596 Dr. A. Russell on the Convection of Heat 



We have, therefore, 



3 A 30\ . 3 A 30 



c 

 and thus 



a..^w-{^(*g)+£(*g)>i^D*. 



D0 3 /,30\ B /,30 



c Dt 



o*i V 3*i/ 3* 2 \ 0*2/ 



Since we suppose that the liquid is flowing in the direction 

 AA', we have, when the steady state is reached, 



D0_ 30 3* 2 _ 30 



c vt ~ c -ds 2 '~-dt- c? av 



and hence, assuming k constant, we get 



3*2 cq \3*i 2 3*2 j 



As <? varies with both ^ and 5 2 , it appears at first sight as 

 if it would be very difficult to obtain a solution of this 

 equation. If, however, we alter the variables from s 1 and s 2 

 to a and /3, the equation simplifies in a remarkable way. 



We have 



30_303« 303/3 

 3*i ~~3* 3*i 3/3 3*i' 



and thus 



, 3«y« ag y/8 



A similar equation holds for ^r— -„. 



3* 2 " 

 Noticing that 



3*^3/3 = 1 3^ = __M = o 

 3*i 3*2 V 3*2 3*i 



V 2 « = 0, and V 2 £ = 0> 



we get 



3*0 , W_£/W^BV\ 



3*i 2 + 3* 2 2 " V 2 \3* 2 + '3/3 2 / 



We also have 



30 _30 q 

 3* 2 - 3/3 V* 



