= Flow-dependent migration beyond an equilibrium polymer depletion sketch
An equilibrium polymer-depletion profile describes how chain conformations and wall exclusion affect concentration without imposed flow. It is not, by itself, a prediction of the concentration field in a flowing polymer solution.
Flow deforms and orients polymers, and hydrodynamic interactions with a nearby wall can produce cross-stream migration. Brownian diffusion opposes the resulting concentration gradients. In a simplified one-dimensional description with constant diffusivity, $J_z=c v_{\mathrm{mig}}-D\,\partial_zc$ shows why a steady profile requires migration and diffusion information in addition to the equilibrium depletion length. A rigid-polymer calculation in simple shear predicts migration away from the wall and a flow-dependent depletion layer; this is a specified model, not a universal direction for every polymer and flow.https://journals.aps.org/pre/abstract/10.1103/PhysRevE.76.040801[Cross-stream migration in dilute solutions of rigid polymers undergoing rectilinear flow near a wall, abstract: shear-dependent migration and depletion]{ref}
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