Experimentally Determining the Mobility Matrix of a Helical Flagellum as a Function of Boundary Distance*

Previous laboratory studies that used macroscopic robots to model swimming with a helix at low Reynolds number assumed that the linear system of equations representing the motion of a solid body could be determined by exploring each degree of freedom independently (Rodenborn et al., PNAS 2013). However, our recent work finds that this methodology is more complicated when the robot is near a boundary. Axial rotation of the helix creates an attractive force towards the boundary because it pumps fluid parallel to the boundary. This force creates an asymmetry in our data between CW and CCW rotation of the helix and is predicted only if the helix is rotating and translating (Lauga et al. Biophys. J. 2006). However, we can still determine the coefficient of the mobility matrix by averaging the CW and CCW values, and the asymmetry allows us to also determine the attractive force matrix element by taking the difference between the values. We also find that the elements of the mobility matrix determined in this manner display an approximately exponential decrease as a function of the boundary distance with an interaction length about four times the helical radius.