A new proof on axisymmetric equilibria of a three-dimensional Smoluchowski equation

We consider equilibrium solutions of the Smoluchowski equation for rodlike nematic polymers with a Maier–Saupe excluded volume potential. The purpose of this paper is to present a new and simplified proof of classical well-known results: (1) all equilibria are axisymmetric and (2) modulo rotational symmetry, the number and type of axisymmetric equilibria are characterized with respect to the strength of the excluded volume potential. These results confirm the phase diagram of equilibria obtained previously by numerical simulations (Faraoni et al 1999 J. Rheol. 43 829–43, Forest et al 2004 Rheol. Acta 43 17–37, Larson and Ottinger 1991 Macromolecules 24 6270–82).