Dynamics of filaments: modelling the dynamics of driven microfilaments

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Abstract
We describe a method for simulating the inertialess dynamics of a flexible filament immersed in a fluid. Typically, this regime is appropriate for filaments a few micrometres or less in size (flagella that propel micro-organisms for example). We apply the model to two systems; a filament that is wiggled at one end and planar swimming motion characteristic of simple spermatozoa. For the former we find qualitative agreement with theory. The shape is determined by a balance between bending and viscous forces and there is an optimal balance that maximizes the propulsion generated by this mechanism. Quantitatively we find less satisfactory agreement. For the spermatozoa, assuming a relatively naive bending mechanism in the form of a travelling force quadrupole wave, the model generates waveforms in very good agreement with experiment. This is only true, however, if the bending forces acting on the filament are large compared with the viscous forces. Experimental measurements of the tail stiffness imply this should not be the case. We discuss the implications of this observation in the context of the sperm's swimming mechanism.

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