  We study a simplified Ericksen-Leslie system modeling the flow of nematic
liquid crystals with partially free boundary conditions. It is a coupled system
between the Navier-Stokes equation for the fluid velocity with a transported
heat flow of harmonic maps, and both of these parabolic equations are critical
for analysis in two dimensions. The boundary conditions are physically natural
and they correspond to the Navier slip boundary condition with zero friction
for the velocity field and a Plateau-Neumann type boundary condition for the
map. In this paper we construct smooth solutions of this coupled system that
blow up in finite time at any finitely many given points on the boundary or in
the interior of the domain.
