Elastic non-linearity of a filament leads to three-period and chaotic solutions in Stokes flow

The flow of Newtonian fluid at a low Reynolds number is generally regular and time-reversible due to the absence of nonlinear effects. For example, if the fluid is sheared by its boundary motion that is reversed, all the fluid elements return to their initial positions. Consequently, mixing in microchannels happens solely due to molecular diffusion and is very slow. Here, we show, numerically, that the introduction of a single, freely-floating, flexible filament in a time-periodic linear shear flow can break reversibility and give rise to chaos due to elastic nonlinearities if the bending rigidity of the filament is within a carefully chosen range. Within this range, not only the shape of the filament is spatiotemporally chaotic, but also the flow is an efficient mixer.

We model the filament using the bead-rod model. We consider two different models for the viscous forces : (a) they are modeled by the Rotne--Pregor tensor. This incorporates the hydrodynamic interaction between every pair of beads. (b) we consider only the diagonal term of the Rotne--Pregor tensor which makes the viscous forces local. In both of these cases, we find the same qualitative result: the shape of a stiff filament is time-invariant -- either straight or buckled for large enough bending rigidity; it undergoes a period-n bifurcation (n = 2,3, 4, etc) as the filament is made softer; becomes spatiotemporally chaotic for even softer filaments. For case (a) but not for (b) we find that the chaos is suppressed if bending rigidity is decreased further.