Long-Range Correlations in Sedimentation

Abstract

Using particle image velocimetry we have measured the sedimentation dynamics of non-Brownian colloidal spheres over a wide range of low concentrations ( $\phi \le 0.05$) and sample cell sizes $W$. Fluctuations in the settling velocity show universal large-scale finite-range correlations, in the form of swirls, which depend simply on volume fraction and particle radius and follow $〈\delta V_z\left(0\right)\delta V_z\left(z\right){〉}^{1/2}\simeq {2.0V}_{\mathrm{sed}}{\phi }^{1/3}\exp (-z/2\xi )$, where $\xi \simeq 20a{\phi }^{-1\mathrm{}/3}$. In turn, the predicted divergence of the velocity variance with increasing cell size $W$ is cut off in a universal way as $W$ exceeds the swirl size and is well represented by $〈\delta V_z^2\left(0\right){〉}^{1/2}{=2.0V}_{\mathrm{sed}}{\phi }^{1/3}[1-\exp (-W/60a{\phi }^{-1\mathrm{}/3}\left)\right]$.