Unstable Invasion of Sedimenting Granular Suspensions

We investigate the development of mobility inversion and fingering when a granular suspension is injected radially between horizontal parallel plates of a cell filled with a miscible fluid. While the suspension spreads uniformly when the suspension and the displaced fluid densities are exactly matched, even a small density difference is found to result in a dense granular front which develops fingers with angular spacing that increase with granular volume fraction and decrease with injection rate. We show that the timescale over which the instability develops is given by the volume fraction dependent settling timescale of the grains in the cell. We then show that the mobility inversion and the nonequilibrium Korteweg surface tension due to granular volume fraction gradients determine the number of fingers at the onset of the instability in these miscible suspensions.

Figure 1

(a) A schematic of the experimental cell consisting of parallel plates filled with a miscible fluid and the suspension injection system. Quadrant view of the advancing granular suspension with ${\varphi }_g=0.2$ and $Q=0.05{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ at time $t=20\mathrm{s}$, 50 s, 100 s, 150 s when ${\rho }_s={\rho }_d$ (b), and ${\rho }_s=1.07{\rho }_d$ (c). The granular component appears blue and the fluid component in the suspension fluoresces green under combined white-blue-UV lighting. Fingers are observed to develop when ${\rho }_s\ne {\rho }_d$. See movies in Supplemental Material [16].

Figure 2

(a),(b) The radius of the fluid front $r_f$ and granular front $r_g$ as a function of time when ${\rho }_s={\rho }_d$ (a) and ${\rho }_s=1.07{\rho }_d$ (b). Inset: A green color enhanced image used to identify $r_f$, which leads $r_g$. The original image corresponds to $t=100\mathrm{s}$ in Fig. 1. An estimated front of the suspension $R(t)$ plotted assuming Eq. (1) fit ${\alpha }_h=1.22$. (c),(d) The average volume fraction $⟨\varphi ⟩$ as a function of distance $r$ and time $t$ when ${\rho }_s={\rho }_d$ (c) and ${\rho }_s\ne {\rho }_d$ (d). The color map is scaled by the maximum volume fraction $⟨\varphi {⟩}_{\max }=0.25$ in each case to highlight the variation in $⟨\varphi {⟩}_{\theta }$.

Figure 3

(a) Radial cumulative density reveals peaks corresponding to development of fingers denoted by markers plotted in 10 s time intervals. (b) The angular self-correlation function $K_{\varphi }$ when ${\rho }_s=1.07{\rho }_d$. The peak indicated by the arrow corresponds to growth of the maximum amplitude mode.

Figure 4

Various finger patterns observed with a black-and-white camera after a fixed volume of suspension is injected (${\rho }_s=1.07{\rho }_d$). The scale bar is 20 mm.

Figure 5

(a) Systematic variations in ${\theta }_m$ can be observed with $Q$ and ${\varphi }_g$. (b) The observed time when annulus form $t_o$ versus calculated settling time $t_s$ of the grains solving Eq. (3) under various $Q$ and ${\varphi }_g$. The line corresponds to $t_o/t_s=1$. (c) The measured radius $R_c$ when fingers appear. The lines correspond to calculated $t_s$ obtained by solving Eq. (3). (d) The calculated mode number from Eq. (5) is found to capture the measured mode number $n_m$ over the entire range of the $Q$ and ${\varphi }_g$. The line corresponds to slope 1.