Studies on the Motion of Viscous Flows— VI 



transfer of heat, and diffusive substance. Equations (1) are interrelated by the 

 equation of state, which for small variations in concentration and temperatures 

 is: 



, 1 ^P 1 ^/O 



P-- p„(l+a8T+a SO , a -- ^— •«"= ^-- (2) 



The basic solution of Eqs. (1) is that of molecular transfer of heat and salinity 

 across this layer of stationary fluid 



vo = , 77, = g [ pdZ, T, = /3'Z + 7' , C, = fi"Z - 7" , (3) 



.. , 



where the constants 7' and 7" and the gradients /3' and /3" are determined by 

 the boundary conditions. By superimposing the basic solution in Eqs. (3) on 

 small perturbations, we have from Eqs. (1) and (2) the linearized equations for 

 the perturbations 



(a) — = yV^Vj - — grad 77j - g(a'Tj + a"c J k 



div V = , ■-,, , . • 



dTi '; ;.':'- : ■ .":':.. '^. (4) 



(b) — = k'V^T^ - fi'c^ . ■ 



(C) — = k"V2c - /3"a>^ . 



In Eqs. (4), in accordance with Boussinesq's equations (1904) as used by Ray- 

 leigh (1916), the small quantities which arise from variations of density are 

 neglected, with the exception of those which represent the buoyancy force. 

 Taking the divergence of Eq. (4a) by use of Eq. (4b), we obtain 



1 / ^T, 3C.\ 



_L V^TT, = -g a' — + a" — -] ■ (5) 



Po ' \ BZ 3Z/ ^ ' 



Elimination of ttj from Eqs. (4) and (5) gives • -'' .,.■ 



3^ , 1 = -BV,^a'T, + a"c,) , (6) 



where 



^ BZ2 3x2 3y2 



695 



