Soret-driven convection in colloidal suspensions

Convection in colloidal suspensions of solid particles is characterized by the interplay between thermophoresis, sedimentation and Brownian diffusion. Their coupled effects is represented by a dimensionless parameter $\beta$ and experiments by Chang {\it et al.} (2008) have shown that for a given set of experimental parameters, $\beta$ is a function of the particle radius $r_p$ with the function $\beta(r_p)$ having the shape of an inverted parabola with two roots in the range $ 5$ nm $\le r_p \le 125$ nm. We investigate both the linear and non-linear convection in a suspension of solid particles using a particulate medium in a Rayleigh-B\'{e}nard geometry set-up. The analysis focuses on the particle dominated convection regime for which the onset is steady and to disturbances having infinitely long wavelength. For $0 < \beta \ll 1$, which corresponds to particle size near the two roots of $\beta(r)$, we retrieve the instability threshold conditions for the binary mixture model. For $\beta = O\bigl(1\bigr)$, we show that, unlike the binary mixture model, the conditions for instability onset can be mapped to corresponding experimental parameters. A nonlinear evolution equation is derived and its predictions compared to those of a similar equation for the binary mixture case.