Effective longitudinal slip over grooves encapsulated by a nearly inviscid lubricant

TL;DR

This work analyzes the effective longitudinal slip over grooved substrates encapsulated by a lubricant in the limit of a nearly inviscid lubricant ($\mu\ll1$). It develops two complementary frameworks: an interior-problem approach for fixed encapsulation height $b$ (key limit I) yielding an algebraic slip-length scaling $\lambda \sim b/(\mu\phi)$, and a thin-film generalized Philip problem for $b$ of order $\mu$ (key limit II) giving $\lambda$ of order unity determined by a two-region (exterior and thin-film) coupling. The authors derive a detailed phase map in $(b,\phi)$ space, solve the generalized Philip problem numerically, and obtain analytic and asymptotic results for the transition from logarithmic to algebraic small-solid-fraction behavior, including an explicit inner-outer matched asymptotic framework for slightly encapsulated ridges. The work shows that encapsulated SLIPS can produce large slip lengths and significant drag reduction, with the scaling controlled by encapsulation geometry and lubricant thickness, and it connects to classical Philip’s problem while highlighting new singular behaviors absent in unencapsulated or air-filled systems. Overall, the paper advances understanding of how nearly inviscid lubricants drive effective slip in structured surfaces, with implications for designing low-drag microfluidic interfaces.

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.

Figures (9)

• Figure 1: Dimensional schematic.
• Figure 2: Dimensionless schematic.
• Figure 3: (a) Interior problem derived in the limit $\mu\to0$ with $b$ fixed. (b) Generalized Philip problem derived in the limit $\mu\to0$ with $\beta=b/\mu$ fixed.
• Figure 4: "Phase map" of the different regimes in the $(b,\phi)$ parameter space (for $\mu\ll1$).
• Figure 5: Solutions of the generalized Philip problem, showing the leading-order slip length $\Lambda$ as a function of the ridge semi-width (or apparent solid fraction) $\phi$, for several values of $\beta=b/\phi$. Symbols: numerical solutions. Solid curve: Philip's result \ref{['Philip']} for $\beta=0$. Dash-dotted curves: the algebraic approximation $\Lambda\sim \beta/\phi$, valid for $\beta\gg\phi$ (see subsection \ref{['ssec:algebraicPhilip']}).