  Recent experiments in one and two-dimensional microfluidic arrays of droplets
containing Belousov -Zhabotinsky reactants show a rich variety of spatial
patterns [J. Phys. Chem. Lett. 1, 1241-1246 (2010)]. The dominant coupling
between these droplets is inhibitory. Motivated by this experimental system, we
study repulsively coupled Kuramoto oscillators with nearest neighbor
interactions, on a linear chain as well as a ring in one dimension, and on a
triangular lattice in two dimensions. In one dimension, we show using linear
stability analysis as well as numerical study, that the stable phase patterns
depend on the geometry of the lattice. We show that a transition to the ordered
state does not exist in the thermodynamic limit. In two dimensions, we show
that the geometry of the lattice constrains the phase difference between two
neighbouring oscillators to 120 degrees. We report the existence of domains
with either clockwise or anti-clockwise helicity, leading to defects in the
lattice. We study the time dependence of these domains and show that at large
coupling strengths, the domains freeze due to frequency synchronization.
Signatures of the above phenomena can be seen in the spatial correlation
functions.
