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

TL;DR

This study models how soluble surfactants affect drag reduction in microfluidic channels with superhydrophobic surfaces, revealing complex interactions involving Marangoni effects, surfactant strength, and exchange rates.

Contribution

It introduces an asymptotic model for surfactant influence on drag reduction in superhydrophobic channels at high Péclet numbers, with analytical solutions for various regimes.

Findings

Strong Marangoni effects immobilize the interface.

Drag reduction depends on surfactant strength and exchange rate.

Asymptotic solutions predict drag reduction across parameter space.

Abstract

Motivated by microfluidic applications, we investigate drag reduction in laminar pressure-driven flows in channels with streamwise-periodic superhydrophobic surfaces (SHSs) 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\'{e}clet number is large. Using asymptotic and numerical techniques, we determine the influence of surfactant on drag reduction in terms of the relative strength of advection, diffusion, Marangoni effects and bulk-surface exchange. In scenarios with strong 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. The quasi-stagnant cap has an intricate structure with an upstream slip region, followed by intermediate inner regions, and a quasi-stagnant region that is mediated by weak bulk diffusion. The quasi-stagnant region differs from the immobile region of a classical stagnant cap, observed for instance in surfactant-laden air bubbles in water, by displaying weak slip. As exchange weakens, the bulk and interface decouple: the surfactant distribution is linear when the surfactant is strong, whilst it forms a classical stagnant cap when the surfactant is weak. The asymptotic solutions offer closed-form predictions of drag reduction across much of the parameter space, providing practical utility and enhancing understanding of surfactant dynamics in flows over SHSs.