  Synchronization of two actuated sheets serves as a simple model for the
interaction between flagellated microswimmers. Various factors, including
inertia, sheet elasticity, and fluid viscoelasticity, have been suggested to
facilitate the synchronization of two sheets; however, the importance of
different contributions to this process still remains unclear. We perform a
systematic investigation of competing effects of inertia, sheet elasticity, and
fluid viscoelasticity on the synchronization of two sheets. Characteristic time
$\tau^\mathrm{s}$ for the synchronization caused by inertial effects is
inversely proportional to sheet Reynolds number $\mathrm{Re}$, such that
$\tau^\mathrm{s} \omega \propto \mathrm{Re}^{-1}$ with $\omega$ being the wave
frequency. Synchronization toward stable in-phase or opposite-phase
configuration of two sheets is determined by the competition of inertial
effects, sheet elasticity, fluid compressibility and viscoelasticity.
Interestingly, fluid viscoelasticity results in strong synchronization forces
for large beating amplitudes and Deborah numbers $\mathrm{De} > 1$, which
dominates over other factors and favors the in-phase configuration. Therefore,
our results show that fluid viscoelasticity can dramatically enhance
synchronization of microswimmers. Our investigation deciphers the importance of
different competing effects for the synchronization of two actuated sheets,
leading to a better understanding of interactions between microswimmers and
their collective behavior.
