Brownian Granular Flows Down Heaps

Open Access

Brownian Granular Flows Down Heaps

Antoine Bérut, Olivier Pouliquen, and Yoël Forterre

Phys. Rev. Lett. 123, 248005 – Published 12 December 2019

Abstract

We study the avalanche dynamics of a pile of micrometer-sized silica grains in water-filled microfluidic drums. Contrary to what is expected for classical granular materials, avalanches do not stop at a finite angle of repose. After a first rapid phase during which the angle of the pile relaxes to an angle $θ_{c}$ , a creep regime is observed where the pile slowly flows until the free surface reaches the horizontal. This relaxation is logarithmic in time and strongly depends on the ratio between the weight of the grains and the thermal agitation (gravitational Péclet number). We propose a simple one-dimensional model based on Kramers’ escape rate to describe these Brownian granular avalanches, which reproduces the main observations.

Received 17 July 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.248005

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Brownian motion Granular avalanches Granular flows Jamming Noise

Granular fluids Hard sphere colloids Microfluidic devices

Polymers & Soft MatterFluid Dynamics

Authors & Affiliations

Antoine Bérut ^*, Olivier Pouliquen ^†, and Yoël Forterre ^‡

Aix Marseille Univ, CNRS, IUSTI, Marseille 13013, France

^*antoine.berut@univ-lyon1.fr Present address: Univ Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622 Villeurbanne, France.
^†olivier.pouliquen@univ-amu.fr
^‡yoel.forterre@univ-amu.fr

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Issue

Vol. 123, Iss. 24 — 13 December 2019

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Images

Figure 1
Brownian granular piles made of silica microparticles in water-filled microfluidic drums. (a) Sample preparation before (left) and after (right) sealing. (b) Sketch of the experimental setup with inclined microscope and rotating stage to trigger avalanches ( $D = 100 μ m$ , $W = 50 μ m$ ). (c) Wide view of the water-filled PDMS drums half filled with $4.4 μ m$ silica particles under gravity. (d) Vertical thermal fluctuations of a single silica particle (diameter $d = 4.4 μ m$ ) at the top of the pile. (e) Amplitude of vertical fluctuations as function of the inverse gravitational Péclet number for various particle size (Inset: typical image used to track single particles at the top of the pile). (f) Avalanche generation and time evolution of the pile angle after an initial tilting $θ_{start} = 45 °$ for silica beads of diameter $d = 4.4 μ m$ . The blue curve is an average over 10 drums.
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Figure 2
Avalanche and creep behavior for a single particle size. (a) Temporal evolution of the pile angle with different initial angles $θ_{start}$ ranging from 5° to 50° for $d = 2.68 μ m$ in pure water. Inset: Zoom on the end of the trajectories. (b) Semi-logarithm representation showing a first fast avalanching regime (red dashed line) followed by a slow creep regime (green dashed line) for $θ_{start} = 30 °$ and $d = 2.36 μ m$ in pure water. The threshold angle $θ_{c}$ separating the two regimes is about 8°. Inset: Comparison with the same particles in a NaCl ionic solution ( $10^{- 2} mol L^{- 1}$ ) to screen the electrostatic repulsion between particles. In this case $θ_{c}$ is about 15°.
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Figure 3
Thermal creep regime. (a) Long-time temporal evolution of the pile angle for various particle sizes corresponding to $Pe = 397$ ( $d = 4.40 μ m$ ), $Pe = 55$ ( $d = 2.68 μ m$ ), $Pe = 33$ ( $d = 2.36 μ m$ ), $Pe = 19$ ( $d = 2.06 μ m$ ), and $Pe = 6$ ( $d = 1.55 μ m$ ) ( $θ_{start} = 15 °$ ). Inset: Long-time behavior for $4.40 μ m$ particles, with lower initial angle. (b) Heat map of particle displacement in the creep regime for two particle sizes. The map is obtained by computing image differences with a small time step (1 s for $1.55 μ m$ particles and 2 s for $4.40 μ m$ particles) and averaging over 1 min. The light colors correspond to an area where the particles are moving.
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Figure 4
Thermally activated avalanche model. (a) Sketch of the potential felt by a particle at the top of the inclined pile. (b) Avalanches dynamics in the creep regimes predicted by the model for various values of Pe, as a function of $\log (t / τ)$ ( $α = 2.64$ ). (c) Comparison of the logarithmic regime slopes between the experimental data and the model with $α = 2.64$ . Each experimental data corresponds to a sample. Horizontal error bars come from the dispersion of the particle’s diameters. Vertical error bars are given by the dispersion of the slopes measured in the different drums of each sample.
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