Shear alignment and tensorial Taylor--Aris dispersion of Brownian rods in a circular tube

Abstract

Brownian rods disperse in pressure-driven flow through a coupling between axial shear, anisotropic translational diffusion and Jeffery--Brownian rotation. Classical tube Taylor--Aris theory treats transverse mixing as a scalar process, and existing passive-rod reductions have mainly addressed planar geometries. A circular tube adds two ingredients: the shear strength varies with radius and freely rotating rods sample a three-dimensional orientation space. We formulate a tensorial Taylor--Aris theory for dilute axisymmetric rods in Poiseuille flow by solving the local steady orientation Fokker--Planck problem and using its second moments to close a conservative axisymmetric transport equation. The long-wave reduction shows how each part of the diffusion tensor enters the one-dimensional limit. The radial diffusivity sets the invariant cross-sectional measure and the cell problem for the leading Taylor coefficient; the radial--axial component produces an inverse-P{\'e}clet correction to the migration speed; the axial component gives the direct diffusivity. The central mechanism is the streamwise alignment generated in high-shear annular layers. Alignment reduces radial diffusivity there, shifts the long-time sampling of the velocity profile toward slower streamlines, and amplifies the radial cell response. In strong shear this raises the Taylor coefficient by about \(23\%\) for aspect ratio \(p=1000\) and by about \(30\%\) in the infinitely slender limit, approaching the fully aligned bound. Direct simulations of the full tensorial equation validate the asymptotic coefficients. The same radial mixing operator also gives a Sturm--Liouville spectral model that tracks finite-time relaxation from different radial injections to the long-time Taylor regime.