611 



with respect to the C axis, and that both hg and 

 V Q vanish at infinity, as desired. 



The maximum vorticity is 0.09 and is at the 

 point 



is integrated and will introduce negligible errors 

 in the result. After (40) is substituted into 

 Eq. (39) , the latter is solved by repeated use of 

 the following formula for various values of n: 



n = 1//2 , C = 0, 



at which both V and W are zero. The maximum vertical 

 velocity is 1/6it and is at the origin. The maximum 

 absolute value of ^g is 0.63817/6ir, which occurs 

 at 9 = or TT, r = 1.1225. 



dr^ 



2r 



dr 



+ 2 



n -r 

 r e 



9 2 



[n{n - 2) - 2(n - l)r^]e-^ . 



The result for f is put into Eq. (38) , and we have 



THE SECOND APPROXIMATION 



The equation for hj is 



Lhi = Von + Voc' 



(35) 



where Vq and W are the velocity components from 

 the first approximation. The right-hand side of 

 Eq. (35) can be written in polar coordinates as 



r ae 



dr 



Hence Eq. (35) can be written as 



,2 _^2 



Lh, 



sin -r' ,, 

 e (1 - e 



where L, in its polar-coordinate form, is 



/ l93 

 V630 



4320 



61 

 1260 



„2 



2 ^ 11 

 r + 



1260 



3024 



(41) 



The function H, is tabulated in Table 1. A look 

 at hj given by Eq. (36) then reveals that the 

 temperature is increased in the upper half of the 

 r\-t, plane and decreased in the lower-half plane, 

 making the isotherms more widely spaced in the 

 upper-half plane and more crowded in the lower-half 

 plane. 



The tabulated values of H, show a very smooth 

 variation of Hj with r, verifying the expectation 

 that the local irregular variation of (40) is 

 diffused away when Eq. (39) is solved with (40) 

 replacing its right-hand side. 



The next step is to solve 



3^ 18 18^ ^3 

 3r2 r 3r r^ 39'' 3r 



Writing 



(L - l)5i = -h. + Vo?o^ + WgSoi;. 



(42) 



A simplification is possible before we attempt to 

 solve Eq. (42) . Differentiating Eq. (35) , we have 



3Tr 



Hi (r). 



(36) 



(L + 2)hj^ = Vghg^^ + W^hg^^ + Vo^hg^ 



we have 



(1 



(37) 



+ "on^^oc- 



(43) 



if we write Lj^ for L with the operator 3^/392 in 

 L replaced by -n . 



To solve Eq. (37), we let 



Hj = r~ f, 



so that Eq. (37) becomes 



(38) 



Let 



^1 3 

 Then Eq. (42) becomes 



(44) 



(L - l)q + (L + 2) ^ = J (Vghg^^ + Wghg^^) , (45) 



f" + ( 2r - - ) f ' + 2f = -e (1 



). (39) 



Then we approximate the right-hand side of this 

 equation by 



2 -^ 

 -re 



^2 „*+ ^6 „8 

 '^ ~ ~ '^ 6 ii""*" 240 



(40) 



^0 3 ^OiT 

 By virtue of (43), Eq. (45) becomes 



1 



(L - l)q 



3 (Vg^hg^ + Wonho^)- 



But 



(46) 



The greatest error occurs at r = 1.8, but it is 

 less than 6.5^ of the maximum value of the quantity 

 approximated. Up to r = 1.2 the approximation is 

 excellent. It is expected that the local errors 

 around r = 1.8 will be diffused out when Eq. (39) 



g^ = (fg)cn, Wg^ = -(<fg)nn, 



ct 

 velocity field (V, , Wg^^) , and we can write Eq. (46) 



so that fg is a stream function for the fictitious 



