Effective mobilities for thin-film flows on micropillar arrays

Micropillar-textured surfaces provide versatile platforms for controlling wetting, imbibition, and droplet dynamics, and also serve as model systems for engineered textured surfaces used in applications ranging from thermal management and microfluidics to semiconductor processing. While spontaneous hemiwicking in such geometries has been extensively studied, a unified framework for thin-film modeling that captures both capillary-driven and Marangoni-driven transport across arbitrary film thicknesses has remained lacking. Here, we derive explicit analytical mobility expressions for lubrication-based modeling of wetting dynamics on periodic micropillar arrays. Using an equivalent microchannel formulation and a Fourier series solution of the Stokes equations, we obtain exact mobilities for both pressure-driven and Marangoni-driven flows, valid for film thicknesses both below and above the pillar height. The resulting formulation captures the interplay between film thickness, pillar height, and the lateral confinement scale, and connects confined transport, overtopped films, and effective slip behavior within a single framework. In particular, Marangoni-driven transport exhibits a stronger saturation with increasing pillar height than pressure-driven transport once lateral confinement dominates. These mobility laws recover known hemiwicking limits and provide the constitutive ingredient needed to extend lubrication theory to complex wetting dynamics on textured substrates.