  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.
