591 



Comparing (31) and (33) and using (27), we get 



4" 1 



D 



11 ^D 



Alt 



D^ 



+ A5^ 



;34) 



where^ A3, Ai, , and A5 may depend on s. Equating 

 coefficients in (34) we get (29) again and the 

 following 



dAb 9 d 

 q, + D — ^ - N^D :;^ (D + h) 



dt 



dt 



-"3 



2 3 

 2 4 



qi h 



3 



D(Ab) 



(35) 



dD 



h (D + 2h) 



dt ^1 D , 2 



(Vo - q,t) 



(03 + 204) 



2 2 1 2 



q.^h** (D + 2^)'* 



(V 



- qjt) 



= 



(D + ih) 



dt ■ dt ""^1 D ~~2. 



(Vo - qjt) 



, dD ^ dh . h 

 3 -rrr + TT - 2q 



(41) 



D dAb ^ „ dD 

 2 dt ■*" 2h dt 



dAb 



dt 



D-^Ab dh 



(D + h) 



3 3 

 2 h 



q, h 



a^^ 



D(Ab) 



3 3 

 2, h 



q, h 



6h dt 



■6h^^= "5 



D(Ab) 



(36) 



(37) 



3/4 



where a^ = Aj_/Bi2B3'"^ (i = 3,4,5). Equations (35)- 

 (37) , (29) , and (11) are five equations in the three 

 unknowns. They determine the solution to the first 

 order for large Ri , although we must make sure that 

 all equations are satisfied to that order. In this 

 regard, if we use (35)-(37) , (29) , and the deriva- 

 tive of (11) , i.e., (10), we may consider these as 

 five homogeneous linear, algebraic equations in five 

 unknowns dAb/dt, dO/dt, dh/dt, qj , and qj ^' ^j^S/^/q 

 (Ab) . The determinant of these equations vanishes 

 and we satisfy compatibility. 



6. HOMOGENEOUS CASE (N = 0) 



If N = 0, the upper fluid is homogeneous and (11) 

 becomes 



1 2 

 (D + 2h)Ab = Vq - q t 



(38) 



3 7 



6a5 



6a q 



- 2aq 



qj^h^D + |h)' 



2 2 ( 

 D (Vo - q^t) 



= (42) 



Two effects occur in (40) and (41) . 

 them by adding the two equations . 



3 7 



We may separate 

 We get 



d 

 dt 



(D + h) 



2ah + 6ac 



qi 



2uh 



D + 



,v§ 



(43) 



11' 



The term on the right of (43) expresses the upward 

 motion of the boundary between (intermittently) 

 turbulent and non-turbulent fluid due to turbulence 

 in the interfacial layer causing entrainment of the 

 upper, non- turbulent fluid. On the other hand, the 

 second terms in (40) and (41) express the upward 

 motion of the boundary between fully turbulent and 

 intermittently turbulent fluid (and the consequent 

 decrease of h) due to heating alone. This contribu- 

 tion to the entrainment velocity is proportional to 

 the interfacial thickness, h, and disappears when 

 the common approximation is made that h = 0. 



Let us find an approximate solution to (40) -(42). 

 If we let Do and ho be the values of D and h at t = 

 0, we make the following definitions: 



ho h u 



hi 



Do 



Do 



-= Di 

 Do ^ 



where Vq is a constant related to initial conditions. 

 We use (35) and (38) to eliminate Ab in (29), (36), 

 and (37) . We get 



3 3 



dt 



(Ab) = - -^ - a^ 



q, h 



2^,'* (D + ^h) 



(v;^ 



:^ 



1 



dh _^ qih (D + 2h) 

 dt "*" D 



«3 



(V^ 



6ac 



qit) 



3 7 

 2v,'t 



7 

 1 4 



h\ qj h (D + -h) 



(Vo - qit) 



(39) 



(40) 



_^For arbitrary Ri , the quantities Aj may depend on Ri. As 

 Ri ->■ «>, however, A^ will approach "constants" which may, of 

 course, be zero. 



2 2 



3 3 



qi Dp 



v2 



1 

 3 



"0 Do 3 



Then equations (40) -(42) may be written 



dh. 



h, (Di + J hi) 

 .A. £ 5 



dT Di (1 



6t) 



aq - 5at 



1 

 Di 



ill 



Dl y 



7 7 

 t» It 



M (Dl + jhi) 5 



Did - 6t) 



i 



dD^ 

 dT 



(Dj + ihj) 



(1 



6t) 



(03 + 20^) 



h^SD^ +ih^ 



7 7 



D^l 



Z 



(44) 



= 



(45) 



(46) 



