^{1}, Christophe Ancey

^{1}and Gaël Epely-Chauvin

^{1,a)}

### Abstract

We present flume experiments showing plastic behavior for perfectly density-matched suspensions of non-Brownian particles within a Newtonian fluid. In contrast with most earlier experimental investigations (carried out using coaxial cylinder rheometers), we obtained our rheological information by studying thin films of suspension flowing down an inclined flume. Using particles with the same refractive index as the interstitial fluid made it possible to measure the velocity field far from the wall using a laser-optical system. At long times, a stick-slip regime occurred as soon as the fluid pressure dropped sufficiently for the particle pressure to become compressive. Our explanation was that the drop in fluid pressure combined with the surface tension caused the flow to come to rest by significantly increasing flow resistance. However, the reason why the fluid pressure diffused through the pores during the stick phases escaped our understanding of suspension rheology.

The work presented here was supported by the Swiss National Science Foundation under Grant No. 200021-105193/1 (a project called “Transient free-surface flows of concentrated suspensions Application to geophysical flows,” funded by an R’Equip grant), the competence center in Mobile Information and Communication Systems (a center supported by the Swiss National Science Foundation under Grant No. 5005-67322, MICS project), the competence center in Environmental Sciences (TRAMM and APUNCH projects), and specific funds provided by EPFL (*vice-présidence à la recherche*). We are grateful to Belinda Bates for copyediting the paper.

I. INTRODUCTION

II. EXPERIMENTAL

III. EXPERIMENTAL RESULTS

A. Outline

B. Analysis of Run I

C. Fracture regime

D. Plastic regime

IV. INTERPRETATION

A. Analysis and comparison with similar works

B. Jamming as an explanation of stick-slip motion

V. CONCLUSION

### Key Topics

- Suspensions
- 42.0
- Fracture mechanics
- 9.0
- Free surface
- 9.0
- Granular flow
- 8.0
- Kinematics
- 8.0

## Figures

(a) Position of the front as a function of time for Runs B and D. We also show the similarity solution . (b) Detail of the front position as a function of time for Run D. The released mass was 6 kg for both runs. The solids fraction was 0.580 (Run B) or 0.595 (Run D). For the sake of comparison, the data were scaled: and with the following scales: L * = 2.55 m (distance from flume entrance to point of measurement), T * = L */U *, , and H * L * = A. We have also introduced μ(ϕ) = μ f (1 − ϕ/ϕ m )−β (Krieger-Dougherty's viscosity function) with β = 2 and ϕ m = 0.625 (maximum solids fraction).

(a) Position of the front as a function of time for Runs B and D. We also show the similarity solution . (b) Detail of the front position as a function of time for Run D. The released mass was 6 kg for both runs. The solids fraction was 0.580 (Run B) or 0.595 (Run D). For the sake of comparison, the data were scaled: and with the following scales: L * = 2.55 m (distance from flume entrance to point of measurement), T * = L */U *, , and H * L * = A. We have also introduced μ(ϕ) = μ f (1 − ϕ/ϕ m )−β (Krieger-Dougherty's viscosity function) with β = 2 and ϕ m = 0.625 (maximum solids fraction).

Evolution of the flow depth h (dashed line) and interstitial fluid pressure head h p = P f /(ρg cos θ) (solid line, red online). The dot-and-dash line (blue online) represents the pore pressure decrease attributed to pore pressure diffusion. Measurements were taken at z = 5 cm (centerline of the flume) and downstream coordinate x = 255 cm.

Evolution of the flow depth h (dashed line) and interstitial fluid pressure head h p = P f /(ρg cos θ) (solid line, red online). The dot-and-dash line (blue online) represents the pore pressure decrease attributed to pore pressure diffusion. Measurements were taken at z = 5 cm (centerline of the flume) and downstream coordinate x = 255 cm.

Evolution of the depth-averaged velocity for Run I. Measurements were taken at z = 5 cm (centerline of the flume) and downstream coordinate x = 255 cm. Every 9 min, we had to clear the cache memory of the camera, an operation that lasted 3 min and during which we could not take measurements. This explains why there was no velocity record for 9 ⩽ t ⩽ 12 min, 21 ⩽ t ⩽ 24 min, and 33 ⩽ t ⩽ 36 min.

Evolution of the depth-averaged velocity for Run I. Measurements were taken at z = 5 cm (centerline of the flume) and downstream coordinate x = 255 cm. Every 9 min, we had to clear the cache memory of the camera, an operation that lasted 3 min and during which we could not take measurements. This explains why there was no velocity record for 9 ⩽ t ⩽ 12 min, 21 ⩽ t ⩽ 24 min, and 33 ⩽ t ⩽ 36 min.

Time variations in the ratios P f /Σ yy and for two realizations of Run I. Measurements (bottom fluid pressure and flow depth) taken at x = 255 cm. The normal stress Σ yy was computed as Σ yy = ρgh cos θ. The steady state (time-averaged) velocities were mm/s (a) and mm/s (b). The stick-slip regime occurred at t = 398 s (a) and at t = 337 s (b).

Time variations in the ratios P f /Σ yy and for two realizations of Run I. Measurements (bottom fluid pressure and flow depth) taken at x = 255 cm. The normal stress Σ yy was computed as Σ yy = ρgh cos θ. The steady state (time-averaged) velocities were mm/s (a) and mm/s (b). The stick-slip regime occurred at t = 398 s (a) and at t = 337 s (b).

Picture of a fracture viewed from the top. The flume width is 10 cm, the typical length of the fracture is 10–15 cm (flow from right to left).

Picture of a fracture viewed from the top. The flume width is 10 cm, the typical length of the fracture is 10–15 cm (flow from right to left).

Snapshots showing the velocity field within the flowing suspension during the fracture process (flow from right to left) for Run I. We show only the velocity norm . The time increment between successive images was 0.5 s. Measurements were made at x = 255 cm along the centerline of the flume (z = 5 cm).

Snapshots showing the velocity field within the flowing suspension during the fracture process (flow from right to left) for Run I. We show only the velocity norm . The time increment between successive images was 0.5 s. Measurements were made at x = 255 cm along the centerline of the flume (z = 5 cm).

Cross-stream flow depth profiles for Run I. They were taken at different times and two different positions along the channel x = 66 cm (thin line) and x = 205 cm (thick line).

Cross-stream flow depth profiles for Run I. They were taken at different times and two different positions along the channel x = 66 cm (thin line) and x = 205 cm (thick line).

Picture of the free surface taken 60 min after release for Run I. Left: picture taken at x ∼ 150 cm. At the bottom of the image a rivulet draining the interstitial fluid can be identified. Other ripples are formed by the slow creeping of the bulk (flow from right to left). Right: a view of the upper end of the reservoir, fluid seepage has dried the bulk and only the solid frame remains at rest (flow from right to left).

Picture of the free surface taken 60 min after release for Run I. Left: picture taken at x ∼ 150 cm. At the bottom of the image a rivulet draining the interstitial fluid can be identified. Other ripples are formed by the slow creeping of the bulk (flow from right to left). Right: a view of the upper end of the reservoir, fluid seepage has dried the bulk and only the solid frame remains at rest (flow from right to left).

Time variations in the flow variables over one typical phase of slipping: the dashed (red online) curve represents the depth-averaged streamwise velocity while the thin solid line is the normal component . The (blue online) curve marked with dots is the basal fluid pressure head h p while the curve flagged with gray squares is the flow depth. Dots labeled from (a) to (i) refer to the times at which the velocity profiles of Figure 10 have been plotted: (a) Δt = 1.05 s, (b) Δt = 1.20 s, (c) Δt = 1.35 s, (d) Δt = 1.50 s, (e) Δt = 1.60 s, (f) Δt = 4.00 s, (g) Δt = 10 s, (h) Δt = 11.35 s, and (i) Δt = 11.75 s. Time Δt = 0 corresponds to 623 s after the initial release. Measurements were taken on Run I at the centerline (z = 5 cm) at downstream coordinate x = 255 cm.

Time variations in the flow variables over one typical phase of slipping: the dashed (red online) curve represents the depth-averaged streamwise velocity while the thin solid line is the normal component . The (blue online) curve marked with dots is the basal fluid pressure head h p while the curve flagged with gray squares is the flow depth. Dots labeled from (a) to (i) refer to the times at which the velocity profiles of Figure 10 have been plotted: (a) Δt = 1.05 s, (b) Δt = 1.20 s, (c) Δt = 1.35 s, (d) Δt = 1.50 s, (e) Δt = 1.60 s, (f) Δt = 4.00 s, (g) Δt = 10 s, (h) Δt = 11.35 s, and (i) Δt = 11.75 s. Time Δt = 0 corresponds to 623 s after the initial release. Measurements were taken on Run I at the centerline (z = 5 cm) at downstream coordinate x = 255 cm.

Detail of Fig. 9 . We split the depth-averaged velocities in two: bottom region (y = 0 to y = h/2) and top region (y = h/2 to y = h). Dashed lines show the bottom region whereas the solid line shows the top region; the thick (red online) line represents the streamwise velocity u. The thin (black online) line is the normal velocity component v. We also plotted the excess pore pressure head. The dots indicate the times at which the u(x, y, z, t) velocities were taken. Time Δt = 0 corresponds to 623 s after the initial release.

Detail of Fig. 9 . We split the depth-averaged velocities in two: bottom region (y = 0 to y = h/2) and top region (y = h/2 to y = h). Dashed lines show the bottom region whereas the solid line shows the top region; the thick (red online) line represents the streamwise velocity u. The thin (black online) line is the normal velocity component v. We also plotted the excess pore pressure head. The dots indicate the times at which the u(x, y, z, t) velocities were taken. Time Δt = 0 corresponds to 623 s after the initial release.

Velocity profiles u(x, y, z, t) at times (upper panel) (a) Δt = 1.05 s; (b) Δt = 1.20 s, (c) Δt = 1.35 s, (d) Δt = 1.50 s, (lower panel) (e) Δt = 1.60 s, (f) Δt = 4.00 s, (g) Δt = 10 s, (h) Δt = 11.35 s, and (i) Δt = 11.75 s. Time t = 0 corresponds to 623 s after the initial release. The vertical dashed lines represent the mean (depth-averaged) velocities. Measurements were taken on run I at the centerline (z = 5 cm) and downstream coordinate x = 255 cm.

Velocity profiles u(x, y, z, t) at times (upper panel) (a) Δt = 1.05 s; (b) Δt = 1.20 s, (c) Δt = 1.35 s, (d) Δt = 1.50 s, (lower panel) (e) Δt = 1.60 s, (f) Δt = 4.00 s, (g) Δt = 10 s, (h) Δt = 11.35 s, and (i) Δt = 11.75 s. Time t = 0 corresponds to 623 s after the initial release. The vertical dashed lines represent the mean (depth-averaged) velocities. Measurements were taken on run I at the centerline (z = 5 cm) and downstream coordinate x = 255 cm.

Time evolution of the free-surface velocities for Run I. The velocities were taken simultaneously at different places along the centerline of the channel x = 66 cm, 127 cm, 205 cm, and 280 cm.

Time evolution of the free-surface velocities for Run I. The velocities were taken simultaneously at different places along the centerline of the channel x = 66 cm, 127 cm, 205 cm, and 280 cm.

Details of the bottom pore pressure recorded along the channel at four different places for Run I: x = 238.3 cm, 256.4 cm, 274.2 cm, and 292.3 cm. The vertical dot-and-dash lines show the propagation of pressure waves with a celerity of c = 40 ± 5 cm/s.

Details of the bottom pore pressure recorded along the channel at four different places for Run I: x = 238.3 cm, 256.4 cm, 274.2 cm, and 292.3 cm. The vertical dot-and-dash lines show the propagation of pressure waves with a celerity of c = 40 ± 5 cm/s.

A stick-slip cycle: the excess pore pressure head h p − h = P f /(ρg cos θ) − h as a function of the streamwise depth-averaged velocity , the trajectory is counterclockwise. We have extracted a single cycle (Run I).

A stick-slip cycle: the excess pore pressure head h p − h = P f /(ρg cos θ) − h as a function of the streamwise depth-averaged velocity , the trajectory is counterclockwise. We have extracted a single cycle (Run I).

## Tables

Features of the different runs: mean solids fraction ϕ, initial mass m, particle size distribution, the estimate of the bulk viscosity using the Krieger-Dougherty relation μ(ϕ) = μ f (1 − ϕ/ϕ m )−β (with β = 2 and ϕ m = 0.625), the characteristic flow depth H *, the velocity and time scales and T * = L */U *, and the flow Reynolds number Re = ρU * H */μ. For all runs, the particle density, fluid density, fluid viscosity, and flume inclination were kept constant: ρ f = ρ p = 1.184 g/cm3, μ f = 0.124 Pa s, θ = 25°. The length scale is L * = 2.55 m (the distance from the flume inlet to the main observation window). The flume length is L = 3 m. The flow depth scale is H * = A/L *, where A = m/(ρW) denotes the initial volume per unit width (with W = 10 cm the flume width). The nomenclature is the same as that used in Paper I. 61

Features of the different runs: mean solids fraction ϕ, initial mass m, particle size distribution, the estimate of the bulk viscosity using the Krieger-Dougherty relation μ(ϕ) = μ f (1 − ϕ/ϕ m )−β (with β = 2 and ϕ m = 0.625), the characteristic flow depth H *, the velocity and time scales and T * = L */U *, and the flow Reynolds number Re = ρU * H */μ. For all runs, the particle density, fluid density, fluid viscosity, and flume inclination were kept constant: ρ f = ρ p = 1.184 g/cm3, μ f = 0.124 Pa s, θ = 25°. The length scale is L * = 2.55 m (the distance from the flume inlet to the main observation window). The flume length is L = 3 m. The flow depth scale is H * = A/L *, where A = m/(ρW) denotes the initial volume per unit width (with W = 10 cm the flume width). The nomenclature is the same as that used in Paper I. 61

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