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 ( $\ensuremath{\varphi}\ensuremath{\le}0.05$) 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 $〈\ensuremath{\delta}{V}_{z}(0)\ensuremath{\delta}{V}_{z}(z){〉}^{1/2}\ensuremath{\simeq}{2.0V}_{\mathrm{sed}}{\ensuremath{\varphi}}^{1/3}\mathrm{exp}(\ensuremath{-}z/2\ensuremath{\xi})$, where $\ensuremath{\xi}\ensuremath{\simeq}20a{\ensuremath{\varphi}}^{\ensuremath{-}1/3}$. 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 $〈\ensuremath{\delta}{V}_{z}^{2}(0){〉}^{1/2}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.0V}_{\mathrm{sed}}{\ensuremath{\varphi}}^{1/3}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}W/60a{\ensuremath{\varphi}}^{\ensuremath{-}1/3})]$.