Reciprocal swimming at intermediate Reynolds number

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. We asymptotically expand the Navier-Stokes equations in the small amplitude limit to obtain a system of linear PDEs. Using a combination of numerical and analytical methods we solve the system to obtain the dimer's swim speed and show that there are two mechanisms that give rise to motion: an effective slip velocity on the boundary and Reynolds stresses in the bulk. 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) the geometric asymmetry stemming from the other's presence. We can thus unify and explain behaviors 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.