Backflow-Induced Inertial Arrest of Velocity Fluctuations in Sedimenting Suspensions

Abstract

A self-contained hydrodynamic theory is proposed to reconcile the discrepancy between divergent Stokesian velocity fluctuations and finite experimental measurements in sedimenting suspensions. We show that the compensating backflow induces non-negligible inertia, giving rise to an emergent screening length $\xi \sim a{\varphi }^{-1/3}R{e}_p^{-1/3}$ far exceeding the mean interparticle spacing $a{\varphi }^{-1/3}$ even at vanishingly small particle Reynolds numbers. This backflow inertial screening, together with finite-time viscous diffusion, arrests the indefinite spatiotemporal growth of large-scale velocity fluctuations. The resulting velocity fluctuations scale as $\delta u\sim {\varphi }^{1/3}{V}_{s}R{e}_p^{-1/6}$, together with the viscous correlation time ${\tau }_c={\xi }^2/\nu$, reproducing the well-known hydrodynamic self-diffusivity scaling $D_H\sim V_{s}a$. The theory predicts the prefactors of these scaling laws without adjustable parameters, in good quantitative agreement with experimental measurements. It also successfully captures the experimentally observed crossover from the finite-correlation regime to the finite-system regime as the screening length becomes comparable to the system size.