An asymptotic model of the Poisson--Nernst--Planck--Stokes system for ion transport in narrow channels

Abstract

Ion transport through narrow channels is determined by the interaction between electrochemical and hydrodynamic effects, which are influenced by the channel geometry, ion concentrations, pressure and potential gradients, and surface charges. Understanding the mechanisms that control electrokinetic phenomena such as ion selectivity and flow transitions is crucial for elucidating biological functions and for further developing the design of artificial nanofluidic systems. On the continuum scale, these processes are described by the coupled Poisson-Nernst-Planck-Stokes equations (PNPS). However, direct numerical simulations in two or three dimensions are computationally intensive and provide only limited insights into the underlying physical and mathematical structure. Taking advantage of the small aspect ratio characteristic of nanopores, we derive a systematic asymptotic reduction of the PNPS boundary value problem. In contrast to existing one-dimensional reductions, which assume a Debye length much smaller than the channel radius, our analysis identifies a distinct asymptotic regime in which the Debye length is comparable to the channel width. This framework extends the applicability of reduced PNPS models and recovers previous approximations as limiting cases. The resulting model provides clarity and predictability for a wide range of settings. We demonstrate the influence of geometry and flow on ion transport in trumpet-shaped nanopores, flow transitions that occur due to electrostatic and hydrodynamic forces, and the conductivity properties of a protein-based channel.