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 study the stability of density stratified flow between co-rotating vertical cylinders with rotation rates Ωo < Ωi and radius ratio ri/ro = 0.877, where subscripts o and i refer to the outer and inner cylinders. Just as in stellar and planetary accretion disks, the flow has rotation, anticyclonic shear, and a stabilizing density gradient parallel to the rotation axis. The primary instability of the laminar state leads not to axisymmetric Taylor vortex flow but to the non-axisymmetric stratorotational instability (SRI), so named by Shalybkov and Rüdiger (2005). The present work extends the range of Reynolds numbers and buoyancy frequencies (N = (−g/ρ)(∂ρ/∂z)) examined in the previous experiments by Boubnov and Hopfinger (1997) and Le Bars and Le Gal (2007). Our observations reveal that the axial wavelength of the SRI instability increases nearly linearly with Froude number, F r = Ωi/N . For small outer cylinder Reynolds number, the SRI occurs for inner inner Reynolds number larger than for the axisymmetric Taylor vortex flow (i.e., the SRI is more stable). For somewhat larger outer Reynolds numbers the SRI occurs for smaller inner Reynolds numbers than Taylor vortex flow and even below the Rayleigh stability line for an inviscid fluid. Shalybkov and Rüdiger (2005) proposed that the laminar state of a stably stratified rotating shear flow should be stable for Ωo/Ωi > ri/ro, but we find that this stability criterion is violated for N sufficiently large; however, the destabilizing effect of the density stratification diminishes as the Reynolds number increases. At large Reynolds number the primary instability leads not to the SRI but to a previously unreported nonperiodic state that mixes the fluid.

The generation of internal gravity waves by tidal flow over topography is an important oceanic process that redistributes tidal energy in the ocean. Internal waves reflect from boundaries, creating harmonics and mixing. We use laboratory experiments and two-dimensional numerical simulations of the Navier-Stokes equations to determine the value of the topographic slope that gives the most intense generation of second harmonic waves in the reflection process. The results from our experiments and simulations agree well but differ markedly from theoretical predictions by S. A. Thorpe ͓"On the reflection of a train of finite amplitude waves from a uniform slope," J. Fluid Mech. 178, 279 ͑1987͔͒ and A. Tabaei et al. ͓"Nonlinear effects in reflecting and colliding internal wave beams," J. Fluid Mech. 526, 217 ͑2005͔͒, except for nearly inviscid, weakly nonlinear flow. However, even for weakly nonlinear flow ͑where the Dauxois-Young amplitude parameter value is only 0.01͒, we find that the ratio of the reflected wave number to the incoming wave number is very different from the prediction of weakly nonlinear theory. Further, we observe that for incident beams with a wide range of angles, frequencies, and intensities, the second harmonic beam produced in reflection has a maximum intensity when its width is the same as the width of the incident beam. This observation yields a prediction for the angle corresponding to the maximum in second harmonic intensity that is in excellent accord with our results from experiments and numerical simulations.

We study the propulsion matrix of bacterial flagella numerically using slender body theory and the regularized Stokeslet method in a biologically relevant parameter regime. All three independent elements of the matrix are measured by computing propulsive force and torque generated by a rotating flagellum, and the drag force on a translating flagellum. Numerical results are compared with the predictions of resistive force theory, which is often used to interpret micro-organism propulsion. Neglecting hydrodynamic interactions between different parts of a flagellum in resistive force theory leads to both qualitative and quantitative discrepancies between the theoretical prediction of resistive force theory and the numerical results. We improve the original theory by empirically incorporating the effects of hydrodynamic interactions and propose new expressions for propulsive matrix elements that are accurate over the parameter regime explored.

The presence of a nearby boundary is likely to be important in the life cycle and evolution of motile flagellate bacteria. This has led many authors to employ numerical simulations to model near-surface bacterial motion and compute hydrodynamic boundary effects. A common choice has been the method of images for regularized Stokeslets (MIRS); however, the method requires discretization sizes and regularization parameters that are not specified by any theory. To determine appropriate regularization parameters for given discretization choices in MIRS, we conducted dynamically similar macroscopic experiments and fit the simulations to the data. In the experiments, we measured the torque on cylinders and helices of different wavelengths as they rotated in a viscous fluid at various distances to a boundary. We found that differences between experiments and optimized simulations were less than 5% when using surface discretizations for cylinders and centerline discretizations for helices. Having determined optimal regularization parameters, we used MIRS to simulate an idealized free-swimming bacterium constructed of a cylindrical cell body and a helical flagellum moving near a boundary. We assessed the swimming performance of many bacterial morphologies by computing swimming speed, motor rotation rate, Purcell's propulsive efficiency, energy cost per distance, and a new metabolic energy cost defined to be the energy cost per body mass per distance. All five measures predicted the same optimal flagellar wavelength independently of body size and surface proximity. Although the measures disagreed on the optimal body size, they all predicted that body size is an important factor in the energy cost of bacterial motility near and far from a surface.