Sinking dynamics and splitting of a granular droplet

Recent experimental results have shown that binary granular materials fluidized by combined vibration and gas flow exhibit Rayleigh-Taylor-like instabilities that manifest themselves in rising plumes, rising bubbles, and the sinking and splitting of granular droplets. This work explores the physics behind the splitting of a granular droplet that is composed of smaller and denser particles in a bed of larger and lighter particles. During its sinking motion, a granular droplet undergoes a series of binary splits resembling the fragmentation of a liquid droplet falling in a miscible fluid. However, different physical mechanisms cause a granular droplet to split. By applying particle image velocimetry and numerical simulations, we demonstrate that the droplet of high-density particles causes the formation of an immobilized zone underneath the droplet. This zone obstructs the downwards motion of the droplet and causes the droplet to spread and ultimately to split. The resulting fragments sink at inclined trajectories around the immobilized zone until another splitting event is initiated. The occurrence of consecutive splitting events is explained by the reformation of an immobilized zone underneath the droplet fragments. Our investigations identified three requirements for a granular droplet to split: (1) frictional interparticle contacts, (2) a higher density of the particles composing the granular droplet compared to the bulk particles, and (3) a minimal granular droplet diameter.

Figure 1

Comparison between experimental measurements (top row) and simulation results (bottom row) on the dynamics of the splitting of a granular droplet ($D=50\mathrm{mm}$) in a vibro-gas-fluidized bed using $f=10\mathrm{Hz}$, $A=1\mathrm{mm}$, and $U/U_{\mathrm{mf}}=0.95$ (exp.) or $U/U_{\mathrm{mf}}=0.97$ (sim.). (a) Time series showing the evolution of a granular droplet during fragmentation. Bulk particles are black; droplet particle are white (particle properties according to Table 1). In the experiments, white tracer particles were added to the bulk phase for PIV measurements. (b) Velocity field of the bulk particles around the granular droplet as extracted from the red-framed region in (a) at 1.7 s. The granular droplet is white. The bulk particle phase is color coded according to the velocity magnitude of the bulk particles. The velocity data were averaged over a period of 0.3 s to reduce noise in the PIV data.

Figure 2

The dynamics of the bulk and granular droplet particles at different stages of the vibration cycle (phase φ). Each row illustrates one of the five stages of motion identified with increasing φ from (a) to (e). The sketch in the left top corner of each row indicates the vertical displacement of the bed at the given phase (red dot) and the extent of the corresponding stage of motion (red part of the sine curve). In the first column, the droplet particles are gray and the normal contact force ${\mathit{F}}_{\mathrm{c},\mathrm{n}}$ acting between contacting bulk particles is shown as black lines. The opacity of the black lines increases with increasing contact force magnitude. In the second column, the granular droplet is shown as a gray contour and the particle velocity $\mathit{v}$ is visualized via color coding and vector fields. Here, $\mathit{v}$ is calculated in the reference frame of the vibrating bed, i.e., the $y$ component of $\mathit{v}$ is reduced by the instantaneous vertical velocity of the confining walls. All results are obtained in a CFD-DEM simulation using $A=1\mathrm{mm}$, $f=10\mathrm{Hz}$, and $U/U_{\mathrm{mf}}=0.97$. The third column sketches the overall motion of the granular droplet (gray) and the bulk particle phase (green); light green indicates a mobilized and dark green an immobilized bulk phase.

Figure 3

Detachment process of the right daughter droplet and the initiation of a secondary split. The droplet particles are shown in gray; frames in the top row show the particle-velocity field at the mobilization stage ($\phi =\pi$), whereas the frames at the bottom row show the normal contact forces between the bulk particles and are taken in the corresponding spreading stage ($\phi =3/2\pi$). Each frame shows the same phase in the vibration cycle. The simulation was performed using $A=1\mathrm{mm}$, $f=10\mathrm{Hz}$, and $U/U_{\mathrm{mf}}=0.92$. $U/U_{\mathrm{mf}}$ is smaller than in Fig. 2 to make the second binary split occur faster.

Figure 4

Influence of the coefficient of friction μ on the sinking and fragmentation dynamics of a granular droplet of $D=50\mathrm{mm}$. (a) Time series of granular droplet dynamics as obtained by CFD-DEM simulations using $A=1\mathrm{mm}$, $f=10\mathrm{Hz}$, and $U/U_{\mathrm{mf}}=0.84$. Top row: simulation with a friction coefficient of $\mu =0.3$; Bottom row: friction coefficient $\mu =0$. (b) Velocity of the bulk particles surrounding the granular droplet (white area) in the framed regions at $t=1.6\mathrm{s}$ (top row) and $t=0.4\mathrm{s}$ (bottom row). The background color quantifies the magnitude of the bulk particle velocity |$\mathit{v}$|. The velocity data are time averaged over 0.1 s. The simulation used a comparatively low value of $U/U_{\mathrm{mf}}$ to reduce the emergence of large gas bubbles in the zero-friction simulation.

Figure 5

Numerical study of the effect of the density ratio ${\rho }_{\mathrm{d}}/{\rho }_{\mathrm{b}}$ and size ratio $d_{\mathrm{d}}/d_{\mathrm{b}}$ between the droplet and the bulk particles on the granular droplet dynamics. Within a row, $d_{\mathrm{d}}/d_{\mathrm{b}}$ is constant (top to bottom: 0.645, 1, 1.55) and within a column ${\rho }_{\mathrm{d}}/{\rho }_{\mathrm{b}}$ is constant (left to right: 2.4, 1). In all simulations, $d_{\mathrm{b}}=1.7\mathrm{mm}$, ${\rho }_b=2500\mathrm{kg}/{\mathrm{m}}^3$, $A=1\mathrm{mm}$, $f=10\mathrm{Hz}$, and $U/U_{\mathrm{mf}}=0.92$. Each simulation result (a)–(f) is visualized in two panels: the left panel displays the instantaneous shape of the granular droplet (white) surrounded by the bulk particles (black) after a given time $t$. The red square marks the initial position of the granular droplet. The right panel shows the normalized superficial velocity of the gas flow $|{\varepsilon }_{\mathrm{f}}{u}_{\mathrm{f}}|/U$ (color coded) around the same granular droplet at $t$. The gas velocity data were averaged over one vibration period (0.1 s).

Figure 6

Splitting angle as a function of vibration strength, fluidization level, and droplet diameter: (a) Splitting angle α of a granular droplet as a function of vertical distance traveled $y_{\mathrm{h}}$. The solid blue line represents the average splitting angle and the blue-shaded area represents the standard deviation based on three repetitions. The dashed red line is fitted using $\alpha \left(y_h\right)=k_1{e}^{-k_2{y}_h}+{\alpha }_{\mathrm{asym}}$. The inset shows the trajectories of the two daughter droplets evolving from a granular droplet of size $D=30\mathrm{mm}$ for $U/U_{\mathrm{mf}}=0.98$, $f=30\mathrm{Hz}$, and $A=0.05\mathrm{mm}$. The red arc shows the construction of α at $y_{\mathrm{h}}=8.5\mathrm{cm}$. (b) Effect of vibration strength on the asymptotic angle; the black squares plot ${\alpha }_{\mathrm{asym}}$ for $D=30\mathrm{mm}$ and $f=30\mathrm{Hz}$ with varying $A$ and the red triangles plot ${\alpha }_{\mathrm{asym}}$ for $D=30\mathrm{mm}$ and $A=0.05\mathrm{mm}$ for varying $f$. (c) Effect of $U/U_{\mathrm{mf}}$ on ${\alpha }_{\mathrm{asym}}$ for $D=30\mathrm{mm}$, $f=30\mathrm{Hz}$, and $\mathrm{\Gamma }=0.18$. (d) Effect of granular droplet size $D$ on ${\alpha }_{\mathrm{asym}}$ for $f=30\mathrm{Hz}$, $\mathrm{\Gamma }=0.18$, and $U/U_{\mathrm{mf}}=0.96$. The error bars in (b), (c), and (d) represent the standard deviation based on three repetitions.

Figure 7

Vertical and horizontal velocities of the sinking daughter droplets during their first split as a function of vibration strength (a) and gas flow (b). The results represent the averaged velocities of the leading edges as obtained by digital image analysis of the experiments (see S1 in Ref. [23]). The dashed lines in the background plot the asymptotic angle through ${\alpha }_{\mathrm{asym}}=2{tan}^{-1}\left(\frac{v_{\mathrm{hor}}}{v_{\mathrm{vert}}}\right)$. The error bars show the standard deviation based on three experimental runs.