Load-dependent resistive-force theory

The passive rotation of rigid helical filaments is the strategy employed by flagellated bacteria (and some artificial microswimmers) to swim at low Reynolds numbers. In his classical 1976 paper, Lighthill calculated, for the force-free swimming of a rotating helix with no load attached (e.g. with no cell body), the 'optimal' resistance coefficients that, in a local resistive-force theory, most closely reproduce predictions from the nonlocal slender-body theory. These resistance coefficients have since been used ubiquitously in the literature, regardless of whether the conditions under which they were originally derived hold. Here we revisit the problem in the case where a load is attached to the rotating helical filament. We show that the optimal resistance coefficients depend in fact on the size of the load, and highlight and improve upon the growing inaccuracy of Lighthill's coefficients as the load increases. We also provide a physical explanation for the origin of this surprising load-dependence.