  Hydrodynamic interactions between two identical elastic dumbbells settling
under gravity in a viscous fluid at low-Reynolds-number are investigated within
the point-particle model. Evolution of a benchmark initial configuration is
studied, in which the dumbbells are vertical and their centres are aligned
horizontally. Rigid dumbbells and pairs of separate beads starting from the
same positions tumble periodically while settling down. We find that elasticity
(which breaks time-reversal symmetry of the motion) significantly affects the
system's dynamics. This is remarkable taking into account that elastic forces
are always much smaller than gravity. We observe oscillating motion of the
elastic dumbbells, which tumble and change their length non-periodically.
Independently of the value of the spring constant, a horizontal hydrodynamic
repulsion appears between the dumbbells - their centres of mass move apart from
each other horizontally. The shift is fast for moderate values of the spring
constant k, and slows down when k tends to zero or to infinity; in these
limiting cases we recover the periodic dynamics reported in the literature. For
moderate values of the spring constant, and different initial configurations,
we observe the existence of a universal time-dependent solution to which the
system converges after an initial relaxation phase. The tumbling time and the
width of the trajectories in the centre-of-mass frame increase with time. In
addition to its fundamental significance, the benchmark solution presented here
is important to understand general features of systems with larger number of
elastic particles, at regular and random configurations.
