In this paper, we study a coupled dynamic system describing nematic
liquid crystal flows. The system was motivated by the Ericksen-Leslie
equations modeling the flow of nematic liquid crystals. The purpose of
studying the simplified system is to understand the flow properties of
more complicated materials, the material configurations , as well as
the interactions between them.  Unlike in the previous studies where
the Dirichlet boundary conditions are prescribed, we consider here the
free-slip boundary conditions which possess a number of distinct
advantages.  The results in this paper form the analytical background
for the forthcoming numerical simulations of the system.