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 λ, 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 Stokeslet method. The experimental results differ qualitatively and quantitatively from the predictions of resistive force theory. The difference is especially large for and/or , 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 λ, 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.

We precisely measure the force-free swimming speed of a rotating helix in viscous and viscoelastic fluids. The fluids are highly viscous to replicate the low Reynolds number environment of microorganisms. The helix, a macroscopic scale model for the bacterial flagellar filament, is rigid and rotated at a constant rate while simultaneously translated along its axis. By adjusting the translation speed to make the net hydrodynamic force vanish, we measure the forcefree swimming speed as a function of helix rotation rate, helix geometry, and fluid properties. We compare our measurements of the force-free swimming speed of a helix in a high-molecular weight silicone oil with predictions for the swimming speed in a Newtonian fluid, calculated using slender-body theories and a boundary-element method. The excellent agreement between theory and experiment in the Newtonian case verifies the high accuracy of our experiments. For the viscoelastic fluid, we use a polymer solution of polyisobutylene dissolved in polybutene. This solution is a Boger fluid, a viscoselastic fluid with a shear-rateindependent viscosity. The elasticity is dominated by a single relaxation time. When the relaxation time is short compared to the rotation period, the viscoelastic swimming speed is close to the viscous swimming speed. As the relaxation time increases, the viscoelastic swimming speed increases relative to the viscous speed, reaching a peak when the relaxation time is comparable to the rotation period. As the relaxation time is further increased, the viscoelastic swimming speed decreases and eventually falls below the viscous swimming speed. motility | propulsion | rheology S mall motile organisms often swim in complex fluids. Mammalian spermatozoa beat their flagella to move through cervical fluid (1). The Lyme disease spirochete Borrelia burgdorferi flexes and rotates its body to move through the extracellular matrix of our skin (2). The nematode Caenorhabditis elegans undulates its body to move through soil saturated with water (3). While there is an extensive framework for understanding the mechanics of swimming of small organisms in purely viscous Newtonian liquids such as water, our understanding of the basic principles of swimming in non-Newtonian fluids is still in its infancy (4). The behavior of complex fluids is varied, and there are many possible nonNewtonian effects a swimmer could encounter, including elastic response of the fluid, shear-dependent viscosity, adhesion to suspended particles or fibers, or the permeability of a porous medium. In this article we focus on swimming in an elastic liquid, and our goal is to determine how the speed of a model bacterial swimmer is changed by elastic effects.It is known that helically shaped bacteria such as Leptospira or B. burgdorferi swim more rapidly in solutions with methylcellulose than in nonviscoelastic solutions of the same viscosity (2, 5). On the other hand, C. elegans, which moves using planar undulations of its body, swims more slowly in a viscoelastic fluid than in a vi...

The shear viscosity of a two-dimensional (2D) liquid was calculated using molecular dynamics simulations with a Yukawa potential. The viscosity has a minimum at a Coulomb coupling parameter Gamma of about 17, arising from the temperature dependence of the kinetic and potential contributions. Previous calculations of 2D viscosity were less extensive as well as for a different potential. The stress autocorrelation function was found to decay rapidly, contrary to earlier work. These results are useful for 2D condensed matter systems and are compared to a dusty plasma experiment.