  We investigate, numerically, the effects of externally imposed material flows
on the structure and temporal evolution of liquid crystal skyrmions. The
dynamics of a 2D system of skyrmions is modeled using the Ericksen-Leslie
theory, which is based on two coupled equations, one for material flow and the
other for the director field. As the time scales of the velocity and director
fields differ by several orders of magnitude for realistic values of the system
parameters, we have simplified the calculations by assuming that the velocity
relaxes instantaneously when compared to the relaxation of the director field.
Thus, we have used a finite-differences method known as artificial
compressibility with adaptive time step to solve the velocity field and a
fourth-order Runge-Kutta method for the director field. We characterized the
skyrmion shape or configuration as a function of the time and the average
velocity of the flow field. We found that for velocities above a certain
threshold, the skyrmions stretch in the direction perpendicular to the flow, by
contrast to the regime of weak flows where the skyrmions stretch along the
streamlines of the flow field. These two regimes are separated by an abrupt
(first-order) dynamical transition, which is robust with respect to e.g., the
liquid crystal elastic anisotropy. Additionally, we have found how the presence
of a second skyrmion affects the evolution of the shape of the skyrmions, by
comparing the evolution of pairs of skyrmions to the evolution of a
single-skyrmion.
