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Nonparallel solutions of extended nematic polymers under an external field. (English) Zbl 1136.35015
Summary: We continue the study on equilibria of the Smoluchowski equation for dilute solutions of rigid extended (dipolar) nematics and dispersions under an imposed electric or magnetic field [Q. Wang, S. Sircar and H. Zhou, Commun. Math. Sci. 3, No. 4, 605–620 (2005; Zbl 1091.35017)]. We first provide an alternative proof for the theorem that all equilibria are dipolar with the polarity vector parallel to the external field direction if the strength of the permanent dipole \((\mu)\) is larger than or equal to the product of the external field \((E)\) and the anisotropy parameter \((\alpha_0)\) (i.e. \(\mu\geq |\alpha_0|E)\). Then, we show that when \(\mu<|\alpha_0|E\), there is a critical value \(\alpha^*\geq 1\) for the intermolecular dipole-dipole interaction strength \((\alpha)\) such that all equilibria are either isotropic or parallel to the external field if \(\alpha\leq \alpha^*\); but nonparallel dipolar equilibria emerge when \(\alpha>\alpha^*\). The nonparallel equilibria are analyzed and the asymptotic behavior of \(\alpha^*\) is studied. Finally, the asymptotic results are validated by direct numerical simulations.

MSC:
35C15 Integral representations of solutions to PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
82D60 Statistical mechanical studies of polymers
76A15 Liquid crystals
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
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