Fluid inertia and the scallop theorem

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. Recent works that investigated dimer models with finite inertia show interesting results (e.g. switches in 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 a simple swimmer, an asymmetric spherical dimer of oscillating length, at small amplitudes at intermediate Reynolds numbers. 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: due to boundary conditions (effective slip velocity) and due to 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 geometrical 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.