Finite wavelength selection for the linear instability of a suspension of settling spheroids

Abstract

AbstractThe instability of an initially homogeneous suspension of spheroids, settling due to gravity, is reconsidered. For non-spherical particles, previous studies in the literature report that normal-mode density perturbations of maximum growth rate are those of arbitrarily large, horizontal wavelength. Using the ‘mixture theory’ for two-phase flow we show that the maximum growth rate for horizontal density perturbations is obtained for a finite wavelength if the inertia of the bulk motion associated with the normal-mode density perturbation is accounted for. We find that for long wavelengths, $\lambda \ensuremath{\rightarrow} \infty $, the growth rate approaches zero as ${\lambda }^{\ensuremath{-} 2/ 3} $. The maximum growth rate is obtained for $\lambda \ensuremath{\sim} d/ \sqrt{{\ensuremath{\alpha} }_{0} { \mathit{Re}}_{L}^{1/ 2} \raisebox {-5pt}{}} $, where $d$ is the axis perpendicular to the axis of rotational symmetry of the spheroid, ${\ensuremath{\alpha} }_{0} $ is the undisturbed volume fraction of particles and ${\mathit{Re}}_{L} $, typically ${\ll }1$, is a Reynolds number of the bulk motion on a typical length scale $L\ensuremath{\sim} d/ \sqrt{{\ensuremath{\alpha} }_{0} } $ and a velocity scale on the order of the undisturbed settling speed. The theoretical results for the wavelength selection agree qualitatively well with previous experimental results in the literature of measured correlation lengths of vertical streamers in settling fibre suspensions.