Propulsion of microorganisms by a helical flagellum

Many bacteria (e.g. \textit{E. coli} and \textit{Salmonella}) swim by rotating rigid helical flagella, which are typically several $\mu$m long and 0.4 $\mu$m in diameter. We investigate this propulsion in laboratory measurements on macroscopic rotating helices (typical diameter, 12 mm) in a fluid with viscosity $10^{5}$ times that of water; thus the Reynolds number in the experiments is much less than unity, just as for bacteria. We measure the propulsive force and torque generated by a rotating flagellum, and the drag force on a translating flagellum; thus we can determine all elements of the propulsion matrices along the axial direction. We also compute force, torque and drag using the regularized Stokeslets method of Cortez et al. (2005). Our experimental and numerical results are in excellent agreement. However, these results differ significantly from the predictions of resistive force theories developed by Gray and Hancock (1953) and Lighthill (1975). The difference between our measurements and resistive force theory is especially large for helices with small pitch/diameter ratios, which is the regime of many bacteria.