Two-Component Benard Steady Convection with Surface Adsorption

Abstract

The Bénard-Marangoni instability of a two-component liquid layer open to air has received great attention in the last twenty years [1–2]. A series of studies made by Velarde and collaborators [3–8] analyzed a two-component liquid model, where buoyancy, surface tension, surface deformation, heat and mass transfer as well as Soret effect in the volume were all taken into account. For steady transitions, the sufficient condition for instability can be approximated by the simple relation

$$\text{M/M}_\text{c} + \text{E/E}_\text{c} + \text{R/R}_\text{c} = 0$$

(1.1)

where M, E and R are Marangoni, elasticity (solutal) Marangoni and Rayleigh numbers respectively. M_c is the critical value taken at E = R = 0 and, correspondingly, are E_c and R_c. Three different mechanisms, described by M, E and R, interact with each other, and affect the stability of the system. When E > 0, one has, (∂R/∂E)_M < 0, which means that the larger the elasticity number, the lower the stability of the fluid. The elasticity (solutal) Marangoni number plays a destabilizing role. However, no surface adsorption has been considered in these papers. For a multi-component liquid with surfactant solute, the equilibrium solute concentration in the surface is, usually, much higher than in the bulk [9], thus leading to solute accumulation at the open surface as the exchange of solute between surface and bulk are through adsorption-desorption process.