Squirmer dynamics near a boundary

Abstract

The boundary behavior of axisymmetric microswimming squirmers is theoretically explored within an inertialess Newtonian fluid for a no-slip interface and also a free surface in the small capillary number limit, preventing leading-order surface deformation. Such squirmers are commonly presented as abridged models of ciliates, colonial algae, and Janus particles and we investigate the case of low-mode axisymmetric tangential surface deformations with, in addition, the consideration of a rotlet dipole to represent torque-motor swimmers such as flagellated bacteria. The resulting boundary dynamics reduces to a phase plane in the angle of attack and distance from the boundary, with a simplifying time-reversal duality. Stable swimming adjacent to a no-slip boundary is demonstrated via the presence of stable fixed points and, more generally, all types of fixed points as well as stable and unstable limit cycles occur adjacent to a no-slip boundary with variations in the tangential deformations. Nonetheless, there are constraints on swimmer behavior-for instance, swimmers characterized as pushers are never observed to exhibit stable limit cycles. All such generalities for no-slip boundaries are consistent with observations and more geometrically faithful simulations to date, suggesting the tangential squirmer is a relatively simple framework to enable predications and classifications for the complexities associated with axisymmetric boundary swimming. However, in the presence of a free surface, with asymptotically small capillary number, and thus negligible leading-order surface deformation, no stable surface swimming is predicted across the parameter space considered. While this is in contrast to experimental observations, for example, the free-surface accumulation of sterlet sperm, extensive surfactants are present, most likely invalidating the low capillary number assumption. In turn, this suggests the necessity of surface deformation for stable free-surface three-dimensional finite-size microswimming, as previously highlighted in a two-dimensional mathematical study of singularity swimmers [Crowdy et al., J. Fluid Mech. 681, 24 (2011)].

Fluid flow and sperm guidance: a simulation study of hydrodynamic sperm rheotaxis

“…In our validation studies in the absence of a background flow, as presented in the electronic supplementary material, we have observed that sperm with this beat pattern typically boundary accumulate with a significant trajectory curvature for the beat-period averaged path near a no-slip surface. In particular, this mode of boundary accumulation differs from the hydrodynamic boundary capture theoretically explored in [17,28] and much of [16], where the repulsive surface potential was not taken into consideration. Thus, we often work in the regime where the virtual cell reaches far closer to the surface, so that shear does not wash the cell downstream, and hence the accumulation heights in this study, which generally satisfy h , 0.05L, are much smaller than that predicted by simply the hydrodynamic interaction of the cell and the boundary, in the absence of a surface repulsion potential.…”

“…One must note that the solutions differ significantly according to the boundary conditions imposed at z ¼ 0, with the imposition of no-slip referred to as the Blakelet solution, because the Regularized Stokeslet method uses solutions to Stokes' equations known as Blakelets [27] in the numerical procedures. For analogous reasons, the solutions associated with a fixed tangential stress, matching that of the background shear flow, and no normal velocity at z ¼ 0 are referred to as Imagelet solutions [28], and the solutions with no constraints at z ¼ 0 are referred to as Stokeslet solutions [29]. Note that the imposition of a fixed tangential stress and zero normal velocity at z ¼ 0 corresponding to the Imagelet solutions, may not be relevant in most physical situations, and similarly for the Stokeslet solution given the surface repulsion potential force is retained.…”

Dynamics of a treadmilling microswimmer near a no-slip wall in simple shear

“…The induction of the flow, by which we mean the imposition of a controlled background ambient flow, is widely used as a A treadmilling microswimmer near a no-slip wall in simple shear 649 oscillatory periodic motion of the swimmer along the wall, and this behaviour was later shown to be qualitatively the same as that described by analytical solutions that fully describe such a circular treadmiller near the wall without any need for a singularity approximation (Crowdy 2011). Qualitatively similar nonlinear periodic motions have been found in a three-sphere swimmer (Or, Zhang & Murray 2011), a spheroidal squirmer (Ishimoto & Gaffney 2013) and in experiments , which provides evidence that even idealized two-dimensional models can provide useful insights into more complicated three-dimensional dynamics in certain situations, especially when the physical source of the observed dynamics is not clear. More recently, simple two-dimensional swimmer models have been used to study self-diffusiophoretic Janus particles near a wall (Crowdy 2013), swimmer-swimmer interactions in a film (Clarke, Finn & MacDonald 2014) and wall-bounded motion of swimmers incorporating viscoelastic effects (Yazdi, Ardekani & Borham 2014.…”

Microswimmers near surfaces

“…The squirmer model has also been employed to study the effects of the hydrodynamic near field on the swimming motion near surfaces [44,45].…”