Extended Data Fig. 3: Design space for scale-free vertical tracking using a “hydrodynamic treadmill”: Effect of transient fluid shear and chamber curvature.
a, Rotational stage acceleration leads to a non-uniform velocity profile, which at the scale of the object is locally a simple shear flow. This shear flow (with shear rate \(\dot \gamma\)), can be decomposed into an extensional and vortical component, both of which perturb the object’s equilibrium orientation, in opposing directions. A gravitactic effect, due to a displaced center-of-mass and center-of-buoyancy, stabilizes the orientation and aligns it with gravity over a timescale τgrav. A balance of these two effects is quantified by the ratio of time-scales τshear|min/τgrav, where τshear|min is the inverse of the maximum possible shear rate during a chamber acceleration. b, Plot of τshear|min/τgrav with respect to stage acceleration and the gravitactic moment arm relative to object size. There is a broad region where the object orientation is highly stable (\(\tau _{shear|min}/\tau _{grav} \gg 1\)), implying that tracking has a negligible effect on orientation. The upper bound for stage acceleration is set by imposing the condition \(\tau _{shear|min}/\tau _{grav} > 100\) (solid red curve), and the lower bound is set by the condition Δτ>0 in Extended Data Fig. 2c (cross-over contours shown for τobj = 0, 2, 4). c, Top, tracking a linear motion using a circular chamber implies that, for a non-zero vertical tracking error (Δz), there is a radial drift induced in the object’s motion. Bottom, this drift velocity is given by udrift(t)/uobj=Δz/R(t), where R(t) is the radial position of the object at time t. d, Ratio of radial drift velocity to object’s speed plotted as a function of the radius of the fluidic chamber center-line ((Ri+Ro)/2) and the vertical tracking error, showing the cross-over (red solid line) where the ratio exceeds 1%.
