Drag reduction in surfactant-contaminated superhydrophobic channels at high P\'eclet numbers

Motivated by microfluidic applications, we investigate drag reduction in laminar pressure-driven flows in channels with streamwise-periodic superhydrophobic surfaces (SHSs) that are contaminated with soluble surfactant. We develop a model in the long-wave and weak-diffusion limit, where the streamwise SHS period is large compared to the channel height and the P\'eclet number is large. Employing asymptotic and numerical techniques, we determine the drag due to surfactant in terms of the relative strength of diffusion, Marangoni effects and bulk-surface exchange. In scenarios with strong bulk-surface exchange, the drag reduction exhibits a complex dependence on the thickness of the bulk-concentration boundary layer and surfactant strength. Strong Marangoni effects immobilise the interface through a linear surfactant distribution, whereas weak Marangoni effects yield a quasi-stagnant cap with an intricate asymptotic structure that is mediated by weak bulk diffusion. As bulk-surface exchange weakens, the bulk and interface decouple: the surfactant distribution is linear when the surfactant is strong and forms a classical stagnant cap when the surfactant is weak. The asymptotic solutions offer closed-form predictions of drag reduction across much of parameter space, providing practical utility and enhancing understanding of surfactant dynamics in flows over SHSs.