Enhanced slip properties of lubricant-infused grooves

We ascertain the enhanced slip properties for a liquid flow over lubricant-infused unidirectional surfaces. This situation reflects many practical settings involving liquid flows past superhydrophobic grooves filled with gas, or past grooves infused with another, immiscible, liquid of smaller or equal viscosity, i.e., where the ratio of lubricant and liquid viscosities, $\mu \le 1$. To maximize the slippage, we consider deep grooves aligned with the flow. The (normalized by a texture period $L$) effective slip length, $b_{\mathrm{eff}}$, is found as an expansion to first order in protrusion angle $\theta$ about a solution for a flat liquid-lubricant interface. Our results show a significant increase in $b_{\mathrm{eff}}$ with the area fraction of lubricant, $\varphi$, and a strong decrease with $\mu$. By contrast, only little influence of $\theta$ on $b_{\mathrm{eff}}$ is observed. Convex meniscus slightly enhances and concave meniscus slightly reduces $b_{\mathrm{eff}}$ relative to the case of a flat liquid-lubricant interface. The largest correction for $\theta$ is found when $\mu =0$, it decreases with $\mu$, and disappears at $\mu =1$. Finally, we show that lubricant-infused surfaces of small $\theta$ can be modeled as flat with patterns of local slip boundary conditions, and that the (scaled with $L$) local slip length at the liquid-lubricant interface is a universal function of $\varphi$ and $\mu$ only.

Figure 1

(a) Longitudinal semi-infinite shear flow of liquid of viscosity ${\mu }^{*}$ over a periodic array of rectangular grooves containing a lubricant of viscosity ${\mu }_l^{*}$. (b) Single period window for the grooves.

Figure 2

(a) Viscous flow near a flat liquid-lubricant interface located at $z=0$. The leading-order velocities $u^0$ and $u_l^0$ at this boundary are equal; (b) schematic illustration of $u^1$ and $u_l^1$ at the lubricant-meniscus interface, $z=\theta \eta$.

Figure 3

Velocities, $u^1$ (a) and $u_l^1$ (b), vs. $y/\varphi$ computed within the two-phase approach at fixed $\mu =0.2$ and $\varphi =0.5$ (circles), 0.75 (triangles), 0.9 (diamonds). Solid curves plot the results obtained within the single-phase model.

Figure 4

Velocity $u^1$ calculated at $\varphi =0.5$. Circles show results for $\mu =0$. Solid, dashed, dash-dotted, and dotted curves plot results for $\mu =0.02$, 0.2, 0.5, and 1.

Figure 5

(a) Normalized velocities, $u^1/u_{max}^P$, calculated for $\mu =0.02$. Solid curves from top to bottom plot results for $\varphi =0.9$, 0.75 and 0.5. (b) Corresponding normalized velocities, $u/u_{max}^P$, computed with $\theta =\pi /6$. Symbols show $u^0/u_{max}^P$.

Figure 6

(a) The ratio $b_{\mathrm{eff}}^1/b_{\mathrm{eff}}^0$ as a function of $\varphi$. Solid, dashed, dash-dotted, and dotted curves correspond to $\mu =0$, 0.02, 0.2, and 0.5. Symbols show calculations from Eq. (28). (b) The same, but as a function of $\mu$. Solid, dashed, dash-dotted curves correspond to $\varphi =0.9$, 0.75, and 0.5.

Figure 7

(a) Effective slip length vs. lubricant fraction $\varphi$. Calculations are made for $\theta =\pi /6$. Solid, dashed, dashed-dotted curves correspond to $\mu =0$, 0.02, and 0.2. Symbols show $b_{\mathrm{eff}}^P$ given by Eq. (1). (b) The same as a function of $\mu$. From top to bottom $\varphi =0.9$, 0.75, and 0.5. Solid and dashed curves plot results obtained for $\theta =\pi /6$ and 0. Symbols denote $b_{\mathrm{eff}}^{BV}$ calculated from Eq. (2) using $b_c=0.323\varphi /\mu$. (c) The same as a function of $\theta$ calculated with $\mu =0.2$. From top to bottom $\varphi =0.9$, 0.75, and 0.5.

Figure 8

(a) The first-order slip length, $b_{\mathrm{eff}}^1$, for the case of $\mu =0$ plotted against $\varphi$. Solid curve is calculated from Eq. (B4), symbols show $b_{\mathrm{eff}}^1$ given by Eq. (28). (b) The same $b_{\mathrm{eff}}^1$, normalized by $b_{\mathrm{eff}}^P$.