y^ IT- 101 kh 



We expect the boundary layer thickness to have the same 

 order of magnitude in the higher order solutions as in the zero- 



1/0 



order solution, viz*, e ' -^, Thus, in order to find the first- 

 order interior solution, \'ie neglect all the terms with subscript 

 b since they are negligible in the interior. Thus immediately, 

 V^., the interior portion of V-^^, is known and is (from (27)) 



V r:_[v. -U. -yH.l 

 li "- oix oiy "^ 01 T 



= °.^^i:i(_x + r - £-^'^^)[cos nsy + (nsy-9n^s^) sin nsy! 



(29) 



From (28) and (29) the interior portion of U-,, U-j^., can 



be computed directly, giving 



2 -I /T 



U = - g cos^ r_ x„ + x(r - e^-^)] [2nsy sin nsy + 

 li 9^2 



2 2 2 ^ "^ 



+ (n s y + On-'s-^ + 2)cos nsy ] + C (y^-r) 



where C (y,T) is arbitrary and must be evaluated by applying 

 the boundary conditions to the complete solution, i.e., inter- 

 ior solution plus boundary layer contribution. 



Before proceeding with the boundary layer analysis we 

 can simplify equation (27) to some extent. Near x = r, 

 ^ox ^ 0(e~-^'^^), U = 0(£-^/3), and H = 9"^0(e-^^^). Near 



"I 



X = 0, Vq^ = 0(e~2/3), U^y -t 0(1), and H^y = 9~-^0(l). Thus in 



each case we are justified in using only the contribution of the 



-2/3 -?/^ -1 

 V term provided e >>1 and e " -^>> 9 # As will be shown 

 ox 



later, vjhen the appropriate dimensional constants are substi- 

 tuted, the error involved in neglecting the other terms is 



