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

TL;DR

This work analyzes slip over a liquid-infused grating in the singular limit of an almost inviscid interior lubricant. Using matched asymptotics and conformal mappings, it shows the leading slip length scales as $\lambda \sim \mu^{-1/2}$ due to exponentially small boundary layers around ridge tips, and provides a closed-form first correction $\lambda_0(h)$ expressed via elliptic integrals; in the distinguished shallow-groove limit with $H=h/\sqrt{\mu}$ fixed, the scaling becomes $\lambda \sim \mu^{-1/2}\,\frac{H}{1+H}$ with a further logarithmic correction. The analysis is supported by a boundary-integral numerical scheme that agrees well with the asymptotic predictions. The results clarify when the intuitive $\mu^{-1}$ scaling applies (only for extremely shallow grooves) and illuminate the role of tip-localized flows in drag balance for SLIPS and related microstructured surfaces.

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 $λ$ of the effective slip length to the semi-period is a function of the ratio $μ$ of the interior to exterior viscosities and the ratio $h$ of the grooves depth to the semi-period. We focus on the limit $μ\ll1$, 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 $μ^{-1/2}$ - not $μ^{-1}$ 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 $λ\approx μ^{-1/2}$; 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 $h=O(μ^{1/2})$, upon transitioning to a lubrication geometry. We accordingly conduct a companion asymptotic analysis in the distinguished limit of small $μ$ and fixed $H=h/μ^{1/2}$, which gives $λ\approx μ^{-1/2}H/(1+H)$ as well as a closed-form leading-order correction to that approximation; the intuitive $μ^{-1}$ 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.

Figures (7)

• Figure 1: Schematics showing (a,b) dimensional and (c,d) dimensionless quantities. In (c), only the model domain $0 < x < 1$ is shown.
• Figure 2: Example numerical results. In (a) and (b), the left-hand panels show contour plots of $w^\pm(x,y)$ with contour spacing $\Delta w = 0.2$ (thin contours) and $\Delta w = 1$ (thick contours); the no-slip walls [$w^- = 0$, see \ref{['noslip both']}] are marked with thick black lines. The right-hand panels show the $w^\pm$ profile at various values of $x$. The interior-to-exterior viscosity ratio is $\mu = 0.1$, and the groove depth is (a) $h = 2$ (corresponding to grooves of square cross section) and (b) $h = 1/2$. Panel (c) shows the resulting slip length $\lambda$ as function of $\mu$, for various values of $h$.
• Figure 3: (a) Slip length $\lambda$ as a function of viscosity ratio $\mu$ for deep grooves ($h = \infty$), comparing (symbols) numerical results with (dashed curve) the leading-order approximation \ref{['lo result']} and (solid curve) the two-term approximation \ref{['two-term slip for inf deep']}. (b) Deviation from the leading-order asymptotic prediction, comparing numerical results and the asymptotic prediction $-\lambda_0$ obtained from \ref{['lam0 inf h']}.
• Figure 4: Variation with groove depth $h$ of $\mu^{-1/2} - \lambda$, the (negative of the) slip-length deviation from the leading-order asymptotic prediction (cf. Fig. \ref{['fig:lambda_deep']}). The symbols show numerical results for three values of the viscosity ratio $\mu$. The curves show asymptotic results for $-\lambda_0$: solid, exact result \ref{['lam0 arb h']}; dashed, small-$h$ approximation \ref{['lam0 small h']}; and dash-dotted, large-$h$ approximation \ref{['lam0 large h']}.
• Figure 5: Variation of the slip length (scaled as $\lambda \mu^{1/2}$) with scaled groove depth $H = h/\mu^{1/2}$ for various values of the viscosity ratio $\mu$. Symbols show numerical results, with open symbols indicating $h>1$. The thick solid curve portrays the leading-order result \ref{['lam dist 1-term']} in the distinguished limit $\mu,h \ll 1$ with fixed $H$. The thick dashed curve shows the leading-order result \ref{['lo result']} in the original limit $\mu \ll 1$ with fixed $h$. The corresponding thin curves show the two-term approximations \ref{['lam dist 2-term']} and \ref{['lam 2-term result']}, evaluated for $\mu = 0.01$. The thick dotted curve shows the leading-order result \ref{['slip shallow']} in the ultra-shallow-groove limit $h \ll 1$ with fixed $\mu$. (Under the rescalings of the axes, the leading-order approximations are independent of $\mu$.)