Steady state solutions of the Smoluchowski equation for rigid nematic polymers under imposed fields. (English) Zbl 1091.35017

Summary: We solve the Smoluchowski equation for steady state solutions of rigid nematic polymers and suspensions under imposed elongational flow, magnetic or electric fields, respectively. Under the three imposed fields, we show that (1) the Smoluchowski equation can be cast into a generic form, (2) the external field must be parallel to one of the eigenvectors of the second moment tensor in steady states, and (3) the steady state solution of the Smoluchowski equation (probability density function or simply pdf) is of the Boltzmann type parameterized by material parameters and two order parameters governed by two algebraic-integral equations. Then, we present a complete bifurcation diagram of the order parameters with respect to the material parameters by solving the algebraic-integral equations. The stability of the pdf solutions is inferred from the minimum of the free energy density. The solution method is extended to dilute solutions of dipolar, rigid nematic polymers under an imposed electric field. The first moment of the steady state pdf is shown to be parallel to the external field direction at sufficiently strong permanent dipole or relatively weak dipole-dipole interaction. In this case, the steady solution of the Smoluchowski equation is parameterized by one order parameter and material parameters in the Boltzmann form. Otherwise, the first moment is not necessarily parallel to the external field direction.


35C15 Integral representations of solutions to PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
82D60 Statistical mechanics of polymers
35B32 Bifurcations in context of PDEs
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