Hydrodynamic Reversability

A Dynamical Phase Transition

How do physical systems organize themselves when subject to some external stress? Biological systems evolve in a manner governed in part by natural selection. But what are the principles that govern the evolution of physical systems, especially those far from equilibrium? Which state or states are selected from the vast array of states available to the system?

We examine some of these questions by looking at how suspensions of non-Brownian particles in a viscous liquid organize themselves when the suspension is sheared back and forth. In the absence of particles, shearing a viscous liquid in one direction and then reversing the shear causes all fluid elements to return to their initial positions (provided the Reynolds number is much less than 1), as illustrated in the the dramatic film clip in Figure 1 made by G.I. Taylor several decades ago.

According to the Stokes equations that govern low-Reynolds number hydrodynamics, when large (200 μm) neutrally buoyant particles are added, the particles should follow the sheared fluid and return to their initial positions when the shear is reversed. However, we find that the particles do not return to their initial positions for the first several times the system is sheared back and forth. Instead, the particles exhibit small random displacements from their initial positions after each shear reversal (Fig. 2a). Eventually, if the amplitude of the shear strain is not larger than some threshold, the particles do all return to their initial positions and the systems essentially stops evolving (Fig. 2b).

We have developed a computer model that captures the essential features of these phenomena, including the existence of a threshold strain amplitude below which the particles organized and stop evolving and above which the system continues to evolve. The behavior of the system near the threshold strain exhibits critical behavior indicative of a non-equilibrium critical phase transition. Ongoing studies are examining the nature of this transition, including how particles organize themselves, the effects of dynamical correlations, different flow types (e.g. planar shear vs extensional flow), and hydrodynamics.

 

Figure 1: A dyed fluid element in a viscous liquid is sheared in one direction, which stretches the dyed fluid element into a thin ribbon-like structure. When the shear is reversed, the fluid element returns to its initial spherical configuration illustrating the reversibility of low-Reynolds number flow.

 

 

Figure 2: Stroboscopic movies showing particle positions at the end of each shear cycle. (a) Above threshold where particles exhibit small random displacements from one cycle to the next. (b) Below threshold where particles return to their original positions after each cycle (after particles have organized themselves and stopped diffusing). Images on left show dyed tracer particles in a sea of index-matched, and hence invisible, particles. Images on right show all particles in a 2-d sheet illuminated by a laser sheet.

[1] “Chaos and threshold for irreversibility in sheared suspensions” D. J. Pine, J. P. Gollub, J. F. Brady and A. M. Leshansky, Nature 438, 997-1000 (2005)
[2] “Random organization in periodically driven systems” L. Corté, P. M. Chaikin, J. P. Gollub, and D. J. Pine, Nature Physics 4, 420-424, (2008).