Lie be r and Rintel 



properties, interact in a simple way. The stability or instability of a fluid sub- 

 jected to such forces, with respect to convective vortex perturbations, was 

 found to depend on the value of an interaction number, defined as the sum of 

 the parameters measuring the stabilizing or destabilizing action of the separate 

 agents. As a typical example of such an interaction, the case of the stability of 

 a nongravitating fluid confined between two cylinders rotating with different angu- 

 lar velocities and maintained at different temperatures, was conjectured (Lieber, 

 1957) and then initially treated analytically (Lieber, 1959). This treatment in 

 which was originally projected the idea concerning the effect of simultaneously 

 impressed gradients of macroscopic-state parameters on hydrodynamic stability, 

 was restricted to a small gap between cylinders rotating with nearly equal ve- 

 locity in the same direction. This result was extended in order to examine the 

 effects of the gap as well as of angular velocities, differing both in magnitude 

 and sign and reported in a doctoral dissertation (Rintel, 1961). The results of 

 an approximate free- surface theory (Lieber and Rintel, 1965) for the case of 

 counter rotating cylinders provided a basis for a unified presentation of the re- 

 sults. In all of these cases the interaction number is the sum of the Taylor and 

 Rayleigh numbers, and its critical value is found to be independent of the kine- 

 matic parameters (Lieber and Rintel, 1962). These results are to be presented 

 in a comprehensive paper accommodating some recent experimental results and 

 in subsequent analytical investigations using the same ideas. 



The phenomenon examined in this paper belongs to the same class of phe- 

 nomena and can therefore be considered as another model of this class. The 

 two agents interacting in this case are the buoyancy forces, generated by the 

 gradient of temperature and the gradient of concentration of the diffusive sub- 

 stance. The simplicity of the present model facilitates a schematic representa- 

 tion of the stabilizing or destabilizing action of the various agents. 



THE CRITICAL CONDITIONS 



The differential equations associated with the problem are those of Navier- 

 Stokes, the continuity and molecular transfer of heat, and the diffusive substance: 



dv ^2 1 



— + (v • grad) V = v\J v ~ ~ grad tt - k g 



div V = , 



dT 

 dt 



dc 



+ V • grad T = k'V^T , (1) 



+ V • grad c = k"V c . 

 dt 



In these v designates the velocity vector, -n the pressure, p the density, g the 

 gravitational constant, T the temperature, c the concentration of diffusive sub- 

 stance, and v,k' , and k" are respectively the coefficients of viscosity, molecular 



