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

Abstract

Superhydrophobic surfaces (SHSs) have the potential to reduce drag at solid boundaries. However, multiple independent studies have recently shown that small amounts of surfactant, naturally present in the environment, can induce Marangoni forces that increase drag, at least in the laminar regime. To obtain accurate drag predictions, one must solve the mass, momentum, bulk surfactant and interfacial surfactant conservation equations. This requires expensive simulations, thus preventing surfactant from being widely considered in SHS studies. To address this issue, we propose a theory for steady, pressure-driven, laminar, two-dimensional flow in a periodic SHS channel with soluble surfactant. We linearise the coupling between flow and surfactant, under the assumption of small concentration, finding a scaling prediction for the local slip length. To obtain the drag reduction and interfacial shear, we find a series solution for the velocity field by assuming Stokes flow in the bulk and uniform interfacial shear. We find how the slip and drag depend on the nine dimensionless groups that together characterize the surfactant transport near SHSs, the gas fraction and the normalized interface length. Our model agrees with numerical simulations spanning orders of magnitude in each dimensionless group. The simulations also provide the constants in the scaling theory. Our model significantly improves predictions relative to a surfactant-free one, which can otherwise overestimate slip and underestimate drag by several orders of magnitude. Our slip length model can provide the boundary condition in other simulations, thereby accounting for surfactant effects without having to solve the full problem.

Influence of insoluble surfactants on shear flow over a surface in Cassie state at large Péclet numbers

“…For a thorough discussion of the influence of soluble surfactants (Gibbs monolayers) under a wide range of flow configurations we refer to the article by Landel et al. (2020), which our model complements by an analytically solvable case.…”

“…A subsequent theoretical investigation of the observed phenomena by Landel et al. (2020) mainly focuses on soluble surfactants, presenting a scaling theory for the effective slip length at the surface and comparing with numerical simulations for pressure-driven flow between superhydrophobic plates for a wide range of flow parameters. Here, we complement this study by deriving results for the case of insoluble surfactants driven by shear flow.…”

Superhydrophobic drag reduction in turbulent flows: a critical review

“…In addition, if the water flows with an appreciable speed, the shear stress (Wang et al 2015;Waxler et al 2015a) and pressure fluctuation (Piao and Park 2015;Seo et al 2015) may deplete the plastron dynamically, making high Reynolds number flows exceedingly more challenging for the plastron to survive, as the scarcity of successful SHPo drag reduction at high Reynolds number flows in the literature indicates. Furthermore, even when the plastron is intact, the surfactants in the flowing water (e.g., diffused out from the synthesized surface or naturally existing in the environmental water) may immobilize the water-air interface and deteriorate or negate the drag reduction (Kim & Hidrovo 2012;Peaudecerf et al 2017;Song et al 2018;Landel et al 2020;Li et al 2020a, b). The stability of plastron will be further discussed in Sect.…”

Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows

“…Landel et al. [17] and references therein). The present study includes surfactant effects in a nonlinear study of the underlying phenomena described above.…”