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 Re ~ 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 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 PDEs. 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, counter-intuitive and rich motility is in the parameter space of finite inertia of particles and fluid.

39 pages, 9 figures, in review at JFM

References in corpus (8)

• The hydrodynamics of swimming microorganisms
• Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies
• Continuous breakdown of Purcell's scallop theorem with inertia
• Measurements of the steady streaming flow around oscillating spheres using 3D particle tracking velocimetry
• The Scallop Theorem and Swimming at the Mesoscale
• Reciprocal locomotion of dense swimmers in Stokes flow
• Flow rectification in loopy network models of bird lungs
• Effect of fluid inertia on the motion of a collinear swimmer

Cited by in corpus (7)

• Inertia drives concentration-wave turbulence in swimmer suspensions
• Measurements and modelling of induced flow in collective vertical migration
• Inertia-driven propulsion of asymmetric spinner-dimers at moderate Reynolds numbers
• Motility and rotation of multi-timescale microswimmers in linear background flows
• From Propulsion to Suction: Unraveling Thrust Reversal in Propellers at Intermediate Reynolds Numbers
• Transition in steady streaming and pumping caused by a sphere oscillating in a viscous incompressible fluid
• Forces and symmetry breaking of a living meso-swimmer