Chaos and threshold for irreversibility in sheared suspensions

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Abstract
According to the laws of fluid motion, when a simple fluid or suspension of particles is slowly stirred then unstirred — imagine a spoon in a jar of honey — all parts of the system should miraculously return to their starting points. This is a consequence of the time-reversible equations of motion, at least for two-dimensional flows. But in more complex flows, such as those in three-dimensional or rigorously stirred systems, this delicate effect is destroyed. An investigation of a slowly sheared suspension of solid particles now reveals the microscopic processes behind this transition to irreversible behaviour. Beyond a concentration-dependent threshold strain, irreversibility sets in as a result of chaotic collisions between the particles. Systems governed by time reversible equations of motion often give rise to irreversible behaviour1,2,3. The transition from reversible to irreversible behaviour is fundamental to statistical physics, but has not been observed experimentally in many-body systems. The flow of a newtonian fluid at low Reynolds number can be reversible: for example, if the fluid between concentric cylinders is sheared by boundary motion that is subsequently reversed, then all fluid elements return to their starting positions4. Similarly, slowly sheared suspensions of solid particles, which occur widely in nature and science5, are governed by time reversible equations of motion. Here we report an experiment showing precisely how time reversibility6 fails for slowly sheared suspensions. We find that there is a concentration dependent threshold for the deformation or strain beyond which particles do not return to their starting configurations after one or more cycles. Instead, their displacements follow the statistics of an anisotropic random walk7. By comparing the experimental results with numerical simulations, we demonstrate that the threshold strain is associated with a pronounced growth in the Lyapunov exponent (a measure of the strength of chaotic particle interactions). The comparison illuminates the connections between chaos, reversibility and predictability.