Efficient models for micro-swimmers

We propose minimal models of one-, two- and three-dimensional micro-swimmers at low Reynolds number with a periodic non-reciprocal motion. These swimmers are either "pushers" or "pullers" of fluid along the swimming axis, or combination of the two, depending on the history of the swimming motion. We show this with a linear three-bead swimmer by analytically evaluating the migration speed and the strength of the dipolar flow induced by its swimming motion. It is found that the distance traveled per cycle and the dipolar flow can be obtained from an integral over the area enclosed by the trajectory of the cycle projected onto a cross-plot of the two distances between beads. Two- and three-dimensional model swimmers can tumble by breaking symmetry of the swimming motion with respect to the swimming axis, as occurs in the tumbling motion of Escherichia coli or Chlamydomonas, which desynchronize the motions of their flagella to reorient the swimming direction. We also propose a five-bead model of a "corkscrew swimmer", i.e. with a helical flagellum and a rotary motor attached to the cell body. Our five-bead swimmer is attracted to a nearby wall, where it swims clockwise as observed in experiments with bacteria with helical flagella.