Mixing of a granular layer falling through a fluid

We analyze the granular Rayleigh-Taylor instability of densely packed grains immersed in a compressible or an incompressible fluid using numerical simulations and two types of experiments. The simulations are based on a two-dimensional (2D) molecular dynamics model and the experiments have been carried out in systems of grains immersed in water/glycerol (incompressible fluid) and in air (compressible fluid). The variation of the interstitial fluid is shown to generate different dynamical patterns and mixing properties of the granular systems. The results have been quantified using 2D autocorrelation functions, the power spectrum of the velocity field and velocity field histograms. Excellent agreement is found between the numerical simulations and the experiments.

Figure 1

(Color online) The experimental setup. The cell consists of two rectangular glass plates. Silicone sealing assures the cell to be leak proof. The front view shows a picture taken during an experiment with water/glycerol.

Figure 2

(Color online) For the CIV technique the gray-level of two pictures in a time sequence is compared. In red (dark gray) a sample area is defined in the first picture. In the following picture it is then tried to find a similar area with the same gray-level distribution within the search area shown in blue (light gray).

Figure 3

(Color online) Test of the CIV technique: the left picture shows the velocity field derived directly from the simulation data in comparison with the right picture where the velocity field is obtained by the CIV technique from two pictures of the simulation. Both pictures show a section from the middle of the cell where most particles are moving.

Figure 4

(Color online) Histograms of the velocity field derived by the CIV technique and the velocity field derived directly from the simulation data.

Figure 5

Assuming a Poiseuille flow, a tangential stress between the plates and the particle fluid flow is deduced. This is a fluid-mediated friction, present even without any interparticle contact or particle-plate contact.

Figure 6

Stress and pressure acting on the surface $dA$ of a unit volume $dV$.

Figure 7

(Color online) Comparison of the relative to the grains acceleration and relative to the grains velocity of the fluid to the absolute acceleration and absolute velocity of the grains. A system with plate friction due to a Poiseuille flow and added fluid mass to the particles (blue curves with circles) is compared to a system without friction due to a Poiseuille flow and without added fluid mass (red curves with squares).

Figure 8

The Janssen effect: the in-plane $P_g^{\parallel }$ stress is being deflected by the particles resulting in a normal stress $P_g^{\perp }$ on the plates.

Figure 9

Experiments showing different sedimentation patterns for the cases of air and water/glycerol.

Figure 10

(Color online) Simulations showing different sedimentation patterns for the cases of air and water/glycerol.

Figure 11

(Color online) The three rows in figure (a) and (b) show the time evolution of the simulations (a) and experiments (b) with air from left to right at times $t=0.05\text{s}$, $t=0.19\text{s}$, $t=0.36\text{s}$, $t=0.46\text{s}$, and $t=0.6\text{s}$. From top to bottom the rows illustrate the $u_x/\text{max}\left(\left|u_x\right|\right)$ component and the $u_y/\text{max}\left(\left|u_y\right|\right)$ component of the grain velocity vector field and the bottom row shows the density field ${\rho }_d/\text{max}\left({\rho }_d\right)$. All quantities have been normalized by its maximum absolute value.

Figure 12

(Color online) The three rows in figure (a) and (b) show the time evolution of the simulations (a) and experiments (b) with water/glycerol from left to right at times $t=0.8\text{s}$, $t=4.6\text{s}$, $t=8.4\text{s}$, $t=12.2\text{s}$, and $t=16.0\text{s}$. From top to bottom the rows illustrate the $u_x/\text{max}\left(\left|u_x\right|\right)$ component and the $u_y/\text{max}\left(\left|u_y\right|\right)$ component of the grain velocity vector field and the bottom row shows the density field ${\rho }_d/\text{max}\left({\rho }_d\right)$. All quantities have been normalized by its maximum absolute value.

Figure 13

(Color online) Histograms of the $x$ and $y$ component $\alpha$ of the velocity field after 0.2, 0.3, and 0.5 s from top to bottom in the case of air. The histograms show good agreement between simulations and experiments.

Figure 14

(Color online) Histograms of the $x$ and $y$ component $\alpha$ of the velocity field after 6, 10, and 15 s from top to bottom in the case of water/glycerol. The histograms show good agreement between simulations and experiments.

Figure 15

(Color online) In the first rows a time sequence of the normalized autocorrelation of the $x$ component of the velocity field $C_x\left(\mathbf{d}\right)$ is shown with $\mathbf{d}=\left(\genfrac{}{}{0ex}{}{x}{y}\right)$. The second rows show a time sequence of the normalized autocorrelation of the $y$ component of the velocity field $C_y\left(\mathbf{d}\right)$. In figure (a) and (b) simulations and experiments with grains and air are presented. Simulations and experiments with grains and water/glycerol are shown in the figures (c) and (d). The time steps from left to right correspond to Figs. 11, 12.

Figure 16

(Color online) The average wave number $⟨k⟩$ as a function of time for air and water/glycerol for the velocity field $\mathbf{u}$ and the density field ${\rho }_d$.

Figure 17

(Color online) The standard deviation ${\sigma }_k$ as a function of time for air and water/glycerol for the velocity field $\mathbf{u}$ and the density field ${\rho }_d$.

Figure 18

(Color online) The velocity $y$ component multiplied by the fluid viscosity averaged over the first ten lowest particles in time.

Figure 19

(Color online) ${\mathbf{a}}_{\mathbf{f}}=\mathbf{\nabla }P/\left({\rho }_m{\rho }_s\right)$ is the acceleration on the particles due to the fluid in time compared to the gravitational acceleration ${\mathbf{a}}_{\mathbf{g}}$. The average runs over the first ten lowest particles.