h* for a given 

 Reynolds number as the number below which the value of h* cannot be chosen. 

 The asymptotic behavior is terminated for r > 340. Below and near this value 

 of r, the solution for Re = 0.05 does not seem to change significantly with r. 

 The distance r = 340 relates to the results of the first iteration computations. 

 It can and does happen that at such a limiting distance the second iteration cal- 

 culations involve numerical breakdowns. To obviate this, a distance such as 

 r = 275 is chosen as h* such that the second iteration calculations can be car- 

 ried out successfully. In carrying these out with two or three other radii, it is 

 seen that the second iteration solution retains the asymptotic behavior. Hence 

 we take the physically infinite distance h* as the distance 275 at which mean- 

 ingful information is possible but beyond which the solution breaks down nu- 

 merically. It should be noted that the value of the physically infinite distance 

 then depends on the precision and transformation with which the computations 

 are carried out. This, of course, is true up to a point. However, improved pre- 

 cision and new transformations simply extend the range in which the asymptotic 

 behavior is obtained by shifting the point at which the numerical breakdown oc- 

 curs. And if the extended portion of the range is such that no significant change 

 occurs in the solution evaluated with r = 275, then improved precision and/or a 

 new transformation are unnecessary. Otherwise, improved precision and/or a 

 new transformation, if possible, are desirable. Herein lies the significance and 

 the strength from a numerical point of view of the idea of a physically infinite 

 distance. It should be noted that because h* is selected so that no numerical 

 breakdown takes place in either iterations, it is possible that the solution cor- 

 responding to this h* may give a first iteration drag value CDi which is higher 

 than the least value indicated by figures such as 3B. Figure 3D gives for the 

 solution evaluated at r = 275, a plot of ERi, the error involved in the simplify- 

 ing assumption B^Cr) = for n > 3, against S, where < S < H and RT = 

 h* = e^^"-i = 275, so that H = 1/c log (1 + log h* ) = 0.9448. Figures 3E and 

 3F show the harmonic components of the pressure fields due to first and second 

 iterations respectively. The plot of PRESS-PREC2 in Fig. 3F shows the distribution 

 of total pressure around the cylinder except for a constant term PREC2 which, 

 due to numerical errors, could not be evaluated accurately. It is, however, a 

 small value. 



The same features displayed by the set of graphs for Re = 0.05 are also 

 present in all the other sets of graphs for the remaining 17 values of Re. They 



577 



