Studies on the Motion of Viscous Flows — VI 



by Goroff (1960) have shown that overstable oscillations do not result in sig- 

 nificant changes of heat transfer (and thus presumably no significant changes 

 in the transfer of diffusive substance). He also found that for the critical value 

 of the characteristic parameter predicted by the use of the principle of ex- 

 change of stabilities, there evolves convective motion, which is superimposed 

 on the overstable oscillations. Since this convective motion is accompanied by 

 transfer properties and is therefore of more interest, in terms of Goroff s ob- 

 servation we simplify the analysis by using the principle of exchange of sta- 

 bility. Consequently, according to the present notation, and consistent with 

 Eddington's motivation for introducing the notion, overstability is here con- 

 sidered as a case of stability, although in the formal mathematical sense it is 

 a case of instability. Subsequently we will show that the overstable oscillations 

 indeed provide the stabilizing mechanism as a case of stability. The differen- 

 tial equations for the state of transition to convective motion are 



(a) (D2-k2)a)* = -k2 !!L (a'ATr* + a"AC- C*) , ' '■ ' - ^Jn • ' . : 



(b) (D-k2)T^ 



k'AT 



(9) 



(c) (D-k2)C* = Jll_ /3"a)* . 

 k"AC 



Operating by (D^ - k 2) on Eq. (9a) after elimination of c* and r* by use of the 

 remaining Eqs. (9), we obtain 



(D2-k2)a;* = k^Raw* , (10) 



where . ■ - " 



h^ga'/S' .■■_,..,- 



R, - r; + r: , r: 



R! 



k'v 



h'*ga"yS" 

 k"v 



(11) 



is the generalized Rayleigh number. In terms of a virtual temperature gradient 



a' k" 



dz 



this Rayleigh number can be represented as 



k i^ dz 



697 



