^{1}and David Saintillan

^{1,a)}

### Abstract

The dynamics and transport properties of Brownian semiflexible filaments suspended in a two-dimensional array of counter-rotating Taylor-Green vortices are investigated using numerical simulations based on slender-body theory for low-Reynolds-number hydrodynamics. Such a flow setup has been previously proposed to capture some of the dynamics of biological polymers in motility assays. A buckling instability permits elastic filaments to migrate across such a cellular lattice in a “Brownian-like” manner even in the athermal limit. However, thermal fluctuations alter these dynamics qualitatively by driving polymers across streamlines, leading to their frequent trapping within vortical cells. As a result, thermal fluctuations, characterized here by the persistence length, are shown to lead to subdiffusive transport at long times, and this qualitative shift in behavior is substantiated by the slow decay of waiting-time distributions as a consequence of trapping events during which the filaments remain in a particular cell for extended periods of time. Velocity and mass distributions of polymers reveal statistically preferred positions within a unit cell that further corroborate this systematic shift from transport to trapping with increasing fluctuations. Comparisons to results from a continuum model for the complementary case of rigid Brownian rods in such a flow also highlight the role of elastic flexibility in dictating the nature of polymer transport.

We thank Anke Lindner, Olivia du Roure, Nawal Quennouz, Raymond E. Goldstein, and Michael J. Shelley for enlightening discussions on various aspects of this work. We also gratefully acknowledge funding from NSF CAREER Grant CBET-1150590, from an FMC Educational Fund Fellowship, and from the University of Illinois Campus Research Board.

I. INTRODUCTION

II. POLYMER MODEL

A. Slender-body equations

B. Non-dimensionalization

C. Cellular flow and the bucklinginstability

III. RESULTS AND DISCUSSION

A. Polymertransport: Diffusion vs subdiffusion

B. Velocity distributions

C. Mass distribution of polymers

D. Cellular transport of rigid rods: A continuum approach

IV. CONCLUDING REMARKS

### Key Topics

- Polymers
- 65.0
- Buckling
- 29.0
- Elasticity
- 24.0
- Flow instabilities
- 17.0
- Polymer flows
- 17.0

## Figures

Snapshots from a simulation with 80 000, ℓ p /L = 1000, and w = 1, showing the buckling instability at the stagnation points at cell corners.

Snapshots from a simulation with 80 000, ℓ p /L = 1000, and w = 1, showing the buckling instability at the stagnation points at cell corners.

Center-of-mass trajectories of polymers: (a)–(d) correspond to four different cell sizes W at the same 20 000 and ℓ p /L = 1000, each showing four different trajectories; (e) shows the case of a single polymer with W = πL and 20 000, but ℓ p /L = 10 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812794.1]doi: 10.1063/1.4812794.1.

Center-of-mass trajectories of polymers: (a)–(d) correspond to four different cell sizes W at the same 20 000 and ℓ p /L = 1000, each showing four different trajectories; (e) shows the case of a single polymer with W = πL and 20 000, but ℓ p /L = 10 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812794.1]doi: 10.1063/1.4812794.1.

(a) Mean square displacement d 2(t) of the filament center-of-mass as a function of time at 20 000 for three values of ℓ p /L, where the long-time behavior follows an approximate power law d 2(t) ∝ t α. (b) Dependence of the exponent α on the reduced persistence length ℓ p /L. (c) Distribution of waiting times, defined as periods of time spent by a filament inside a given cell, in simulations with 20 000 and ℓ p /L = 10.

(a) Mean square displacement d 2(t) of the filament center-of-mass as a function of time at 20 000 for three values of ℓ p /L, where the long-time behavior follows an approximate power law d 2(t) ∝ t α. (b) Dependence of the exponent α on the reduced persistence length ℓ p /L. (c) Distribution of waiting times, defined as periods of time spent by a filament inside a given cell, in simulations with 20 000 and ℓ p /L = 10.

(a) Frequency distribution of the magnitude of the center-of-mass velocity for various values of ℓ p /L at 20 000. (b) Contours of the fluid velocity magnitude inside a given cell, indicating that the value of corresponds approximately to the largest ring of uniform velocity within the cell.

(a) Frequency distribution of the magnitude of the center-of-mass velocity for various values of ℓ p /L at 20 000. (b) Contours of the fluid velocity magnitude inside a given cell, indicating that the value of corresponds approximately to the largest ring of uniform velocity within the cell.

Probability distributions of mass of the polymer in a unit cell at 20 000 for different values of ℓ p /L. Probabilities are normalized to be unity for a uniform distribution.

Probability distributions of mass of the polymer in a unit cell at 20 000 for different values of ℓ p /L. Probabilities are normalized to be unity for a uniform distribution.

Steady-state solution of the Fokker-Planck equation (13) at Pe = 10 000 for rigid rods of aspect ratio ε = 0.01.

Steady-state solution of the Fokker-Planck equation (13) at Pe = 10 000 for rigid rods of aspect ratio ε = 0.01.

Characteristic distributions of filament configurations in a unit cell with (a) and ℓ p /L = 1000 and (b) 10 000 and ℓ p /L = 100.

Characteristic distributions of filament configurations in a unit cell with (a) and ℓ p /L = 1000 and (b) 10 000 and ℓ p /L = 100.

## Multimedia

Article metrics loading...

Full text loading...

Commenting has been disabled for this content