u(z,t) 



)(z£;t) + Q(z,-^;t) 1 



21 



2/ 



1 + 2 



n=l 



Mm 



\ ' / • COS TTW 



(z-%) 



(7) 



fz+9 



n=l 



Application to a Tank 



Here instead of the ring, consider a rod of length / with the boundary 

 conditions 



du 



Tz 



0; 2 = 0,/ 



(8) 



expressing the fact that there is zero flow of solute at the ends. If the rod is 

 reflected at z = 0, with the reflected part having the same initial distribution of 

 solute and the free ends joined, it is seen that the boundary conditions (8) are 

 automatically satisfied, and the problem is reduced to that of the ring. Thus, 

 immediately 



u =pBi [Q(z£;t) + B(z,-£;ti\ d^ 



..j/a-i.-W'. 



COSTTtt + COSTTn — 



I I 



(9) 



d^ (10) 



The theta function given by expression (7) does not converge rapidly for 

 very small values of kt. We shall transform it to a very rapidly converging series, 

 but first we must put it into a suitable form. Consider 



Q(z,+^;t) = 



i + 2 |y(^Kcos™«£' 



Let 



Then 



ra=l 



TTlk 24^ 



72 ' X* - o; 



)(z,+^;t) = — 

 2/ 



+ 2 Y e' W 



cos 2-nn x 



n=l 



2/ Q< ^ } 



(11) 



(12) 



(13) 



18 



