  A model describing the evolution of a liquid crystal substance in the nematic
phase is investigated in terms of three basic state variables: the {\it
absolute temperature} $\teta$, the {\it velocity field} $\ub$, and the {\it
director field} $\bd$, representing preferred orientation of molecules in a
neighborhood of any point of a reference domain. The time evolution of the
velocity field is governed by the incompressible Navier-Stokes system, with a
non-isotropic stress tensor depending on the gradients of the velocity and of
the director field $\bd$, where the transport (viscosity) coefficients vary
with temperature. The dynamics of $\bd$ is described by means of a parabolic
equation of Ginzburg-Landau type, with a suitable penalization term to relax
the constraint $|\bd | = 1$. The system is supplemented by a heat equation,
where the heat flux is given by a variant of Fourier's law, depending also on
the director field $\bd$. The proposed model is shown compatible with
\emph{First and Second laws} of thermodynamics, and the existence of
global-in-time weak solutions for the resulting PDE system is established,
without any essential restriction on the size of the data.
