Self-Organized Criticality in Sheared Suspensions

Recent studies reveal that suspensions of neutrally buoyant non-Brownian particles driven by slow periodic shear can undergo a dynamical phase transition between a fluctuating irreversible steady state and an absorbing reversible state. Using a computer model, we show that such systems exhibit self-organized criticality when a finite particle sedimentation velocity $v_s$ is introduced. Under periodic shear, these systems evolve, without external intervention, towards the shear-dependent critical concentration ${\varphi }_c$ as $v_s$ is reduced. This state is characterized by power-law distributions in the lifetime and size of fluctuating clusters. Experiments exhibit similar behavior and, as $v_s$ is reduced, yield steady-state values of $\varphi$ that tend towards the ${\varphi }_c$ corresponding to the applied shear.

Figure 1

(a) Setup for viscous resuspension under oscillatory shear in a Couette geometry. For a strain amplitude ${\gamma }_0=R\Delta \theta /2g$, the suspension bed rises to a height $h({\gamma }_0)$. (b) The picture shows particles located in the plane of a laser sheet shined tangentially through the gap of a transparent Couette cell. Horizontal stripes are artifacts caused by imperfect matching of particle and fluid refractive indices.

Figure 2

Simulation: (a) Detail of the particle distribution in steady state. Particles are represented by circles of diameter $d$, the interaction distance. Filled symbols represent particles that will collide and diffuse irreversibly in the next shear cycle. (b) Area fraction profiles in steady state. Full lines show profiles in the $A<1$ regime for two strain amplitudes ${\gamma }_0=10$, $A=0.78$ (blue or dark gray) and ${\gamma }_0=3$, $A=0.24$ (red or gray) ($v_s={10}^{-4}d/\mathrm{cycle}$). The dashed line shows the profile in the $A>1$ regime for ${\gamma }_0=3$, $A=2.4$ ($v_s={10}^{-3}d/\mathrm{cycle}$). Profiles are averaged over $2\times {10}^4$ shear cycles. Inset: ${\overline{\varphi }}^{\infty }/{\varphi }_c$ as a function of $A$ for several data sets: ${\gamma }_0=0.5$, 2.66, 3, and 10; $4.5<\kappa <20d^{-1}$; $5\times {10}^{-5}<v_s<{10}^{-2}d/\mathrm{cycle}$.

Figure 3

Simulation: (a) Particles forming an active cluster in space-time coordinates. Symbols represent particle centers with their color changing with time from cluster nucleation. The cluster nucleates at the bottom wall $(x,y,t)\approx (-4,0,0)$ and lasts nearly 900 cycles. (b) Distributions of cluster lifetime (top) and size (bottom) for three different strain amplitudes ${\gamma }_0=10$, $A=0.78$ (◇); ${\gamma }_0=3$, $A=0.24$ (○), and ${\gamma }_0=0.5$, $A=0.12$ (△) ($v_s={10}^{-4}d\times {\mathrm{cycle}}^{-1}$). 200 to 600 clusters are recorded over $9\times {10}^4$ cycles. Lines show power-law fits $P(t)\sim t^{-1.20}$ and $P(s)\sim s^{-1.22}$

Figure 4

Experiment: (a) Particle density profiles in steady state for ${\gamma }_0=2.8$ (left), 1.0 (middle), and 0.2 (right). Profiles are averaged over 50 shear cycles. Error bars correspond to standard deviations. (b) Steady-state volume fraction as a function of ${\gamma }_0$. Solid symbols show mean volume fraction ${\overline{\varphi }}^{\infty }$(${\gamma }_0$) measured in the sedimenting case for two different oscillation periods (●: 25 s, ■: 5 s). Open symbols (○) show the critical line ${\varphi }_c({\gamma }_0)$ as measured with neutrally buoyant suspensions. The line is a power-law fit scaling as $(1/0.6+{\gamma }_0{)}^{-1}$.