Effective longitudinal slip over grooves encapsulated by a nearly inviscid lubricant

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.

💡 Research Summary

The paper investigates the effective longitudinal slip length λ for a periodic array of rectangular grooves that are fully wetted (encapsulated) by a low‑viscosity lubricant, when an exterior shear flow is imposed parallel to the grooves. The central focus is the “nearly‑inviscid” limit where the viscosity ratio μ = μ⁻/μ⁺ (lubricant to exterior fluid) is much smaller than one. In this regime the classical superhydrophobic (Cassie) problem becomes ill‑posed because the exterior fluid would experience a shear‑free interface with no means to support the imposed far‑field stress. Consequently, the slip length diverges as μ→0, and the authors set out to determine the precise scaling.

The governing equations are Laplace’s equation for the streamwise velocity potentials w⁺ (exterior) and w⁻ (lubricant) together with no‑slip on the solid ridges, continuity of velocity and tangential stress at the liquid–liquid interface y = 0, periodic Neumann conditions in x, and the far‑field condition w⁺ ≈ y + λ. Integrating the stress balance yields two integral constraints that relate the average shear at the interface to unity, providing the basis for scaling arguments.

In the μ→0 limit the authors show that both w⁺ and w⁻ scale as μ⁻¹, leading to λ ≈ μ⁻¹ · \tilde λ, where \tilde λ is a dimensionless constant determined solely by an interior problem for the lubricant. This interior problem consists of a Laplace equation in the lubricant domain with Dirichlet condition \tilde w⁻ = 1 at the interface, no‑slip \tilde w⁻ = 0 on the solid, and Neumann symmetry on the side walls. The drag per unit length D(b,φ,h) = ∫₀¹∂\tilde w⁻/∂y|_{y=0}dx is computed, and \tilde λ = 1/D. Thus the first key scaling is

 λ ∼ μ⁻¹ · \tilde λ(b,φ,h)  (Key Limit I).

A second distinguished limit is obtained by keeping the ratio β = b/μ fixed while letting μ→0. In this case the thin lubricant film on the ridge tops can be replaced by a Navier slip condition w⁺ = β ∂w⁺/∂y, while the remainder of the interface is shear‑free. The resulting “generalized Philip problem” reduces to solving Laplace’s equation with mixed boundary conditions: no‑slip on the solid, Navier slip on the thin film, and shear‑free elsewhere. When β→0 the Navier condition collapses to no‑slip, recovering Philip’s classic solution for a superhydrophobic grooved surface, where λ depends only on the solid fraction φ. For large β the solution yields an algebraic scaling

 λ ∼ b/(μ φ)  (Key Limit II),

which matches the small‑b limit of the interior problem. The transition between the two regimes is captured by asymptotic analysis and confirmed numerically: for β ≪ 1 the slip length follows Philip’s logarithmic law λ ≈ (π/2) ln(1/φ), while for β ≫ 1 it grows linearly with b/(μ φ).

The authors further explore the thin‑ridge limit φ ≪ 1. In the classical superhydrophobic case the slip length diverges logarithmically as φ→0. In the encapsulated SLIPS, however, when μ φ ≪ b ≪ φ the algebraic regime λ ∼ b/(μ φ) dominates, whereas for b ≪ μ φ the logarithmic scaling reappears. When b becomes comparable to φ the exterior problem loses validity and the interior problem must be revisited. In the distinguished sub‑limit where both b and φ are small but comparable, the authors find that the algebraic growth of λ is arrested at order μ⁻¹/ln b, reflecting a subtle balance between the thin‑film lubrication and the cavity flow.

Overall, the paper demonstrates that for encapsulated SLIPS the nearly‑inviscid limit is singular: the lubricant flow, even when its viscosity is vanishingly small, controls the effective slip. The slip length can be orders of magnitude larger than in the superhydrophobic case, scaling as μ⁻¹ times a geometry‑dependent factor, or as b/(μ φ) depending on the relative size of the lubricant film thickness b. These results provide a rigorous asymptotic framework for designing micro‑textured surfaces that exploit low‑viscosity lubricants to achieve substantial drag reduction.