Desai and Lieber 



Figures 3A through 20F constitute eighteen sets of figures, each corre- 

 sponding to one of the eighteen discrete values of the Reynolds number at which 

 solutions are evaluated. Each of these sets is composed of six graphs. The 

 number, such as 3 in Fig. 3A, refers to the set belonging to a particular Reyn- 

 olds number, while the alphabetical characters A, B, c, D, E, and F refer to 

 the six graphs in that particular set according to the following scheme. 



Character A: Plots of Yi(j), j =1,2,3, and 4 against r. 



Character B: Plots of the drags CDl, CDPi, and CDVi against r. 



Character C: Plots of PRECI, PRETIM, and PREPIM against r. 



PRETIM = PRETl^g^, PREPIM = PREPi^^^ as defined earlier. 



Character D: Plots of ERi against S(H); < S < H. Here the symbol (H) 

 is used to signify that the range of S depends on H . 



Character E: Plots of PRESi, preti, PREPi, and PRECi against 6 . 



Character F: Plots of PRESS-PREC2, PRES2-PREC2, PREP2, PRET2, PRESl, 

 and PRESI against e. The reason why PRESS-PREC2 and 

 PRES2-PREC2 instead of PRESS, PRES2, and PREC2 are plotted 

 is as explained earlier. 



To examine salient features of these results, let us consider the set of 

 plots corresponding to the Reynolds number Re = 0.05, viz., Figs. 3A through 

 3F. The plots in the first three are all against r. They show the effect on the 

 first iteration solution for a given Re, here 0.05, of applying the boundary con- 

 ditions at various distances from the cylinder. In Fig. 3A the constants Yi (j), 

 J = 1, 2, 3, and 4 increase very rapidly in absolute magnitude below r = 7. 

 In general, they may behave erratically below a certain value of r . From r = 

 7 to r = 300 all the four constants behave asymptotically and tend to a limiting 

 set of values. However, the inherent numerical errors involved in computation 

 with finite precision and a large number of operations cause the calculation to 

 break down for r > 340. Since the drag coefficients depend explicitly on the 

 constants Yi(l) and Yi (2), Fig. 3B displays a similar behavior. So also is 

 the case with PRECI, pretim, and prepim in Fig. 3C. The numerical breakdown 

 which occurs for r > 340 with a double-precision program using the transfor- 

 mation r = ee'^^'-i, takes place for r > 22.198 with a double-precision program 

 using the transformation r = e^^ , and for r > 11 with a single-precision pro- 

 gram using the same later transformation. However, the distance r = 7 at 

 which the values for the constants Yi(J) stop changing rapidly and start behav- 

 ing asymptotically remains the same whatever precision program and trans- 

 formation are used. In fact, the results of computation using the transforma- 

 tion r = e'^^ show that in this case, figures corresponding to 3A, 3B, and 3C 

 have plots which remain the same up to r = 11 for both single- and double- 

 precision computations. However, with a different transformation of values of 

 Yi(j), the corresponding plots differ, as far as magnitude of the constants is 

 concerned. Whereas the numerical breakdown occurs for r > 11 with a single- 

 precision program, it is deferred to r > 22.198 by the use of a double-precision 



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