Effective longitudinal slip over grooves encapsulated by a nearly inviscid lubricant
File(s)
Published version
Author(s)
Schnitzer, Ory
Yariv, Ehud
Type
Journal Article
Abstract
We calculate the effective slip length for a rectangularly grooved periodic surface encapsulated (i.e., fully wetted) by a lubricant fluid and subjected to exterior shear flow parallel to the grooves. Our focus is the limit of a nearly-inviscid lubricant, where the ratio µ of the lubricant viscosity to that of the exterior fluid is small. This limit is singular for an encapsulated surface, indicating
a dominant lubricant-flow effect — a stark contrast to superhydrophobic surfaces where the role of the lubricant is typically negligible. In addition to µ, the ratio λ of the slip length to the grooving semi-period depends on three geometric lengths, all normalized by the semi-period: b, the thickness of the lubricant films wetting the groove ridges, and φ and h, the semi-width and height
of the ridges, respectively. We identify two key limits characterizing the regime µ 1. In the first, with b held fixed, we find λ ∼ µ−1˜ λ(b,φ,h), where the rescaled slip length ˜ λ is determined by an interior lubricant-flow problem. In the second, with b/µ held fixed, we find λ ∼ Λ(b/µ,φ), where Λ is governed by an exterior flow problem in which the thin lubricant films wetting the
ridges are effectively replaced by a Navier-slip condition, while the rest of the interface is shear free. As b/µ → 0, the Navier-slip condition simplifies to no slip, whereby the exterior problem reduces to that for a superhydrophobic grooved surface — famously solved by Philip using complex variables (Z. Angew. Math. Phys., 23 353, 1972). Asymptotic and numerical analysis of the exterior
problem demonstrates (i) the transition from Philip’s solution at b/µ 1 to an algebraic behavior Λ ∼b/(µφ) at b/µ 1, matching with the small-b limit of the interior problem; and (ii) for φ 1, the transition from a logarithmic O(lnφ) scaling for b µφ —familiar from the superhydrophobic
case, where φ constitutes the solid-to-air fraction — to the aforementioned algebraic regime when µφ b φ. As b becomes comparable to φ, the exterior problem loses validity in favor of the interior problem. By analyzing the latter in the distinguished sub-limit where b and φ are comparably small, we demonstrate how, as b/φ is increased, the algebraic growth of λ with b/µ is
arrested at order µ−1/lnb.
a dominant lubricant-flow effect — a stark contrast to superhydrophobic surfaces where the role of the lubricant is typically negligible. In addition to µ, the ratio λ of the slip length to the grooving semi-period depends on three geometric lengths, all normalized by the semi-period: b, the thickness of the lubricant films wetting the groove ridges, and φ and h, the semi-width and height
of the ridges, respectively. We identify two key limits characterizing the regime µ 1. In the first, with b held fixed, we find λ ∼ µ−1˜ λ(b,φ,h), where the rescaled slip length ˜ λ is determined by an interior lubricant-flow problem. In the second, with b/µ held fixed, we find λ ∼ Λ(b/µ,φ), where Λ is governed by an exterior flow problem in which the thin lubricant films wetting the
ridges are effectively replaced by a Navier-slip condition, while the rest of the interface is shear free. As b/µ → 0, the Navier-slip condition simplifies to no slip, whereby the exterior problem reduces to that for a superhydrophobic grooved surface — famously solved by Philip using complex variables (Z. Angew. Math. Phys., 23 353, 1972). Asymptotic and numerical analysis of the exterior
problem demonstrates (i) the transition from Philip’s solution at b/µ 1 to an algebraic behavior Λ ∼b/(µφ) at b/µ 1, matching with the small-b limit of the interior problem; and (ii) for φ 1, the transition from a logarithmic O(lnφ) scaling for b µφ —familiar from the superhydrophobic
case, where φ constitutes the solid-to-air fraction — to the aforementioned algebraic regime when µφ b φ. As b becomes comparable to φ, the exterior problem loses validity in favor of the interior problem. By analyzing the latter in the distinguished sub-limit where b and φ are comparably small, we demonstrate how, as b/φ is increased, the algebraic growth of λ with b/µ is
arrested at order µ−1/lnb.
Date Issued
2026-05-01
Date Acceptance
2026-04-22
Citation
Physical Review Fluids, 2026, 11 (5)
ISSN
2469-990X
Publisher
American Physical Society
Journal / Book Title
Physical Review Fluids
Volume
11
Issue
5
Copyright Statement
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
License URL
Identifier
10.1103/hvpr-9s1q
Publication Status
Published
Article Number
054202
Date Publish Online
2026-05-26