Observation of two branches in the hindered settling function at low Reynolds number

We analyze hindered settling speed versus volume fraction $\varphi$ for dispersions of monodisperse spherical particles sedimenting under gravity, using data from 15 different studies drawn from the literature, as well as 12 measurements of our own. We discuss and analyze the results in terms of popular empirical forms for the hindered settling function, and compare to the known limiting behaviors. A significant finding is that the data fall onto two distinct branches, both of which are well described by a hindered settling function of the Richardson-Zaki form $H\left(\varphi \right)={(1-\varphi )}^n$ but with different exponents: $n=5.6\pm 0.1$ for Brownian systems with Péclet number $\mathrm{Pe}<{\mathrm{Pe}}_c$, and $n=4.48\pm 0.04$ for non-Brownian systems with $\mathrm{Pe}>{\mathrm{Pe}}_c$. The crossover Péclet number is ${\mathrm{Pe}}_c\approx {10}^8$, which is surprisingly large.

Figure 1

A time series of photographs for non-Brownian $d=365\mu \text{m}$ glass spheres with initial volume fraction $\varphi =0.21$ sedimenting in an aqueous glycerol solution. The sample is illuminated from both sides, so the only light which reaches the camera is that which is scattered at close to ${90}^{\circ }$. Thus the dark regions at the bottom are densely packed sediment, the dark regions at the top are depleted of particles, and the bright regions in the middle are uniformly dispersed grains that multiply scatter light toward the camera.

Figure 2

Various expectations for the hindered settling function $H\left(\varphi \right)=v/v_s$ versus volume fraction $\varphi$. See Eq. (5) for formulas and special values. Crosses are experimental data [6, 12]; open circles are simulation data [20]; open plus sign is Kozeny-Carman at $\varphi =0.64$. The Snabre-Mills form is shown with the authors' recommended value of $b=5.6$ for the free parameter.

Figure 3

Kymographs for sedimentation of $d=365\mu \text{m}$ grains at different initial volume fractions, as labeled, constructed from the central 1 cm wide portion in a time series of images as shown in Fig. 1. $\text{Height}=0$ is the bottom of the container. In all cases, the supernatant appears as a dark region which grows down from the top, the packing as a dark region that grows up from the bottom, and the dispersion as a narrowing bright region in between. Interface positions (small points) are determined by peaks in vertical intensity gradients. The uncertainty in location is comparable to symbol size. The solid lines are linear fits to interface position versus time, giving $v$ and $v_c$ for upper and lower interfaces, respectively. The expectation for the position of the lower interface based on the $v$ and Eq. (6) is plotted as a dashed maroon line.

Figure 4

Hindered settling function $H=v/v_s$ for suspensions with uniform volume fraction $\varphi$, where $v$ is the average settling speed and $v_s$ is the single-grain Stokes settling speed. The small solid black circles are our data, based on four different particle sizes (Table 3). All other data are taken from the literature (Table 1 and Supplemental Material [14]): Open red circles are for larger non-Brownian particles with $\mathrm{Pe}>O\left({10}^8\right)$ [11, 12, 13, 30, 31]. Small solid blue squares are for smaller Brownian particles with $\mathrm{Pe}<O\left({10}^8\right)$ [15, 16, 32, 33, 34, 35, 36, 37, 38]. The solid curves are the Richardson-Zaki form $H\left(\varphi \right)={(1-\varphi )}^n$ with exponents as labeled. The dashed curve, nearly indistinguishable from $n=4.5$, is $H=exp[-4.76\varphi -5.75{\varphi }^3]$. The dotted curve is the Kozeny-Carman form $H=18K={(1-\varphi )}^3/\left(10{\varphi }^2\right)$ for sintered spheres; the value for random close packing at $\varphi \to {\varphi }_c=0.64$ is particularly well established [28, 29] and is shown as an open cross. The dashed line in the inset is $(1-6.55\varphi )$ [18].

Figure 5

Hindered settling function versus volume fraction for non-Brownian particles. Individual data sets are specified per Table 1.

Figure 6

Hindered settling function versus volume fraction for Brownian particles. Individual data sets are specified per Table 1. The dashed line in the inset represents Batchelor's $H=1-6.55\varphi$ prediction for the leading behavior [18]. The sparsely dotted curve is for the Snabre-Mills form with the authors' recommended value of $b=5.6$ [22].