Propulsion of microorganisms by a helical flagellum

The swimming of a bacterium or a biomimetic nanobot driven by a rotating helical flagellum is often interpreted using the resistive force theory developed by Gray and Hancock and by Lighthill, but this theory has not been tested for a range of physically relevant parameters. We test resistive force theory in experiments on macroscopic swimmers in a fluid that is highly viscous so the Reynolds number is small compared to unity, just as for swimming microorganisms. The measurements are made for the range of helical wavelengths lambda, radii R, and lengths L relevant to bacterial flagella. The experiments determine thrust, torque, and drag, thus providing a complete description of swimming driven by a rotating helix at low Reynolds number. Complementary numerical simulations are conducted using the resistive force theories, the slender body theories of Lighthill and Johnson, and the regularized Stokes let method. The experimental results differ qualitatively and quantitatively from the predictions of resistive force theory. The difference is especially large for lambda < 6R and/or L > 3 lambda, parameter ranges common for bacteria. In contrast, the predictions of Stokeslet and slender body analyses agree with the laboratory measurements within the experimental uncertainty (a few percent) for all lambda, R, and L. We present code implementing the slender body, regularized Stokeslet, and resistive force theories: thus readers can readily compute force, torque, and drag for any bacterium or nanobot driven by a rotating helical flagellum.