  The mechanism of hydrodynamics-induced pairing of soft particles, namely
closed bilayer membranes (vesicles, a model system for red blood cells) and
drops, is studied numerically with a special attention paid to the role of the
confinement (the particles are within two rigid walls). This study unveils the
complexity of the pairing mechanism due to hydrodynamic interactions. We find
both for vesicles and for drops that two particles attract each other and form
a stable pair at weak confinement if their initial separation is below a
certain value. If the initial separation is beyond that distance, the particles
repel each other and adopt a longer stable interdistance. This means that for
the same confinement we have (at least) two stable branches. To which branch a
pair of particles relaxes with time depends only on the initial configuration.
An unstable branch is found between these two stable branches. At a critical
confinement the stable branch corresponding to the shortest interdistance
merges with the unstable branch in the form of a saddle-node bifurcation. At
this critical confinement we have a finite jump from a solution corresponding
to the continuation of the unbounded case to a solution which is induced by the
presence of walls. The results are summarized in a phase diagram, which proves
to be of a complex nature. The fact that both vesicles and drops have the same
qualitative phase diagram points to the existence of a universal behavior,
highlighting the fact that with regard to pairing the details of mechanical
properties of the deformable particles are unimportant. This offers an
interesting perspective for simple analytical modeling.
