The onset of particle-dominated convection regime in colloidal suspensions

In the experiments of B\'{e}nard convection in a suspension of microparticle by Chang {\it et al.} (2008), a parameter $\beta$ was isolated to model the interplay between the effects of thermophoresis, sedimentation and Brownian diffusion. A plot of $\beta$ as function of the particle radius, $r_p$, for a suspension of aluminum oxide particles in water shows that the function $\beta(r_p)$ has the shape of an inverted parabola with two roots so that $0 < \beta \ll 1$ for $1$ nm $\le r_p \le 5$ nm and for $r_p \approx 50$ nm where thermophoresis and sedimentation are balanced. We consider a particulate medium model to determine the threshold instability conditions. Due to the large particle size, the convection process is characterized by longer diffusion time scale, much smaller Lewis number, $\tau$, and larger separation ratio, $S$, than the binary mixture case. For $0 < \beta \ll 1$ which corresponds to two distinct particle radii, a small wavenumber expansion yields the value $R_c = 720\,\tau/S$. For $\beta = O\bigl(1\bigr)$, threshold stability conditions are depicted as function of the particle size and the height of the fluid cell.\\[4pt] Chang, B.H., Mills, A.F. and Hernandez, E., {\it Int. J. Heat Mass Transfer} {\bf 51} (2008), 1332-1341.