Viscotaxis of beating flagella

Many biological microorganisms and artificial microswimmers react to external cues of environmental gradients by changing their swimming directions. We study here the behavior of eukaryotic flagellated microswimmers in linear viscosity gradients. Motivated by the near-surface motion of many microswimmers, we consider flagellar swimming in two spatial dimensions. We employ a model of flagellum consisting of a semi-flexible filament with a travelling wave of spontaneous curvature to study generic aspects of viscotaxis of actively beating flagella. The propulsion of the flagellum in a fluid due to a hydrodynamic friction anisotropy is described by resistive-force theory. Using numerical simulations and analytical theory, we show that beating flagella exhibit positive viscotaxis, reorienting themselves toward higher viscosity areas. We quantify this behavior by characterization of the dependence of the rotational velocity on gradient strength, beat amplitude, swimming speed, and wave length. We also examine the effects of asymmetric flagellar wave forms, which imply circular trajectories in the absence of viscosity gradients; here, large asymmetry leads to trochoid-like trajectories perpendicular to the gradient in the form of drifting circles. Flagellar deformability strongly reduce the beat amplitude and the viscotatic response. The viscotatic response is shown to be captured by a universal function of the sperm number.