Effect of viscosity on the shaking-induced fluidization in a liquid-immersed granular medium

A liquid-immersed granular medium is shaken vertically under a wide range of accelerations ($\mathrm{\Gamma }$ in dimensionless form) and frequencies $\left(f\right)$ and its fluidization process is studied. The granular medium is formed by settling and consists of two size-graded layers (particle diameter $d$) such that the upper layer is fine grained and is less permeable. When $\mathrm{\Gamma }>{\mathrm{\Gamma }}_{\mathrm{c}}$, a liquid-rich layer formed by the accumulated liquid at the two-layer boundary causes a gravitational instability. The upwellings of the instability are separated horizontally by a distance (wavelength) $\lambda$, and their amplitude grows exponentially with time $[\propto exp\left(pt\right)]$ at a growth rate $p$. We conduct experiments for two liquid viscosity cases such that the particle settling velocity $\left(V_{\mathrm{s}}\right)$ of the same particle differs by a factor of 17. We find that for both cases, ${\mathrm{\Gamma }}_{\mathrm{c}}$ is at a minimum in an optimum frequency band centered at $f\sim 100$ Hz. However, the high-viscosity (HV) case has a smaller ${\mathrm{\Gamma }}_{\mathrm{c}}$, a shorter $\lambda$, and a faster dimensionless growth rate $[p^{\prime }=p/(V_{\mathrm{s}}/d)]$. We also measure granular rheology under an oscillatory shear and find that (i) interparticle friction decreases when the strain amplitude becomes large and (ii) friction is smaller for the HV case. From (i), we infer that the shear strain of the shaking experiments becomes largest at around $f\sim 100$ Hz. We consider that (ii) is a consequence of liquid lubrication and is a reason for a smaller ${\mathrm{\Gamma }}_{\mathrm{c}}$ for the HV case. We show that the low- and high-frequency limits of the optimum frequency band can be explained by introducing critical values of dimensionless jerk (i.e., time derivative of acceleration) $J$ and dimensionless shaking energy $S$. The low-frequency limit corresponds to the requirement that in order to unjam the particles, the period of shaking $\left(1/f\right)$ must be shorter than the time needed for the particles to rearrange by settling $\left(d/V_{\mathrm{s}}\right)$, which also explains why the HV case is fluidized at a lower $f$ compared to the LV case. We apply the results of the linear stability analyses for Rayleigh-Taylor instability. Using the measured $\lambda$ and $p$, we infer that (i) only a thin layer beneath the two-layer boundary is mobile and the rest of the lower layer remains jammed and (ii) the effective viscosity of the upper granular layer relative to the liquid is smaller for the HV case as a result of smaller friction.

Figure 1

(a) An experimental setup of the shaking experiments. The thicknesses of the layers are those at the start of shaking and are averages of all experiments (125 runs). LV and HV indicate low- and high-viscosity cases, respectively. $d$ is the particle diameter. (b) Setup for the oscillatory shear rheology measurements.

Figure 2

Comparison of LV (water) and HV (glycerine solution) cases at a shaking condition of $40.7\pm 0.3\mathrm{m}/{\mathrm{s}}^2 \left(\mathrm{\Gamma }=7.4\pm 0.7\right)$ and 40 Hz (see also Movie 1 in the Supplemental Material [15]). The timings of the two cases are scaled by a factor of $\simeq 17$ ($V_{\mathrm{s}}$ scaling). (a) LV case at $40.5 \mathrm{m}/{\mathrm{s}}^2 \left(\mathrm{\Gamma }=6.9\right)$. (b) HV case at $40.9 \mathrm{m}/{\mathrm{s}}^2 \left(\mathrm{\Gamma }=7.8\right)$. (c) Close-up of a rectangular section at $t=5$ s in (a). $\lambda$ is the instability wavelength, and $w$ is the plume width. (d) Same as (c) but at $t=85.2$ s in (b). Yellow (light) lines at the reference times $t=9.9$, 168 s trace the surface of the granular layer and the two-layer boundary. Superimposed blue (dark) horizontal lines are these heights at $t=0$ s, and vertical arrows indicate the displacements. $\delta h$ is the compaction of the whole granular layer. I, II, and III indicate the three stages defined using the relative amplitude [Eq. (13)].

Figure 3

(a) Displacements (five point running averaged) of the surface (blue) and the two-layer boundary (red) of the granular medium, as a function of time for the LV case shown in Fig. 2. A black arrow indicates compaction $\left(\delta h\right)$. (b) Same as (a) but for the HV case shown in Fig. 2. (c) Amplitude $\left(\delta z\right)$ of the two-layer boundary for the experiment shown in (a). A triangle indicates the maximum amplitude. Colored stars correspond to the same timing as shown in Fig. 2. A black line indicates the exponential fit to the instability growth. I, II, and III indicate the three stages defined using the relative amplitude [Eq. (13)]. (d) Same as (c) but for the experiment shown in (b).

Figure 4

(a) Evolution of the two-layer boundary profile of the LV case shown in Fig. 2 at 1 s intervals between $t=0$ s and $t=5$ s. The profiles are incrementally shifted upward by 2 mm. (b) Amplitude spectra of the profiles shown in (a). Triangles indicate the peaks. The spectra are incrementally shifted upward by 0.2 mm. At $t=5$ s the peak spectral amplitude is 1.6 mm, and its wavelength is $\lambda \simeq$ 20 mm. (c),(d) Same as (a),(b) but for the HV case at 16.8-s intervals between $t=0$ s and $t=84$ s. The profiles are incrementally shifted upward by 5 and 0.5 mm for (c) and (d), respectively. At $t=84$ s the peak spectral amplitude is 2.6 mm and its wavelength is $\lambda \simeq 13$ mm.

Figure 5

Time evolution of the total power (power spectrum summed for the whole wave-number range) of the two-layer boundary for the LV and HV cases shown in Fig. 4. Circles correspond to the timing of each amplitude spectrum shown in Fig. 4. Vertical broken lines indicate the timing when the shaking stops.

Figure 6

Oblique view images of the LV and HV cases at $41.4\pm 1.4\mathrm{m}/{\mathrm{s}}^2 \left(\mathrm{\Gamma }=7.5\pm 0.9\right)$ and 50 Hz (see also Movie 2 in the Supplemental Material [15]). The width of the cell is 99.4 mm. (a) LV case at $40.1 \mathrm{m}/{\mathrm{s}}^2 \left(\mathrm{\Gamma }=6.8\right)$. At $t=1.6$ s, eruption occurs by sand boil. At $t=2.2$ s, about seven red patches emerge at the surface which correspond to the upwellings. (b) Same as (a) but for the HV case at $42.1 \mathrm{m}/{\mathrm{s}}^2 \left(\mathrm{\Gamma }=8.1\right)$. At $t=27$ s, about 11 upwellings appear. Planforms of the cellular structures are evident, with upwellings (red patches) at the centers and downwellings at the rims.

Figure 7

Acceleration dependence of the (a) amplitude $\delta z$ at the two-layer boundary and (b) surface displacement (compaction $\delta h$ indicated by an arrow) of the granular medium for the HV case. Acceleration range is 2.0–$40.9 \mathrm{m}/{\mathrm{s}}^2 \left(0.4\le \mathrm{\Gamma }\le 7.8\right)$ and the frequency is fixed at 40 Hz (see also Movie 3 in the Supplemental Material [15]). Shaking stops at $t=85$ s. Reference time is $t=168$ s. Data for $2.0 \mathrm{m}/{\mathrm{s}}^2$ is in regime IIa (transition with sand boil) and other data are in regime III (flame). Regimes are defined using $\delta z^{\prime }$ [Eq. (13)] (see Sec. 4b2 for details).

Figure 8

Frequency dependence of (a) amplitude $\delta z$ at the two-layer boundary and (b) surface displacement (compaction $\delta h$ indicated by an arrow) of the granular medium for the HV case. Frequency range is 10–2000 Hz and the acceleration is fixed at $8.01\pm 0.49\mathrm{m}/{\mathrm{s}}^2 \left(\mathrm{\Gamma }=1.54\pm 0.09\right)$. The 1000–2000-Hz data are in regime Ib (percolation), 10–20-Hz data are in regime IIa (transition with sand boil), and 30–300-Hz data are in regime III (flame). Regimes are defined using $\delta z^{\prime }$ [Eq. (13)] (see Sec. 4b2 for details).

Figure 9

Regime diagrams of the experiments plotted in the parameter space of shaking frequency and dimensionless acceleration $\mathrm{\Gamma }$ for the LV case (left column, 73 runs) and HV case (right column, 52 runs). Marker shapes indicate the regimes. (a) Relative amplitude $\left(\delta z^{\prime }\right)$ of the LV case at the reference time $t=9.9$ s indicated by the different marker colors (sizes). The ranges of $\delta z^{\prime }$ (mm) are as follows (in the order of increasing marker size): black, $\delta z^{\prime }<0.01$; blue, $0.01\le \delta z^{\prime }<0.05$; purple, $0.05\le \delta z^{\prime }<0.1$; light blue, $0.1\le \delta z^{\prime }<0.15$; green, $0.15\le \delta z^{\prime }<0.6$; orange, $0.6\le \delta z^{\prime }<1.3$; pink, $1.3\le \delta z^{\prime }<2.0$; red, $\delta z^{\prime }\ge 2.0$. The black, blue, and red broken lines indicate ${\mathrm{\Gamma }}_{\mathrm{c}:\min }=0.58,S_{\mathrm{c}}=0.01$, and $J_{\mathrm{c}}=4$, respectively, and the domain defined by $\mathrm{\Gamma }>{\mathrm{\Gamma }}_{\mathrm{c}:\min },S>S_{\mathrm{c}}$, and $J>J_{\mathrm{c}}$, bounded by these three lines, is indicated in blue. (b) Same as (a) but for the HV case at the reference time $t=168$ s. The black, blue, and red broken lines indicate ${\mathrm{\Gamma }}_{\mathrm{c}:\min }=0.13,S_{\mathrm{c}}=0.003$, and $J_{\mathrm{c}}=20$, respectively, and the domain bounded by these three lines is indicated in pink. The three boundaries in thin broken lines are those of the LV case indicated in (a) and are shown for comparison. (c) Dimensionless growth rate $\left(p^{\prime }\right)$ of the LV case indicated by the different marker colors (sizes). The ranges of $p^{\prime }$ are as follows (in the order of increasing marker size): black, $p^{\prime }<0.004$; blue, $0.004\le p^{\prime }<0.006$; light blue, $0.006\le p^{\prime }<0.008$; orange, $0.008\le p^{\prime }<0.02$; pink, $0.02\le p^{\prime }<0.03$; red, $p^{\prime }\ge 0.03$. (d) Same as (c) but for the HV case at $t=168$ s. (e) Compaction $\delta h$ (mm) of the LV case at the reference time $t=9.9$ s indicated by the different marker colors (sizes). The ranges of $\delta h$ (mm) are as follows (in the order of increasing marker size): black, $\delta h<0.5$; blue, $0.5\le \delta h<1.0$; light blue, $1.0\le \delta h<1.5$; orange, $1.5\le \delta h<2.0$; pink, $2.0\le \delta h<2.5$; and red, $\delta h\ge 2.5$. (f) Same as (e) but for the HV case at $t=168$ s.

Figure 10

(a) Dimensionless growth rate $p^{\prime }$ vs relative amplitude $\delta z^{\prime }$ (at the reference time $t=9.9$ s) for the LV case. The marker shapes indicate the regimes, and the colors (sizes) indicate $\delta z^{\prime }$ and are defined the same as in Fig. 9. (b) Same as (a) but for the HV case ($\delta z^{\prime }$ is at the reference time $t=168$ s). (c) Compaction $\delta h$ vs relative amplitude $\delta z^{\prime }$ for the LV case at the reference time $t=9.9$ s. A broken horizontal line indicates the compaction calculated using a permeable flow model [Eq. (26)] at $t=9.9$ s using the parameters for the LV case. (d) Same as (c) but for the HV case at the reference time $t=168$ s. A broken horizontal line was similarly calculated at $t=168$ s using the parameters for the HV case.

Figure 11

Comparison of the $\mathrm{\Gamma }$ dependence of instability and compaction for LV (blue) and HV (red) cases at a frequency of $40\pm 10$ Hz (see Movie 3 in the Supplemental Material [15] for selected examples from the HV case). Marker shapes correspond to the three regimes indicated in Fig. 9. Broken lines indicate the power-law fits. (a) Relative amplitude $\delta z^{\prime }$ at the reference times ($t=9.9$ and 168 s for LV and HV cases, respectively). Broken horizontal lines at $\delta z^{\prime }=0.1$, 0.6 mm define the onsets of regime II (triangles) and regime III (circles), respectively. Power-law exponents of the fits are 1.65 (LV) and 0.87 (HV). (b) Dimensionless growth rate $p^{\prime }$. Power-law exponents of the fits are 1.01 (LV) and 0.21 (HV). (c) Compaction $\delta h$ at reference times. Power-law exponents of the fits are 0.53 (LV) and 0.28 (HV). (d) Wave number at which the spectral amplitude attains a peak value.

Figure 12

(a) Storage modulii $\left(G^{\prime }\right)$ [Eq. (5)] of the liquid-immersed granular medium as a function of peak stress ${\sigma }_{\mathrm{p}}$ measured using the setup shown in Fig. 1. Rightward pointing arrows indicate the incremental increase of ${\sigma }_{\mathrm{p}}$ during the measurements. ${\sigma }_{\mathrm{y}}$ (upward pointing arrow) is the yield stress. ${\sigma }_{\mathrm{f}}$ is the frictional stress calculated using Eq. (14) for LV and HV cases. (b) $G^{\prime }$ plotted as a function of the particle diameter scaled strain ${\gamma }_{\mathrm{p}}$ [Eq. (15)]. (c) Loss tangent $\left(tan\delta \right)$ [Eq. (7)] plotted as a function of ${\gamma }_{\mathrm{p}}$. A horizontal broken line indicates $tan\delta =1$. (d) A schematic Stribeck curve and the three regimes (modified from [19]), as a function of So [Eq. (16)]. (e), (b), and (c) rescaled using So.

Figure 13

(a) Schematic diagram (not to scale) showing a situation in which a low-density (small packing fraction $\varphi$) thin layer (thickness $h$) underlies a high-density thick layer (thickness $D$). The lower layer has a smaller viscosity $\left(\eta \right)$ because of a smaller $\varphi$. A Rayleigh-Taylor instability with a wavelength $\lambda$ and a permeable flow (arrows) occur at the same time. (b) Wavelength $\lambda$ and (c) dimensionless growth rate $p^{\prime }$ as a function of $h$ for LV (blue) and HV (red) cases, calculated using Eqs. (22) and (23). Here the upper to lower layer viscosity ratio $\epsilon ={\eta }_{\mathrm{eff}:\mathrm{upper}}/{\eta }_{\mathrm{eff}:\mathrm{lower}}$ [Eq. (21)] of the LV and HV cases are 117 and 45, respectively. $\lambda$ and $p^{\prime }$ values of the experiments shown in Fig. 2 are indicated by the horizontal broken lines. Large triangles at the intersections of the solid and broken lines indicate the solutions for $h$. (d) Relative viscosity ${\eta }_{\mathrm{r}}$ as a function of packing fraction $\varphi$ calculated using Eq. (20) for three values of maximum packing fraction ${\varphi }_{\max }$. The three broken vertical lines indicate different values of ${\varphi }_{\max }$. Triangles indicate ${\eta }_{\mathrm{r}}$ at $\varphi =0.60$ and 0.49, which are the initial values of $\varphi$ of the upper and lower layers, respectively.

Figure 14

Dimensionless growth time [Eq. (24)] and percolation time [Eq. (25)] as a function of the low-density layer thickness $h$. Solid lines indicate growth time for the LV (HV) case using $\epsilon =117 \left(\epsilon =45\right)$. These $\epsilon$ values are the same as those used in Fig. 13. The broken line indicates percolation time. The measured growth time (dimensionless) in regime III (flame) $\left(22\le {\tau }_{\mathrm{grow}}^{\prime }\le 161\right)$ is indicated by an arrow. Schematic diagram shows the inferred regime transitions with the increase of $h$.