Modeling the drag reduction of surfactant-contaminated superhydrophobic surfaces in laminar flows

Surfactant effects are increasingly being considered in the study of flows over superhydrophobic surfaces (SHS); see for instance Peaudecerf et al. (PNAS, 2017) and Song et al. (PRF, 2018). Even trace amounts of these substances induce adverse Marangoni stresses that can negate the drag reduction of SHS. Consequently, a quantification of these effects is inescapable to ensure their efficacy in realistic conditions, where surfactants are unavoidable. Here we combine recently introduced scaling relations for the transport of surfactant at the plastron with laminar three-dimensional channel flow in typical SHS configurations. We obtain a predictive model for the drag reduction as a function of the eleven dimensionless groups of the problem which is then evaluated against fully resolved numerical simulations and experiments.