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 $\mu$ 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 $\mu$, the ratio $\lambda$ of the slip length to the grooving semiperiod depends on three geometric lengths, all normalized by the semiperiod: $b$, the thickness of the lubricant films wetting the groove ridges, and $\varphi$ and $h$, the semiwidth and height of the ridges, respectively. We identify two key limits characterizing the regime $\mu \ll 1$. In the first, with $b$ held fixed, we find $\lambda \sim {\mu }^{-1}\tilde{\lambda }(b,\varphi ,h)$, where the rescaled slip length $\tilde{\lambda }$ is determined by an interior lubricant-flow problem. In the second, with $b/\mu$ held fixed, we find $\lambda \sim \mathrm{\Lambda }(b/\mu ,\varphi )$, where $\mathrm{\Lambda }$ 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 $b/\mu \to 0$, 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 [Philip, Z. Angew. Math. Phys. 23, 353 (1972)]. Asymptotic and numerical analysis of the exterior problem demonstrates (i) the transition from Philip's solution at $b/\mu \ll 1$ to an algebraic behavior $\mathrm{\Lambda }\sim b/\left(\mu \varphi \right)$ at $b/\mu \gg 1$, matching with the small-$b$ limit of the interior problem, and (ii) for $\varphi \ll 1$, the transition from a logarithmic $O(ln\varphi )$ scaling for $b\ll \mu \varphi$—familiar from the superhydrophobic case, where $\varphi$ constitutes the solid-to-air fraction—to the aforementioned algebraic regime when $\mu \varphi \ll b\ll \varphi$. As $b$ becomes comparable to $\varphi$, the exterior problem loses validity in favor of the interior problem. By analyzing the latter in the distinguished sublimit where $b$ and $\varphi$ are comparably small, we demonstrate how, as $b/\varphi$ is increased, the algebraic growth of $\lambda$ with $b/\mu$ is arrested at order ${\mu }^{-1}/lnb$.

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• Interfacial flows
• Low Reynolds number flows
• Perturbative methods