Expanding holes driven by convectionlike flow in vibrated dense suspensions

Surface instabilities in vertically vibrated suspensions of various powders dispersed in silicone oil are investigated in quasi-two-dimensional (2D) and quasi-one-dimensional (1D) systems. As vibration acceleration exceeded a critical value, the flat surface became unstable against a finite-amplitude perturbation. We found an expanding hole or viscous fingerlike pattern in the quasi-2D system and segregation between dried and wet areas in the quasi-1D system. We show that these instabilities are accompanied by convectionlike flow at their rim and in the quasi-1D system, the height of the convectionlike flow can be scaled by acceleration, vibration frequency, diameter of the dispersed powder, mean density of the suspension, and viscosity of silicone oil. We propose a simple model that accounts for the scaling and concentric motion of the convectionlike flow.

Figure 1

Sequence of instabilities in a quasi-2D system: [(a)–(d)] Hole-expanding sequence at different times $t=0$, 2, 4, and 6 s (where $r=0.2\text{mm}$, $\eta =500{\text{mm}}^2/\text{s}$, $\varphi =0.56$, $\Gamma =230\text{m}/{\text{s}}^2$, and $f=40\text{Hz}$); (e) separated state (where $r=0.2\text{mm}$, $\eta =500{\text{mm}}^2/\text{s}$, $\varphi =0.56$, $\Gamma =290\text{m}/{\text{s}}^2$, and $f=40\text{Hz}$); (f) viscous fingerlike pattern (where $r=0.4\text{mm}$, $\eta =500{\text{mm}}^2/\text{s}$, $\varphi =0.60$, $\Gamma =270\text{m}/{\text{s}}^2$, and $f=60\text{Hz}$).

Figure 2

(Color online) (a) Phase diagram of glass beads (0.2 mm) in silicone oil suspension $\left(500{\text{mm}}^2/\text{s}\right)$, shown as acceleration vs packing fraction. The vibration frequency was 40 Hz. An expanding hole is observed in the circle (red) symbol region and a viscous fingerlike pattern is observed in the triangle (green) symbol region. At packing fraction $\varphi =58\%$, sometimes both expanding hole and viscous fingerlike pattern are observed simultaneously because of the nonuniformity of the packing fraction. (b) Phase diagram of glass beads (0.4 mm) in silicone oil suspension $\left(500{\text{mm}}^2/\text{s}\right)$, shown as acceleration vs frequency. The packing fraction $\varphi =56\%$. In both figures, the expanding hole region (circles) indicates that the hole reaches the wall and the shrinking region (crosses) indicates that the hole starts shrinking. Near the onset of acceleration in the shrinking region, the hole can expand but starts shrinking.

Figure 3

(Color online) (a) Change in area with time for different accelerations. $\Gamma$ varies from 160 to 240, 260, and $290\text{m}/{\text{s}}^2$. (b) Scaling of the first stage of time evolution. We cut the first one second to eliminate differences in the initial condition. In both (a) and (b), $r=0.2\text{mm}$, $\eta =500{\text{mm}}^2/\text{s}$, $\varphi =0.56$, and $f=40\text{Hz}$. The thickness of the layer is $\sim 0.6\text{cm}$. (c) Change in the characteristic time scale $\tau$ with acceleration.

Figure 4

(Color online) Time sequence in a quasi-1D system: (a) lateral view of the time sequence of separation at, from the top, $t=0$, 4, 8, 12, and 16 s; (b) frame format of the separated state and the convectionlike flow. $R=\text{convectionlike}$ flow, $h_a=\text{mean}$ height of the suspension region, $h_p=\text{height}$ of the convection, and $\delta v=\text{velocity}$ of circulation of the convectionlike flow.

Figure 5

(Color online) Time evolution of front propagation and scaling in a quasi-1D cell: (a) time evolution of the front propagation. The vertical axis is the length of the suspension area $R$. The diameter of the glass beads is 0.2 mm; the viscosity of the silicone oil is $100{\text{mm}}^2/\text{s}$; the dashed line is a fitting curve using Eq. (5). (b) Scaling of the height of the convectionlike flow. The large black bullet represents a stable hole in cornstarch and CsClaq suspension from [1]. For all data except cornstarch suspension, the packing fraction $\varphi$ is 56%. The total volume of suspension in each data is different. However, at a sufficient total volume, the height of the suspension becomes almost independent of total volume: $+$ (60 Hz, $100{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); ○ (60 Hz, $100{\text{mm}}^2/\text{s}$, 0.1 mm, $1.83\text{g}/{\text{cm}}^3$); $\ast$ (60 Hz, $100{\text{mm}}^2/\text{s}$, 0.05 mm, $1.83\text{g}/{\text{cm}}^3$); • (blue/gray) (60 Hz, $236{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); $\times$ (60 Hz, $358{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); ◻ (60 Hz, $500{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); ◇ (70 Hz, $100{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); △ (50 Hz, $100{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); ▽ (40 Hz, $100{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); $▷$ (30 Hz, $100{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); $◁$ (20 Hz, $100{\text{mm}}^2/\text{s}$, 0.2 mm, $1.83\text{g}/{\text{cm}}^3$); ◼ (60 Hz, $30{\text{mm}}^2/\text{s}$, 0.1 mm, $0.957\text{g}/{\text{cm}}^3$); ▲ (50 Hz, $30{\text{mm}}^2/\text{s}$, 0.1 mm, $0.957\text{g}/{\text{cm}}^3$); ▼ (40 Hz, $30{\text{mm}}^2/\text{s}$, 0.1 mm, $0.957\text{g}/{\text{cm}}^3$); $▶$ (30 Hz, $30{\text{mm}}^2/\text{s}$, 0.1 mm, $0.957\text{g}/{\text{cm}}^3$); $◀$ (60 Hz, $500{\text{mm}}^2/\text{s}$, 0.2 mm, $5.46\text{g}/{\text{cm}}^3$).