All- 101 ^3 



of magnitude as the functions themselves. The terms multiplied 

 by e may therefore be neglected. 



Let us rewrite equations (22), (23), and (26) as the 

 sum of tv/o parts - one part, v/ith subscript i, having the same 

 order of magnitude throughout the domain (the "interior solution"), 

 the second part, with subscript b, sensibly large near the bound- 

 ary and negligibly small in the interior, (the "boundary layer 

 contribution") 



U . = - ns(l 4- a sm T )cos nsy(- x + r - e ) 



,, . s J 1/3 (x-r) £-^^^ + 



U, = - ns(l +asinT)cos nsy -i e e 



ob ' L "l/5l 



1 /^ -1/3 ■^ ^ 



r . 1/3 ^ .xVle" ^^M/=? ^^^ r^ • .X V^ e" s. ~ ~ 



+ [(£ -'-r)cos(— '-^ )+(V3e - -^)sin( ~ )] e 



2 ^2 



V . = - (1 + a sin T)sin nsy 

 oi -^ _]_/3 



J (x-r)B-^/3^ [cos(fJ5i ) + 



V , = (1 + a sin 'i;)sin nsy S e 2 



ob 



-1/3 1 



-^/^ - ...xV/3e-^/\, -^ 



+ (2r^ \/3)sin(±^12J: )] 



^/3 



e 



I 



1/3 

 OH . = (1 + a sin t)(cos nsy+nsy sin nsy)(- x + r - e ) 



f / . > -1/3 

 J l/3^(''-r)e 



9H , = (1 + a sin T:)nsy sin nsy^i £ e 



CD I 



— 1/3 -1/3 ^^ 



+ [(£^^3_^)cos(£y3,r...,. )-f(/3^^^^--^^)sin(^^^ )]e'' ^ 



2 ,/3 2 y 



