Dr Ory Schnitzer (Imperial College London): Slip over superhydrophobic surfaces in the small-solid-fraction limit
When a micro-structured hydrophobic substrate is immersed in water, it often attains a
“superhydrophobic” state, where gas pockets are trapped in the vacancies of the micro-structure. Superhydrophobic surfaces are of enormous practical interest given their unique wetting properties and their ability to significantly reduce hydrodynamic resistance. The latter effect is often quantified by calculating the apparent slip length of an isolated substrate subjected to simple shear flow. While the slip length λ is normally commensurate with the period L of the micro-structure, it may be considerably larger when the solid-to-air fraction ϕ of the surface is small. This is because the small-solid-fraction limit is in general singular, i.e., λ /L → ∞ as ϕ → 0.
Ybert et al. [1] concluded that the small-solid-fraction singularity is logarithmic for grooved surfaces, λ /L = ��(log(1/ϕ)), whereas it is algebraic for surfaces decorated with pillars, λ /L = ��(1/ϕ1/2). . Motivated by numerical evidence in the literature, I will show that grooved surfaces also exhibit an inverse-square-root singularity, provided that the entrapped bubbles protrude into the liquid at an angle α such that π /2 − α = �� (ϕ1/2) [2,3]. The argument is based on a scaling theory and asymptotic analysis in the small-solid-fraction limit, using matched asymptotic expansions and conformal mapping. The analysis illuminates the transition, with increasing α, from a logarithmic singularity to an algebraic one; it also furnishes closed-form asymptotic approximations for the slip length, separately for each distinguished scaling of the protrusion angle α. The latter formulae are combined to form a single uniformly valid approximation, which is in excellent agreement with numerical simulations over the entire range of positive protrusion angles.
I’ll also revisit the small-solid-fraction theory for surfaces decorated with doubly periodic arrays of pillars [4], previously studied by others for circular pillars and rectangular lattices using a different method [5]. In general, the slip length is an anisotropic 2×2 tensor. The slip length at ��(1/ϕ1/2) is found, independently of the lattice geometry, in terms of the Stokes resistance tensor of a plate particle whose shape is that of the pillar cross-section. The ��(1) correction is found, independently of the pillar cross-section shape, by considering Stokeslet interactions between the effectively point-size pillars; the calculation requires the asymptotics of a general class of lattice sums. While the approximations disagree with [5] at ��(1), they are in excellent agreement with numerical data.
[1] C. Ybert et al., Phys. Fluids, 19 123601, 2007
[2] O. Schnitzer, Phys. Rev. Fluids, 1(5) 052101, 2016
[3] O. Schnitzer, J. Fluid Mech., 820 580-603, 2017
[4] O. Schnitzer & E. Yariv, J. Fluid Mech., 843 637-652, 2018
[5] A. M. J Davis & E. Lauga, J. Fluid Mech., 661 402-411, 20