Rheology of granular rafts

J. Lalieu, A. Seguin, and G. Gauthier

Phys. Rev. E 107, 064901 – Published 29 June 2023

Abstract

Rheology of macroscopic particle-laden interfaces, called “granular rafts,” has been experimentally studied in the simple shear configuration. The shear-stress relation obtained from a classical rheometer exhibits the same behavior as a Bingham fluid, and the viscosity diverges with the surface fraction according to evolutions similar to 2D suspensions. The velocity field of the particles that constitute the granular raft has been measured in the stationary state. These measurements reveal nonlocal rheology similar to dry granular materials. Close to the walls of the rheometer cell, one can observe regions of large local shear rate while in the middle of the cell a quasistatic zone exists. This flowing region, characteristic of granular matter, is described in the framework of an extended kinetic theory showing the evolution of the velocity profile with the imposed shear stress. Measuring the probability density functions of the instantaneous local shear rate, we provide evidence of a balance between positive and negative instantaneous local shear rate. This behavior is the signature of a quasistatic region inside the granular raft.

Received 25 July 2022
Revised 20 October 2022
Accepted 15 June 2023

DOI:https://doi.org/10.1103/PhysRevE.107.064901

Physics Subject Headings (PhySH)

Granular flows Particle-laden flows Rheology

Granular materials Wet granular materials

Polymers & Soft Matter

Authors & Affiliations

J. Lalieu , A. Seguin , and G. Gauthier

Université Paris-Saclay, CNRS, Laboratoire FAST, F-91405 Orsay, France

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Vol. 107, Iss. 6 — June 2023

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Images

Figure 1
(a) Sketch of the experimental setup. (b) Imposed torque $M = 1.5 µ N$ m ( $•$ ) and measured velocity of the cylinder $Ω$ ( $•$ ) as a function of time $t$ for $ϕ = 0.74$ . The dashed line represents the steady regime with $Ω_{\infty} = 54$ mHz. (c) Typical instantaneous velocity field for an imposed torque $M = 6 µ N$ m. The color of the velocity vector represents its norm relatively to the velocity of the inner cylinder ( $R Ω_{\infty} = 17.5$ mm $s^{- 1}$ ).
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Figure 2
(a) Dimensionless shear stress $(τ - τ_{0}) / σ$ as a function of the dimensionless mean strain ${\dot{γ}}_{m} t_{c}$ for different solid fractions: ( $▿$ ) $ϕ = 0.71$ , ( $⋄$ ) $ϕ = 0.74$ , ( $\times$ ) $ϕ = 0.76$ , ( $▵$ ) $ϕ = 0.77$ , and ( $□$ ) $ϕ = 0.79$ . The solid line represents a linear fit $τ - τ_{0} = η_{s} (ϕ) σ t_{c} {\dot{γ}}_{m}$ corresponding to Bingham fluid behavior. (b) Normalized surface viscosity $η_{s} / (σ t_{c})$ and (c) dimensionless yield stress $τ_{0} / σ$ as a function of $ϕ$ . The solid line in (b) is given by $η_{s} / (σ t_{c}) = η_{0} ϕ_{c} {(ϕ_{c} - ϕ)}^{- 2 ϕ_{c}}$ with $ϕ_{c} = 0.82$ and $η_{0} = 2.83$ corresponding to the 2D suspension behavior law [36, 37]. The dashed line in (b) corresponds to the Maron and Pierce law [38, 39] adapted in two dimensions, $η_{s} / (σ t_{c}) = η_{0} ϕ_{c} {(ϕ_{c} - ϕ)}^{- 2}$ with $ϕ_{c} = 0.82$ and fit parameter $η_{0} = 0.81$ .
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Figure 3
(a) Normalized velocity profile $v$ as a function of the distance $s$ inside the gap. Same symbols and colors as in Fig. 2: the colors correspond to the value of $τ$ . The solid lines are given by Eq. (4). Inset: Maximal velocity $V_{M}$ reached at the moving wall, as a function of applied stress $τ$ . (b) $ϕ$ as a function of the distance $s$ inside the gap. Same symbols and colors as in Fig. 2. (c) Characteristic length $δ / e$ as a function of $τ / σ$ . The solid line is obtained from the hydrodynamical model [Eq. (3)] with ${(2 κ_{0} η_{0})}^{1 / 2} = 1.9 \times 10^{- 7}$ Pa $m^{2}$ and ${(ε_{0} η_{0})}^{1 / 2} = 3.3 \times 10^{- 4}$ Pa m. The dashed line represents the saturation of the diffusion length at $δ \approx e / 2$ .
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Figure 4
Dimensionless velocity fluctuations $T^{1 / 2} / (d / t_{c})$ as a function of $I_{c}$ for different imposed $τ$ and for $ϕ = 0.76$ and $d = 140 µ m$ (same symbols as in Fig 2). The symbol ( $\circ$ ) correspond to $χ = 69$ mN $m^{- 1}$ . The solid line represents the best fit of the data $T^{1 / 2} \sim I_{c}^{1 / (2 β - 1)}$ with $β = 1.25 \pm 0.05$ .
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Figure 5
(a) $I_{c}$ as a function of $s$ . The dashed line represents the critical value $I_{c}^{*}$ delimiting the two flow regimes. PDF of instantaneous local shear rate ${\dot{γ}}_{i}$ normalized: by the mean local strain ${\dot{γ}}_{ℓ}$ in the flowing region (b) and quasistatic region (c) and by ${\dot{γ}}_{ℓ}^{3 / 4}$ in the quasistatic region (d). Same colors as in Fig. 3. (e) $I_{c}$ as a function of $s$ for different imposed stress ( $☆$ ) $τ / σ = 0.014$ , ( $☆$ ) $τ / σ = 0.022$ , ( $☆$ ) $τ / σ = 0.041$ and for $ϕ = 0.76$ and $d = 80 µ m$ . The dashed line represents the critical value $I_{c}^{*}$ delimiting the two flow regimes. PDF of instantaneous local shear rate ${\dot{γ}}_{i}$ normalized by the mean local strain ${\dot{γ}}_{ℓ}$ in the flowing region (f) and quasistatic region (g) and by ${\dot{γ}}_{ℓ}^{3 / 4}$ in the quasistatic region (h).
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