Particle velocity fluctuations and correlation lengths in dilute sedimenting suspensions

Abstract

We measure the instantaneous velocity of particles sedimenting in a three-dimensional container at low particulate Reynolds numbers. We aim at characterizing the main specificities of the particle velocity fluctuations. We obtain the local and instantaneous Eulerian velocity field from particle image velocimetry: a thin Yag laser light sheet (about two diameters thick) illuminates the particles from one side of the cell to the other. Our measurements are therefore spatially localized and, together with the squared cross sections of the cells, these are the two main originalities of our instrumentation. Four different cells and three different particle sizes give access to aspect ratios (cell width W over particle radius a) ranging from about 50 up to 800. We confirm the existence of eddy-like structures for the velocity fluctuations. The structure size is found to be almost independent of the volume fraction Φ for 6.25×10−4<Φ<5×10−2 for a fixed aspect ratio W/a, in seeming contradiction with the results of Segrè et al. [Phys. Rev. Lett. 79, 2574 (1997)]. The velocity fluctuations’ profiles are close to parabolic for the smallest aspect ratios but display a plateau value in the central part of the cell for higher aspect ratios. The unexpected result of our experiments is the large distance of influence of the boundaries: in fact, the plateau values are observed to saturate for aspect ratios larger than a few hundreds for Φ=1×10−2, for example. Moreover, the saturated plateau values scale as φ0.45±0.05, in contrast with the φ1/3 scaling observed in most previous works.

Sedimentation dynamics of spherical particles in confined geometries

“…We performed some simulations with a container where L y L z which seem to suggest that in such cases the correlation lengths of R yy ͑r͒ and R yz ͑r͒ are defined by the smaller of these dimensions, which is again in a good agreement with the results in Ref. [35]. The correlation lengths of R zx ͑r͒ and R zy ͑r͒ seem to depend both on L x and L periodic even when the former is much smaller than the latter.…”

“…The shapes of R xx ͑r͒ and R xz ͑r͒ are quite similar to the corresponding SVC's shown in Ref. [35]. For the y component of the particle velocities all correlation lengths increase seemingly linearly with increasing L periodic .…”

“…This time simulations performed with L y L z indicate that the correlation lengths depend on L y even if L y Ͻ L z leading again to a scaling similar to that in Ref. [35]. The scaling of R zz ͑r͒, however, differs significantly from the previous by the fact that the correlation length depends also on ⌽.…”

“…A further reason not to expect similar scaling in the present simulations is that the saturation has been shown to be a result of having walls in the direction of gravity. However, the experiments of Bernard-Michel et al [35], performed in a long tube with smaller square cross section (width less or equal than 800a) show almost linear relation between the correlation length of the SVC and the width of the container with no ⌽ dependence. In all directions our measured R x␤ ͑r͒ depends only on L x provided that L x Ͻ L periodic .…”

Spreading fronts in sedimentation of dilute suspension of spheres

“…The mean sedimentation velocity of a suspension can be predicted theoretically at low particle volume fraction 1 and by using sophisticated computer simulations at larger volume fractions, but the individual movement of particles due to long-range hydrodynamic interactions gives rise to velocity fluctuations that are still not completely understood. [2][3][4][5][6][7][8][9][10][11][12][13] Lately, the study of the spreading of the sedimentation front has regained interest 12,13 as it has been advocated that a small vertical stratification due to the broadening of this front can change the characteristics of the velocity fluctuations. [10][11][12] One can find a few experimental investigations of the spreading of this front in the literature showing the importance of different effects, such as hydrodynamic dispersion, polydispersity of the particles, and hindered settling.…”