Title: The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella,
Abstract: The inertialess fluid-structure interactions of active and passive
inextensible filaments and slender- rods are ubiquitous in nature, from the
dynamics of semi-flexible polymers and cytoskeletal filaments to cellular
mechanics and flagella. The coupling between the geometry of deformation and
the phys- ical interaction governing the dynamics of bio-filaments is complex.
Governing equations negotiate elastohydrodynamical interactions with
non-holonomic constraints arising from the filament inex- tensibility. Such
elastohydrodynamic systems are structurally convoluted, prone to numerical
erros, thus requiring penalization methods and high-order spatiotemporal
propagators. The asymptotic coarse-graining formulation presented here exploits
the momentum balance in the asymptotic limit of small rod-like elements which
are integrated semi-analytically. This greatly simplifies the elas-
tohydrodynamic interactions and overcomes previous numerical instability. The
resulting matricial system is straightforward and intuitive to implement, and
allows for a fast and efficient computation, over than a hundred times faster
than previous schemes. Only basic knowledge of systems of linear equations is
required, and implementation achieved with any solver of choice.
Generalisations for complex interaction of multiple rods, Brownian polymer
dynamics, active filaments and non-local hydrodynamics are also
straightforward. We demonstrate these in four examples commonly found in
biological systems, including the dynamics of filaments and flagella. Three of
these systems are novel in the literature. We additionally provide a Matlab
code that can be used as a basis for further generalisations.