Sedimentation instabilities: Impact of the fluid compressibility and viscosity

The effect of an interstitial fluid on the mixing of sedimenting grains is studied numerically in a closed rectangular Hele-Shaw cell. We investigate the impact of the fluid compressibility and fluid viscosity on the dynamics and structures of the granular Rayleigh-Taylor instability. First we discuss the effect of the fluid compressibility on the initial fluid pressure evolution and on the dynamics of the particles. Here, the emerging patterns do not seem highly affected by the compressibility change studied. To characterize the patterns and motion the combined length of the particle trajectories in relation to the movement of the center of mass is analyzed, and the separation of particle pairs is measured as a function of the fluid viscosity.

Figure 1

A layer of beads falls through a gap in vacuum. Time progresses from left to right. White areas represent areas where the particle density is zero. The gray and black areas represent areas filled with particles, and the stripes are added artificially to better demonstrate the dynamics. From left to right time progresses in equal time steps.

Figure 2

Comparison of simulations with different fluid compressibilities at a fluid viscosity of air: ${\mu }_f\left(\text{air}\right)=0.0182\text{mPa}\text{s}$. Gray and black areas represent areas filled with particles. The stripes are added artificially to better demonstrate the dynamics. From left to right time progresses in equal time steps and from top to bottom the bulk modulus is increased.

Figure 3

(Color online) The $y$ position of the center of mass of all particles is plotted in time for different bulk moduli.

Figure 4

(Color online) The excess path length $r\left(t\right)$ in Eq. (12) is plotted in time. Particles are falling mostly straight downward, and the bulk modulus hardly affects the excess path length.

Figure 5

(Color online) The skin depth for different bulk moduli at time steps connected with the oscillations: $t=0.01\text{s}$ and $t=0.05\text{s}$. At the time that the top layer takes to fall through the fluid $t=0.32\text{s}$, the skin depth is much larger than the system size.

Figure 6

(Color online) The averaged pressure profile as a function of the depth at $t=0.02\text{s}$.

Figure 7

The particle density field of simulations with different viscosities. From left to right time is progressing, and from top to bottom the viscosity is increased. If not specified the axis units are given in centimeters. White areas represent particle-free areas. Bubbles of low particle density can be observed.

Figure 8

(Color online) The average excess particle trajectory in relation to the displacement of the center of mass.

Figure 9

(Color online) $\Delta d$ the average distance of particle pairs in time for low-viscosity fluids in bilogarithmic representation. The power-law fit with an exponent $b=1.0$ shows ballistic behavior.

Figure 10

(Color online) $\Delta d$ the average distance of particle pairs in time for high-viscosity fluids in bilogarithmic representation. The initial separation of the pairs has a diffusive behavior with an exponent close to $b=0.5$ in the dashed line. In the progress a crossover to a turbulent-dispersive behavior is observed with a slope close to $b=1.5$ in the solid line.

Figure 11

(Color online) Bilogarithmic representation of $\Delta d$, the average distance of particle pairs in time, for a layer of 600 particles at a height of 2000 particles up in the packing. The slopes increase systematically with the fluid viscosity.

Figure 12

(Color online) Bilogarithmic representation of $\Delta d$, the average distance of particle pairs in time, for a layer of 600 particles at a height of 4000 particles up in the packing.