Kinetic derivation of a compressible Leslie--Ericksen equation for rarified calamitic gases

Nematic ordering describes the phenomenon where anisotropic molecules tend to locally align, like matches in a matchbox. This ordering can arise in solids (as nematic elastomers), liquids (as liquid crystals), and in gases. In the 1940s, Onsager described how nematic ordering can arise in dilute colloidal suspensions from the molecular point of view. However, the kinetic theory of nonspherical molecules has not, thus far, accounted for phenomena relating to the presence of nematic ordering. In this work we develop a kinetic theory for the behavior of rarified calamitic (rodlike) gases in the presence of nematic ordering. Building on previous work by Curtiss, we derive from kinetic theory the rate of work hypothesis that forms the starting point for Leslie--Ericksen theory. We incorporate ideas from the variational theory of nematic liquid crystals to create a moment closure that preserves the coupling between the laws of linear and angular momentum. The coupling between these laws is a key feature of our theory, in contrast to the kinetic theory proposed by {Curtiss \& Dahler}, where the couple stress tensor is assumed to be zero. This coupling allows the characterization of anisotropic phenomena arising from the nematic ordering. Furthermore, the theory leads to an energy functional that is a compressible variant of the classical Oseen--Frank energy (with a pressure-dependent Frank constant) and to a compressible analogue of the Leslie--Ericksen equations. The emergence of compressible aspects in the theory for nematic fluids enhances our understanding of these complex systems.