Mass transfer to a stationary electrode in an insulating plane under oscillatory flow linear and nonlinear behaviors

Abstract

This study presents a detailed numerical investigation of unsteady mass transfer in a confined shear flow subjected to periodic modulation of wall shear stress. A dedicated finite-volume solver was developed to solve the unsteady convection diffusion equation, explicitly accounting for axial diffusion and time-dependent wall-shear conditions. Simulations were performed over a wide range of Peclet numbers ($10^{-2}\le \mathrm{Pe}\le 10^6$), excitation frequencies, and modulation amplitudes ($0.01\le A_0\le 10$), enabling a systematic exploration of both linear and nonlinear response behaviors. The novelty of this work lies in the direct numerical determination of the diffusion-layer transfer function and in the introduction of a unified frequency-domain framework linking linear and nonlinear behaviors. This approach provides a consistent methodology for identifying the linear transfer function while tracking the onset and evolution of nonlinearities as the wall-shear modulation amplitude increases.

Spectral analysis of the unsteady mass flux, performed using Fourier transform techniques, reveals the emergence of harmonic components, highlighting the nonlinear filtering properties of the diffusion layer. Comparisons with the classical Lévêque asymptotic solution confirm that, at low frequencies and large amplitudes, strong convective enhancement occurs ($\mathrm{THD}\gg 100\text{\%}$), whereas at higher reduced frequencies, the system exhibits a quasi-linear response typical of a low-pass filter, with minimal harmonic content ($\mathrm{THD}<10\text{\%}$). The proposed framework constitutes a robust and generalizable tool for analyzing unsteady convective-diffusive and electrochemical transport phenomena in dynamically actuated systems.