r- .-■..;./;• ' ■:. Desai and Lieber 



As we have already noted in the Introduction, Eqs. (1.34) and (1.35) for n = i 

 together are equivalent only formally to the so-called 'Burger's Equation' ob- 

 tained by various authors. After assuming, a priori, that the flow deviates only 

 slightly from potential flows, they set u = ug + uj, v = vg + v^, \p=\jjQ + \pi and 

 then argue that the nonlinear terms can be neglected to arrive at their govern- 

 ing equations. This procedure of obtaining a linear governing equation has no 

 mathematical rationale except in the sense of a small-perturbation technique. 

 On the other hand, the procedure outlined and argued by us assumes (a) the 

 existence of an infinite sum of functions s^, Sj, Sj, . ., S^, ... such that it 

 converges to the solution S, and (b) that a rearrangement and grouping of terms 

 with subsequent setting of each group individually equal to zero is permissible. 

 The results of our work show that this procedure is valid in the whole domain 

 and, at least, for the range of the Reynolds number investigated. Thus an es- 

 sentially new mathematical justification of the assumptions involved in our pro- 

 cedure, which is here justified by comparison with experiments, must be sought 

 eventually. However, one conclusion that emerges from this procedure and its 

 heuristic justification is that no ideas of small-perturbation theory need be 

 brought into picture. 



There is an important difference in the set of Eqs. (1.28) and the set of 

 Eqs. (1.36) and (1.37). The equations in Eqs. (1.28) may be solved in any order, 

 because none of the operators Lj depend on the knowledge of the solutions to 

 any of the equations in the set. Equations (1.36) and (1.37) must be solved in a 

 definite order, progressing with n from n = onwards, since the operator L^ 

 is determined completely only when the preceding solutions S^, i = i, 2, . . . , n - l 

 are determined. This defines an iterative process. We shall call the set of 

 equations corresponding to a particular n the equations for n-th iteration. 



For an iterative process, we must obtain a solution to Eq. (1.36) to start 

 with. Consequently, it is important to choose the nature of this operator Eg 

 carefully. Either Lq can be made formally identical to L, or it can be made 

 structurally similar to L^, n > 1. If a solution Sg can be obtained to LgSg = 

 with Lg formally identical to L such that it satisfies a maximum number of 

 given conditions, then we may call it, by definition, a close solution to the exact 

 solution which satisfies the equation and all the given conditions. The nonlinear 

 effects which correspond physically to dynamical effects are then taken into ac- 

 count right from the beginning of the process of iteration. Because this is not 

 the case with the second alternative, we should, if we can, make Lg formally 

 identical to L. Fortunately, a potential solution which satisfies all except the 

 no-slip condition meets the requirements for being a close solution. Hence we 

 select Lg such that it is formally identical to L and Sg as the corresponding 

 potential solution satisfying all except the no-slip condition to start the process 

 of iteration. 



If we took Sg as a uniform flow through out the flow field, then it would 

 satisfy the harmonic equation and hence the equation LgSg = 0. The equations 

 for the first iteration would then be Oseen's equations. But, with particular 

 reference to external flows, we see that such a selection for Sg would satisfy 

 only the conditions at infinity and none at the wall— not even the condition that 

 the normal velocity component vanish at the wall. Thus it satisfies one condi- 

 tion less than the corresponding potential solution which satisfies, in addition to 



538 



