Slip over liquid-infused gratings in the singular limit of a nearly inviscid lubricant

Abstract

We consider shear-driven longitudinal flow of an exterior fluid over a periodic array of rectangular grooves filled with an immiscible interior fluid (the "lubricant"), the grooves being formed by infinitely thin ridges protruding from a flat substrate. The ratio $\lambda$ of the effective slip length to the semi-period is a function of the ratio $\mu$ of the interior to exterior viscosities and the ratio $h$ of the grooves depth to the semi-period. We focus on the limit $\mu\ll1$, which is singular for that geometry. We find that the viscous resistance to the imposed shear is dominated by a boundary layer of exponentially small extent about the ridge tips, resulting in the effective slip length scaling as $\mu^{-1/2}$ - not $\mu^{-1}$ as implied by intuitive arguments overlooking the tip contributions (and by proposed approximations in the literature). Analyzing that exponential region in conjunction with an integral force balance, we find the simple asymptotic approximation $\lambda\approx \mu^{-1/2}$; using conformal mappings, we also calculate the leading-order correction to that result, which introduces a dependence upon $h$. The ensuing asymptotic expansion breaks down for $h=O(\mu^{1/2})$, upon transitioning to a lubrication geometry. We accordingly conduct a companion asymptotic analysis in the distinguished limit of small $\mu$ and fixed $H=h/\mu^{1/2}$, which gives $\lambda\approx \mu^{-1/2}H/(1+H)$ as well as a closed-form leading-order correction to that approximation; the intuitive $\mu^{-1}$ scaling is accordingly only relevant to the regime $H\ll1$ corresponding to extremely shallow grooves. We demonstrate excellent agreement between our predictions and numerical solutions constructed using a boundary-integral formulation.