We study a Fourier-spectral method for a dissipative system modeling
the flow of liquid crystals. We first prove its convergence in a
suitable sense, and establish the existence of global week solution of
the original problem and its uniqueness in the two dimensional
case. Then, we derive error estimates which exhibit the spectral
accuracy of the Fourier-spectral method. We also construct a fully
discrete scheme and carry out a complete stability and error analysis
for it.  Finally, we present some illustrative numerical results.