Reciprocal swimming at intermediate Reynolds number

Abstract

In Stokes flow, Purcell's scallop theorem forbids objects with time-reversible (reciprocal) swimming strokes from moving. In the presence of inertia, this restriction is eased and reciprocally deforming bodies can swim. A number of recent works have investigated dimer models that swim reciprocally at intermediate Reynolds numbers ${\textit Re} \approx 1$ –1000. These show interesting results (e.g. switches of the swim direction as a function of inertia) but the results vary and seem to be case specific. Here, we introduce a general model and investigate the behaviour of an asymmetric spherical dimer of oscillating length for small-amplitude motion at intermediate ${\textit {Re}}$ . In our analysis we make the important distinction between particle and fluid inertia, both of which need to be considered separately. We asymptotically expand the Navier–Stokes equations in the small-amplitude limit to obtain a system of linear partial differential equations. Using a combination of numerical (finite element) and analytical (reciprocal theorem, method of reflections) methods we solve the system to obtain the dimer's swim speed and show that there are two mechanisms that give rise to motion: boundary conditions (an effective slip velocity) and Reynolds stresses. Each mechanism is driven by two classes of sphere–sphere interactions, between one sphere's motion and (1) the oscillating background flow induced by the other's motion, and (2) a geometric asymmetry induced by the other's presence. We can thus unify and explain behaviours observed in other works. Our results show how sensitive, counterintuitive and rich motility is in the parameter space of finite inertia of particles and fluid.

Transition in steady streaming and pumping caused by a sphere oscillating in a viscous incompressible fluid

“…We found a similar cancelation in the mean swimming velocity of a deforming sphere 19,30 . The cancelation was found also for the mean swimming velocity of an oscillating two-sphere, as shown by Derr et al 31 . The first term in the expansion (6.13) for the moment K 2 (s) and in the expansion (6.14) for the moment M 2 (s) is equivalent to the result at high frequency found by Riley 8 and verified by Spelman and Lauga 14 .…”

“…It shows an interesting dependence on the scale parameter s = a ωρ/2η, with a flow reversal at the critical value s 0 = 2.85632. Clearly this will have drastic effects in the two-body problem, as shown in the analysis of the two-sphere swimming problem by Derr et al 31 . In particular we expect that the reversal of swim direction seen by Dombrowski et al 32 is related to the flow reversal of the single sphere steady streaming flow.…”

The propulsion direction of nanoparticles trapped in an acoustic field

“…This finding was confirmed by Nadal & Michelin (2020) and Derr et al. (2022). Interestingly, Lippera et al.…”

“…Collis, Chakraborty & Sader (2017 subsequently studied the motion of a general asymmetric particle at arbitrary frequency, and showed that the propulsion direction reverses at a critical frequency. This finding was confirmed by Nadal & Michelin (2020) and Derr et al (2022). Interestingly, Lippera et al (2019) proved that to first-order in particle non-sphericity, translational oscillations alone cannot generate propulsion; contrary to Nadal & Lauga (2014).…”

Non-Stokesian dynamics of magnetic helical nanoswimmers under confinement

“…Therefore, noninertial swimming microorganisms have evolved alternative strategies to propel themselves forward. Among these strategies is the exploitation of drag anisotropy, achieved through a series of time-irreversible motions ( 4 ). One well-known and widely studied example is the flagellar swimming of Escherichia coli , which has a single anterior flagellum, a rotary joint, and a motor to rotate the flagellum ( 5 , 6 ).…”