  To most existing non-equilibrium theories, the modeling of non-isothermal
processes was a hard task. Intrinsic difficulties involved the non-equilibrium
temperature, the coexistence of conserved energy and dissipative entropy, etc.
In this paper, by taking the non-isothermal flow of nematic liquid crystals as
a typical example, we illustrated that thermodynamically consistent models in
either vectorial or tensorial forms could be constructed within the framework
of Conservation-Dissipation Formalism (CDF). And the classical isothermal
Ericksen-Leslie model and Qian-Sheng model were shown to be special cases of
our new vectorial and tensorial models in the isothermal, incompressible and
stationary limit. Most importantly, from above examples, it was learnt that
mathematical modeling based on CDF could easily solve the issues relating with
non-isothermal situations in a systematic way. The first and second laws of
thermodynamics were satisfied simultaneously. The non-equilibrium temperature
was defined self-consistently through the partial derivative of entropy
function. Relaxation-type constitutive relations were constructed, which gave
rise to the classical linear constitutive relations, like Newton's law and
Fourier's law, in stationary limits. Therefore, CDF was expected to have a
broad scope of applications in soft matter physics, especially under the
complicated situations, such as non-isothermal, compressible and nanoscale
systems.
