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Superhydrophobic surfaces (SHSs) have the potential to reduce drag at solidboundaries. However, multiple independent studies have recently shown thatsmall amounts of surfactant, naturally present in the environment, can induceMarangoni forces that increase drag, at least in the laminar regime. To obtainaccurate drag predictions, one must solve the mass, momentum, bulk surfactantand interfacial surfactant conservation equations. This requires expensivesimulations, thus preventing surfactant from being widely considered in SHSstudies. To address this issue, we propose a theory for steady,pressure-driven, laminar, two-dimensional flow in a periodic SHS channel withsoluble surfactant. We linearise the coupling between flow and surfactant,under the assumption of small concentration, finding a scaling prediction forthe local slip length. To obtain the drag reduction and interfacial shear, wefind a series solution for the velocity field by assuming Stokes flow in thebulk and uniform interfacial shear. We find how the slip and drag depend on thenine dimensionless groups that together characterize the surfactant transportnear SHSs, the gas fraction and the normalized interface length. Our modelagrees with numerical simulations spanning orders of magnitude in eachdimensionless group. The simulations also provide the constants in the scalingtheory. Our model significantly improves predictions relative to asurfactant-free one, which can otherwise overestimate slip and underestimatedrag by several orders of magnitude. Our slip length model can provide theboundary condition in other simulations, thereby accounting for surfactanteffects without having to solve the full problem.

A theory for the slip and drag of superhydrophobic surfaces with surfactant