Smith 



Table 3 



Maximum Percent Errors Obtained for Values of the Surface 

 Normal Derivatives for the Case of Dirichlet Conditions 

 on Concentric Spheres 



By suitably combining the above solutions, it is possible to obtain the 

 solution to a problem typical of those of the reentry body application. 

 The boundary conditions are: 



q) (0.8) = 16900 



(P(l) = 618.06 P4(cos 6) + 2793.14 P2(cos 6) - 133.60 . 



Figure 17 compares analytic values of r^ Bcp/Br with those calculated by 

 using the smaller element number. Agreement is good. The quantity 

 3cp/Br is weighted by r ^ to correct for the difference in surface area of 

 the inner and outer walls. If there were no circumferential flux in the 

 porous material the two curves would coincide. The fact that they do 

 not is evidence of appreciable flux in the circumferential direction. The 

 corresponding curve for the larger element number case lies exactly 

 halfway between the two in the figure (half the error). 



In summary, the ability of the existing surface source density program 

 to calculate accurate solutions to the interior Dirichlet problenn using 

 reasonable element numbers has been demonstrated. The present ap- 

 plication is an unusual one. 



Presumably, since the method is not configuration—limited, it could 

 solve similar problems for general shapes with about the same accuracy. 



I will close by citing an instrument problem that is especially interesting 

 because Laplace's equation applies so rigorously. That is not the case with fluid 

 flows, because of the effects of viscosity. In fluid flows the Navier-Stokes 

 equation applies rigorously; what we are doing now is approximating it by using 

 Laplace's equation. 



The problem is that of a superconducting bearing suspended in a magnetic 

 field. Because the bearing is superconducting the field is unable to penetrate 



360 



