Mechanism of anomalous sinking of an intruder in a granular packing close to incipient fluidization

Objects released into a granular packing close to incipient fluidization may float or sink depending on their density. Contrary to intuition, Oshitani et al. [Phys. Rev. Lett. 116, 068001 (2016)] reported that under certain conditions, a lighter sphere can sink further and slower than a heavier one. While this phenomenon has been attributed to a local fluidization around the sinking sphere, its physical mechanisms have not yet been understood. Here, we studied this intriguing phenomenon using both magnetic resonance imaging and discrete particle simulation. Our findings suggest that local fluidization around the sinking sphere and the formation and detachment of gas bubbles play a critical role in driving this anomaly. An analysis of forces acting on the intruder revealed that the upward-directed fluid force acting on a sphere is almost fully counterbalanced by the sum of the net contact forces and the gravitational force acting downward, when the sphere density is close to the bulk density of the granular packing (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}\approx 1$). At the time when bubbles detach from the sphere, the gas pressure gradient experienced by the sphere is slightly attenuated and the sphere is pushed downward by the particle cap located on top of the sphere. Because the deviations from the force equilibrium are small, the sphere sinks slowly. Even after the sphere has reached its final stable depth, local fluidization in combination with bubble formation remains in the proximity of the sphere.

Figure 1

Final sinking depth of intruding spheres in a powder bed with forced aeration ($U_0/U_{\mathrm{mf}}=0.95, D_{\mathrm{sphere}}=30\mathrm{mm}$). Anomalous sphere sinking occurs when the sphere density (${\rho }_{\mathrm{sphere}}$) is close to the bulk density of a particle bed (${\rho }_{\mathrm{bed}}$). The final sinking depth ($h_{\mathrm{final}}$) is normalized by the sphere diameter ($D_{\mathrm{sphere}}$). Obtained from Oshitani et al. [20].

Figure 2

Computational domain. Particles are colored according to their initial vertical position in the bed.

Figure 3

(a) MR images of spheres of different densities (rows) sinking into a bed that is kept at near incipient fluidization ($U_0/U_{\mathrm{mf}}=0.95$). Note that the time increments vary with ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}$. (b) Bubble formation near a partially submerged sphere (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=0.65$). Dashed line and arrows indicate the perimeter of the sphere and bubbles, respectively. (c) Bubble formation in the proximity of a slow sinking sphere (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.00$). The dynamics of the detaching bubbles are clearly observable (see Supplemental Video 1 [44]).

Figure 4

Sphere sinking trajectories at $U_0/U_{\mathrm{mf}}=0.95$. $h$ is the center position of the spheres and increases downward. The initial bed surface was located at $h/d_{\mathrm{sphere}}=0.5$. The heaviest sphere ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=2.50$ reached the bottom inlet of the bed in both the MRI experiment and the simulation.

Figure 5

(a) A time series of 1D horizontal MR images of a sinking sphere with ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.00$. Dark contrasts outside the sphere represent bubbles. This measurement was performed at a temporal resolution of 1.96 ms. (b) Magnified view. Dashed line shows the sphere surface. Contrasts in these figures are adjusted to enhance visibility.

Figure 6

Fluidization-sensitive MR images (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.00, U_0/U_{\mathrm{mf}}=0.95$). During the slow and deep sinking, the sphere is surrounded by fluidized granular material (darker contrast) while gas bubbles detach from the sides of the sphere and reach the free surface. Once this chain of bubbles stops (i.e., bubbles do not reach the top surface of the bed anymore), the cap region above the sphere defluidizes (higher contrast) and the sphere stops sinking. A fluidized region (darker contrast) with some bubble formation remains at the sides and below the sphere. The relatively long acquisition time of 546 ms creates motion blurring for the sinking sphere (see Supplemental Video 3 [44]).

Figure 7

Time-averaged fluidization-sensitive MR images for intruding spheres with ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.00$, 1.20, and 2.50 after reaching their final depths ($U_0/U_{\mathrm{mf}}=0.95$).

Figure 8

(a) Numerical results of spheres of different densities (rows) sinking into a bed that is kept near incipient fluidization ($U_0/U_{\mathrm{mf}}=0.95$). The contours reflect the local void fraction. The results for the density ratios ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=0.65$, 1.00, 1.20, 1.40, and 2.50 are shown. The time increments depend on ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}$ (see Supplemental Video 4 [44]). (b) Three-dimensional presentation of the void structures inside the bed (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=0.65$). The isosurface of the local void fraction ${\varepsilon }_{\mathrm{local}}=0.80$ is plotted. (c) Three-dimensional presentation of the void structures inside the bed (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20$) (see Supplemental Video 5 [44]).

Figure 9

Final sinking depths in MRI experiments and numerical simulations at $U_0/U_{\mathrm{mf}}=0.95$. The blue dashed and green dash-dotted lines show the linear least-squares fittings of the MRI results (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=0.65$ and 1.20) and numerical results (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=0.65$, 1.00, 1.40, 1.50, 1.80, 2.00), excluding the anomalous results, respectively. The triangle and inverted triangle show the sinking depths reached at the first stop (step II) in the MRI experiment ($t=0.28\mathrm{s}$) and numerical simulation ($t=0.29\mathrm{s}$) that will appear in Fig. 14, respectively.

Figure 10

Vertical forces acting on a floating sphere (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=0.65, U_0/U_{\mathrm{mf}}=0.95$). A positive force acts vertically upward.

Figure 11

Schematic of the gas pressure drop in a uniformly packed granular bed with a forced gas flow. A sphere submerged in such a particle bed experiences a vertical, upward-directed fluid force due to the negative pressure gradient in the bed. The black dashed line shows the gas pressure distribution in an undisturbed bed. The red solid line shows a gas pressure distribution disturbed by the presence of a submerged sphere.

Figure 12

Vertical forces and velocity of a rapidly sinking sphere (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.80, U_0/U_{\mathrm{mf}}=0.95$).

Figure 13

Coordination number $z$ and force chains of normal contact forces ${\mathit{f}}_{\mathrm{C},\mathrm{n}}$ scaled by the gravitational force working on a particle. The coordination number shows the number of particles and spheres in contact. Only force chains with $|{\mathit{f}}_{\mathrm{C},\mathrm{n}}|/|m_{\mathrm{p}}\mathit{g}|>19.92$ are depicted (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.80, U_0/U_{\mathrm{mf}}=0.95$).

Figure 14

Semilog plot of the trajectories of slow and deep sinking spheres ($U_0/U_{\mathrm{mf}}=0.95$). The data are the same as shown in Fig. 4, but plotting on the spheres with ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.00$ for the MRI experiment and ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20$ for the numerical simulation. The slow and deep sinking occurs in step III after it stopped once at step II.

Figure 15

(a) Vertical forces acting on a slow and deep sinking sphere (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20, U_0/U_{\mathrm{mf}}=0.95$). (b) Enlarged view of the total force. (c) Sphere sinking velocity as a function of time. In (b), (c), the black lines plot the simulation results and the red lines plot the results smoothed by a Gaussian filter. Note that the velocity range in (c) is very different from that in Fig. 12. The insets in (a), (c) show a magnification of the data in the initial stage of the sinking process ($t\le 0.50$ s). For the highlighted section ($t=2.6–3.0\mathrm{s}$), further detailed discussions will be developed later.

Figure 16

Coordination number $z$ and force chains of normal contact forces (${\mathit{f}}_{\mathrm{C},\mathrm{n}}$) scaled by the gravitational force working on a particle. The coordination number shows the number of particles and spheres in contact. Only force chains with $|{\mathit{f}}_{\mathrm{C},\mathrm{n}}|/|m_{\mathrm{p}}\mathit{g}|>13.28$ are depicted (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20, U_0/U_{\mathrm{mf}}=0.95$).

Figure 17

Sphere velocity and forces acting on the sphere during $t=2.6–3.0\mathrm{s}$ (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20, U_0/U_{\mathrm{mf}}=0.95$). In the highlighted sections, the total force becomes negative and the sphere sinking velocity increases.

Figure 18

(a) Distribution of the local void fraction and gas pressure in a vertical, central plane during $t=2.91–2.96\mathrm{s}$ (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20, U_0/U_{\mathrm{mf}}=0.95$). Gas pressure isolines are plotted with an interval $\mathrm{\Delta }p=25\mathrm{Pa}$. The detached bubble extends the distance between successive isolines near the sphere and induces a relaxation of the fluid force acting on the sphere. (b) Particle velocity in the vertical direction. Downward motion of particles occurs in the cap region at the top of the sphere. Upward particle motion occurs locally when a bubble detaches.

Figure 19

Change in the local void fraction relative to the initial state $\mathrm{\Delta }{\varepsilon }_{\mathrm{local}}\left(t\right)={\varepsilon }_{\mathrm{local}}\left(t\right)-{\varepsilon }_{\mathrm{local}}\left(t=0\right)$ (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20, U_0/U_{\mathrm{mf}}=0.95$). The difference from the initial void fraction was calculated using the local void fraction in each CFD cell fixed in space. Only variations in the range of $\mathrm{\Delta }{\varepsilon }_{\mathrm{local}}=\pm 0.05$ are depicted. An “air cushion” exists always at the bottom of the sphere (see Supplemental Video 6 [44]).

Figure 20

MRI-determined sinking trajectories of ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.00$ spheres at different superficial velocities.

Figure 21

MRI-determined sinking trajectories of spheres with ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20$ for different superficial velocities.

Figure 22

MRI-determined sinking trajectories of spheres with ${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=2.50$ for different superficial velocities.

Figure 23

MRI-determined final sinking depths depending on superficial velocity. The triangle shows the sinking depths reached at the first stop (${\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.00$).

Figure 24

Numerical results of the sinking depths of a sphere as a function of the spring constants ($U_0/U_{\mathrm{mf}}=0.95, {\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=2.50, {\varepsilon }_0=0.398$).

Figure 25

Numerical results of the sinking depths of a sphere as a function of the coefficient of friction ${\mu }_{\mathrm{p}}$ ($U_0/U_{\mathrm{mf}}=0.94, {\rho }_{\mathrm{sphere}}/{\rho }_{\mathrm{bulk}}=1.20, {\varepsilon }_0=0.403$).