Steady transport of active particles under continuous release in confined shear flow

Abstract

Continuous release is a fundamental source condition in transport, yet theoretical treatments have focused on downstream concentration development for passive solutes or fully developed cross-sectional distributions of active particles. We develop a spatial theory for the steady transport of active particles under continuous release in confined shear flows, resolving the concentration field from the inlet to the far field. Based on the Smoluchowski equation, we formulate a boundary-value problem with point-source inlet flux and boundary conditions. Separation of the streamwise coordinate from the cross-sectional variables yields a non-self-adjoint generalized eigenvalue problem, whose spatial modes are superposed to construct the solution. The eigenproblem is solved within a Galerkin spectral framework; downstream-admissible modes are retained and their coefficients determined from the inlet flux through a weighted biorthogonal expansion. Excellent agreement with individual-based simulations validates the theory. For plane Poiseuille flow under convection-dominated conditions, we find pronounced position--orientation coherence for spherical particles in the early developing region: swimming and shear reorientation drive the population through successive angular stages, generating alternating off-centre and centreline accumulation regions downstream, while dispersion progressively weakens this coherence. Particle elongation enhances orientational alignment, producing an early three-peaked vertical profile and stronger, more persistent lateral accumulation downstream, while reducing the coherence of centreward migration and thereby suppressing secondary centreline accumulation. The streamwise marginal concentration varies non-monotonically, reflecting changes in the mean streamwise velocity, with its far-field limit given by the reciprocal of the asymptotic drift velocity.