The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella

Clément Moreau (1, 2, 3) , Laetitia Giraldi (2, 1, 3) , Hermes Gadêlha (4)

1 McTAO - Mathematics for Control, Transport and Applications
2 COMUE UCA - COMUE Université Côte d'Azur (2015-2019)
3 LJAD - Laboratoire Jean Alexandre Dieudonné
4 University of York [York, UK]

The inertialess fluid-structure interaction of active and passive inextensible filaments and slender-rods are central in many processes in nature, from the dynamics of semi-flexible polymers to cy-toskeletal filaments, flagella and artificial swimmers, covering a wide range of scales. The nonlinear coupling between the geometry of deformation and the physical effects invariably result on intricate governing equations that negotiate elastohydrodynamical interactions with non-holonomic constraints, as a direct consequence of the filament inextensibility. This triggers numerical instabili-ties and require penalization methods and high-order spatiotemporal propagators. Here, we exploit the the momentum balance in the asymptotic limit of small rod-like elements which are integrated semi-analytically. This bypasses the nonlinearities arising via the inextensibilty constraint, avoiding in this way the need of Langrange multipliers. The equivalence between the continuous and asymp-totic model allows a direct comparison between the two formalisms. The nature of the asymptotic approximation entails that coarse discretisation is possible while having a higher precision when compared with the continuous approach. We show that the asymptotic model is also numerically stable in situations in which the continuous formalism has a very poor performance, while the coarse structure guarantees faster computations, over than a hundred times faster than previous schemes. The asymptotic model is simple and intuitive to implement, and generalisations for complex interaction of multiple rods, Brownian polymer dynamics, active filaments and non-local hydrodynamics are straightforward. We demonstrate these via four exemplars: transient dynamics, force-displacement buckling instability, magnetic artificial swimmer and cross-linked filament-bundle dynamics, and we additionally provide the Matlab codes.

Mots clés: fluid-structure interaction; biomechanics; filament dynamics; elastohydrodynamics; flagella and cilia; soft matter; buckling instability

Domaines: Dynamique des Fluides [physics.flu-dyn]

Clément Moreau, Laetitia Giraldi, Hermes Gadêlha. The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella. Journal of the Royal Society Interface, 2018, 15 (144), ⟨10.1098/rsif.2018.0235⟩. ⟨hal-01658670v2⟩