Studies on the Motion of Viscous Flows — III 



the conditions at infinity, the condition on the normal velocity component. It is 

 therefore less close and therefore less appropriate than the latter. In fact, 

 there cannot be any solution closer than the potential solution, because if it 

 were otherwise then it would have to satisfy all the conditions, and that would 

 make it the exact solution which we assume not to be the case. 



Two solutions may be equally close from a mathematical point of view, i.e., 

 they both satisfy the given equations and all except one of the given conditions, 

 and yet be different because the conditions they do not satisfy may not be the 

 same. If such is the case, then we have to decide upon the appropriateness of 

 one with respect to the other. For the problems in fluid mechanics, all the con- 

 ditions except the no-slip condition seem self-evident, and hence a solution sat- 

 isfying them would seem to be more appropriate than the solution that does not 

 satisfy one of them and satisfies instead the no-slip condition. If we adhere to 

 this view, then the potential solution is the more appropriate. 



There is another way in which the appropriateness can be meaningfully de- 

 cided. That solution S^ which, first, allows us to determine in some way the 

 number of iterations necessary to secure sufficient convergence and, second, 

 calls for a minimum number of these iterations would certainly be the most ap- 

 propriate. As is shown later, for the problem of the flow around a circular cyl- 

 inder the potential solution which satisfies all but the no- slip condition is the 

 most appropriate according to this criterion. 



If Lq were made structurally similar to Lp, n > l, then the equation 

 LqSq = would represent the set of equations for a creeping motion. Stokes has 

 obtained a solution to these equations for the case of a sphere which satisfies all 

 the conditions on the flow. Other cases of axisymmetric bodies have been ex- 

 plored since then. But one cannot obtain, in all cases, nonsingular solutions to 

 these equations which satisfy all the boundary conditions. The case of a cylinder 

 is one such. However, what is interesting in this case is that the solution which 

 Stokes obtains and discusses in Eq. 130 of Ref. [16] is one which satisfies the 

 harmonic equation. A solution to the harmonic equation cannot satisfy all the 

 conditions on the flow, and if one required all but the no-slip condition to be 

 fulfilled, the streamline field will be identical to that for a potential flow. The 

 pressure field found from the equations of creeping motion would appear con- 

 stant everywhere, while that due to a potential flow is a function of space vari- 

 ables and evidently much closer to the actual pressure field in most of the flow 

 field. This shows that the dynamics of the potential flow as compared to that of 

 a creep flow is much closer to the actual— a strong reason for the selection of 

 Lq such that it is formally identical to L and Sg as the potential flow solution. 



As noted earlier, any solution of the harmonic equation will satisfy the 

 equation LqSq = 0, where Lq is formally identical to L. For the case of the 

 flow around a circular cylinder, the appropriate general form of the stream 

 function which satisfies the harmonic equation is the following: 



i//' = (Ar'i + Br + Cr log^ r + Dr^) sin 6 . (1.38) 



Then 



539 



