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 Newtons law and Fouriers 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.
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