The dynamics of a flexible filament sedimenting in a viscous fluid are
explored analytically and numerically. Compared to the well-studied case of
sedimenting rigid rods, the introduction of filament compliance is shown to
cause a significant alteration in the long-time sedimentation orientation and
filament geometry. A model is developed by balancing viscous, elastic, and
gravitational forces in a slender-body theory for zero-Reynolds-number flows,
and the filament dynamics are characterized by a dimensionless
elasto-gravitation number. Filaments of both non-uniform and uniform
cross-sectional thickness are considered. In the weakly flexible regime, a
multiple-scale asymptotic expansion is used to obtain expressions for filament
translations, rotations, and shapes. These are shown to match excellently with
full numerical simulations. Furthermore, we show that trajectories of
sedimenting flexible filaments, unlike their rigid counterparts, are restricted
to a cloud whose envelope is determined by the elasto-gravitation number. In
the highly flexible regime we show that a filament sedimenting along its long
axis is susceptible to a buckling instability. A linear stability analysis
provides a dispersion relation, illustrating clearly the competing effects of
the compressive stress and the restoring elastic force in the buckling process.
The instability travels as a wave along the filament opposite the direction of
gravity as it grows and the predicted growth rates are shown to compare
favorably with numerical simulations. The linear eigenmodes of the governing
equation are also studied, which agree well with the finite-amplitude buckled
shapes arising in simulations
An analytical expression for the fluctuation-rounded stretch-coil transition of semiflexible polymers in extensional flows is derived. The competition between elasticity and tension is known to cause a buckling instability in filaments placed near hyperbolic stagnation points and the effect of thermal fluctuations on this transition has yet to receive full quantitative treatment. Motivated by the findings of recent experiments as well as our simulations, we solve for the amplitude of the first buckled mode near the onset of the instability. This reveals a stochastic supercritical bifurcation, which is in excellent agreement with full numerical simulations.
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 "Brownianlike" 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.
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