183 



We have obtained four equations, Eqs. (46-49) , 

 for three unknowns, uo,Ci)„j, and itiri , t>"t it can be 

 easily shown that one of them is not independent. 



Changing variables back into the original ones, 

 we get, as the governing equations for C-region, 



^e 3^- h[3^'"2'5l' = ° ' 



^e s^- M^("3qi) = . 



(50) 



(51) 



(52) 



The terms of order 0(1/e ) are neglected in the 

 above equations. 



Separated Retarding Region (D-region) 



Introducing normalized variables, |, fj, C for the 

 separated retarding region in the same manner, the 

 orders of differentiation are assumed, 



1 8 



1 3 



hj8xi L e3g ' h23x2 L e3n 



1 3 



3X3 L £33g ' 



r^ = 0(1) 



3? 



^ = 0(1) 



3n 



rr = 0(1) 



(53) 



(54) 



Velocity and pressure are assumed to be expanded 

 asymptotically , 



, 3u, „ 3u, ^ 3ui 

 "131 ^ ^l3r ^ "13F" 



+ e 



3,*^. I / ^ 3 ^ 3.^ , ^ 3 ,^ 3.^ 

 — (U1U2) + (vi-^ + wi-j^)u2 + (V2T^ + W2-5^)ui 



H 



'3^ 



-k9uivi+ kiv. 



1 api 

 3, 



-(P2+u,^) + -^(uivj) + — (ulw{) 



^'P2+"l 



3n 



(P3+ 2u[u2) + T-r(uJv2 + U2vJ) 

 ,_,.,„,,.,„,, -2k2uivi-k,(ui-vi ) 



£5 UqL 3^2 



„ 3v, „ 3vi „ 3v, 

 "1^^ + ^1^^ + wi^^ 



(59) 



+ £ 



3-^ '^ 3 " 3. , ,^ 3 .^ 3\" 

 — (V1V2) + (uiT^ + wj— )V2+ ("2^ + W2-T:r)vi 



+k2Ui2- kiujvi 



1 3Pj_ 

 £ 3f| 





— (U1V2 + U2V1) + — (P3 + 2VjV^, 



+ ■g^{viW2 + V2wi) + k2(Uj2 - v'2) _ 2kjuivJ 



1 V S^V] , 



£6 UqL 3^2 



(60) 



qi/Uo = e(ui+ui) + e2(u2+U2) + - 

 q2/Uo = £(vi+vi) + £2(v2+V2) + - 

 I3/U0 = £ (wi+wj) + £'*(W2+W2) + 

 (p-p )/pU = epi + £P2 + . . . , 



(55) 



(56) 



where u , u , ... are all time-averaged variables 

 and u , u , ... are fluctuating. Here the fluctu- 

 ating terms of pressure are omitted because they 

 do not appear in the basic equations . 



The vorticity can be also expanded asymptotically. 



Uo/L 



Uq/L 



e2 3£ 



1 3v2 



+ 0(1) 



_1_ 3Ul_ 1 ^ 

 3vi 3ui 



+ 0(1) 



'1 



Uq/l dl 3n 



+ 0(£) 



(57) 



Under these assumptions, the leading terms of the 

 continuity equation are written. 



3pi 3p2 „ 



^ + ET^ + 0(e2) = 0. 



(61) 



The leading terms of Eqs. (59, 60, 61) yield 



Pl{C<n,C) = const. (62) 



Equation (62) means that the pressure is constant 

 throughout the present region as far as 0(e) is 

 concerned. 



Now the second terms of Eqs. (58, 59, 60) yield 



Suj 3ui 



3ui 



,;2> 



^IT¥^ + *i-5T^ + "IT^ = - i7(P2 + "1^) 



3,"i"i. 9,-1-1. 

 - T^(uivi) - — (uiwi) , 



, 3Vi 3v SVj 3 .,., 



Ul^r^r- + V.-r^— + W^r-;^ = - T^(u,Vi) 



3, , -1?, ^/"'"'^ 

 — (P2 + Vi^^) - ^^(vjwi) 



(63) 



(64) 



3Un 



3w, 



35 8n 3c 



(58) 



(65) 



" and the governing equations are 



In the above equations, the molecular viscosity 



