Symmetries of the Doi kinetic theory for nematic polymers of arbitrary aspect ratio: at rest and in linear flows,
M. Gregory Forest, Ruhai Zhou, Qi Wang
The Doi theory has successfully modeled the monodomain shear flow problem for rigid, rod-like nematic polymers. Numerical simulations of the Smoluchowski equation for the orientational probability distribution function (pdf) have predicted: monodoma in attractors in regions of the 2-parameter space of nematic concentration N and shear rate; and bifurcation curves of monodomain transitions. Theoretical work has focused on approximate constructions of pdf solutions in various linear flow regimes. An alternative approach undertaken here is to develop symmetries of the Smoluchowski equation which imply global properties that all solutions must obey, and potentially explain important numerical or experimental phenomena. In our previous work, the authors developed flow-nematic symmetries for mesoscopic tensor approximations of the extended Doi theory for arbitrary molecular aspect ratio. In this paper we generalize these symmetries to the Smoluchowski equation in explicit constructive form: the orientational degeneracy of quiescent nematic liquids is a continuous O(3) symmetry, which is spontaneously broken by flow; in simple shear, a discrete reflection symmetry survives; a rod-discotic symmetry is established between monodisperse nematic liquids of reciprocal aspect ratio; and finally, a (flow, aspect ratio) symmetry relates the linear-flow response of finite aspect-ratio liquids to a related linear flow of an infinite aspect-ratio rod or discotic liquid. These symmetries are evident in solution diagrams from recent kinetic and mesoscopic simulations of the shear flow problem, and further explain deeper symmetries of the stable and unstable manifolds of numerical solution branches.