Effective slip lengths for immobilized superhydrophobic surfaces

Abstract

Analytical solutions are found for both longitudinal and transverse shear flow, at zero Reynolds number, over immobilized superhydrophobic surfaces comprising a periodic array of near-circular menisci penetrating into a no-slip surface and where the menisci are no longer shear-free but are taken to be no-slip zones. Explicit formulae for the associated longitudinal and transverse effective slip lengths are derived; these are then compared with analogous results for superhydrophobic surfaces of the same characteristic geometry but where the menisci are shear-free. The new formulae give results that are consistent with recent experimental observations that have prompted suggestions that menisci that are assumed to be free of shear have in fact been immobilized. Significantly, for transverse shear flow, it is found that at critical downward meniscus protrusion angles of around 47 • , for many surface geometries, it is impossible to distinguish, purely from the effective slip length, between a no-shear and a no-slip boundary condition. We also find that immobilized menisci bowing into the grooves at supercritical angles just below 90 • can be almost twice as slippery to transverse shear as no-shear menisci. The results are relevant to recent discussion as to whether surface immobilization, due to contamination by surfactants or other physical mechanisms, is compromising drag reduction properties expected from an assumed no-shear condition.

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

“…As far as we are aware, the case of Stokes flow in a transverse channel with mixed boundary conditions changing twice (on one or both channel sides), which is reminiscent of the longitudinal-channel work of Philip (1972a), has not been shown to have a closed analytical form in the literature. It would be valuable to re-examine the present problem with conformal mapping tools similar to those used by Crowdy (2016Crowdy ( , 2017a. Figure 3(b) plots curves of u I,c /(2(1 − γ Ma )) versus gas fraction φ, with g as a parameter.…”

Superhydrophobic annular pipes: a theoretical study

“…Those canonical problems are relevant to modelling superhydrophobic surfaces with the no-shear boundaries serving as a good first model of the spanning menisci in the Cassie state, especially when the viscosity of the fluid trapped in the grooves is much lower than that of the working fluid (Asmolov & Vinogradova 2012; Schönecker & Hardt 2013; Crowdy 2017 c ). The larger question of the relevance of this no-shear assumption in practice is a topic of active discussion (Crowdy 2017 b ).…”

A review on slip boundary conditions at the nanoscale: recent development and applications

“…The directional structures induce a direction-dependent effective slip length, which can be quantified by the effective slip length tensor [13,47,56,106]. Although a large number of studies have investigated the effects of anisotropic textures on effective slip lengths, by means of numerical simulations and analytical solutions [13,[107][108][109][110], a general expression, which can account for the variation of the effective slip length with the flow direction for arbitrary textures, is still too complicated to derive. Nonetheless, similar to the case of isotropic textures, as stated in Section 2.4.1, approximate equations for some limiting cases have been proposed.…”