Scaling Behavior for the Onset of Convection in a Colloidal Suspension

We investigate the early stages of mass convection in a colloidal suspension at high solutal Rayleigh number ${\widehat{\mathrm{R}\mathrm{a}}}_s$. From the time evolution of shadowgraph images and by assuming a diffusive growth of the boundary layers we obtain an indirect measurement of the concentration boundary layer thickness ${\delta }^{*}$ at the onset of convection. We show that the dimensionless boundary layer thickness $\delta ={\delta }^{*}/d$ scales as ${\mathrm{R}\mathrm{a}}_s^{-p}$, where ${\mathrm{R}\mathrm{a}}_s={\widehat{\mathrm{R}\mathrm{a}}}_s\delta$ is a modified solutal Rayleigh number for convection which accounts for the actual density unbalance and $d$ is the thickness of the sample layer. This scaling behavior is analogous to that reported at steady state for turbulent convection in simple fluids. We find $p=0.35$, a value compatible with the exponent $1/3$, reported for turbulent heat convection in simple fluids at steady state.

Figure 1

Variance of shadow images sequences plotted as a function of time. The cell thickness is $d=1.0\mathrm{m}\mathrm{m}$. The applied temperature differences and solutal Rayleigh numbers correspond to $\Delta T=34.8\mathrm{K}$ and ${\widehat{\mathrm{R}\mathrm{a}}}_s=1.84\times {10}^7$ (solid line), $\Delta T=23.75\mathrm{K}$ and ${\widehat{\mathrm{R}\mathrm{a}}}_s=1.25\times {10}^7$ (dashed line), and $\Delta T=9.6\mathrm{K}$ and ${\widehat{\mathrm{R}\mathrm{a}}}_s=5.08\times {10}^6$ (dotted line).

Figure 2

Latency time ${\tau }^{*}$ for the onset of the instability plotted as a function of the solutal Rayleigh number ${\widehat{\mathrm{R}\mathrm{a}}}_s$. The solid line is the best fit of the experimental results yielding a power law ${\tau }^{*}=25.3{\widehat{\mathrm{R}\mathrm{a}}}_s^{-0.52}$.

Figure 3

Effective solutal Nusselt number $1/\delta$ plotted as a function of the modified solutal Rayleigh number ${\mathrm{R}\mathrm{a}}_s$. The solid curve represents the best fit of the experimental results yielding a power law $1/\delta =0.051{\mathrm{R}\mathrm{a}}_s^{0.35}$.