Effect of streamwise cross-sectional variation on liquid retention in liquid-infused substrates under an external flow

Lubricant infused in micropatterned substrates will drain in the presence of an external, immiscible shear flow, but this drainage can be retarded by using the substrate geometry to exploit capillary effects at the interface. Recent work [Wexler et al., Phys. Rev. Lett. 114, 168301 (2015)] produced general geometric guidelines for the design of substrates to oppose this shear drainage in rectangular grooves with uniform cross section in the streamwise direction of the external flow. In this study, we generalize the fluid-retention predictions for lubricant-infused grooves with slowly varying cross section and interfacial geometries. Accounting for interfacial deformation is found to reduce the predicted lubricant retention in geometrically uniform grooves, albeit differently from past predictions. Streamwise nonuniformity in groove cross section is shown to provide little benefit to lubricant retention but may potentially cause significant reductions in retention, and thus should be considered carefully when establishing substrate manufacturing tolerances. An optimal groove shape for maximal lubricant-retention length is derived, although its practical use appears limited by its small surface area coverage compared to uniform grooves.

Figure 1

The geometry of a rectangular domain, showing (top) the isometric overview with steady-state retention length of lubricant, $L_{\infty }$, and a single cross section highlighted; and (bottom) a cross-sectional view, showing the radius of curvature, $R\left(x\right)$, contact angle, $\theta \left(x\right)$, and groove half width $a\left(x\right)$ and depth $h\left(x\right)$. The interface and oil in the groove are colored gray. The external, two-dimensional flow profile is illustrated with its associated shear stress, ${\tau }_{\text{ext}}$.

Figure 2

The steady-state geometry of a uniform rectangular grid, corresponding to the calculations performed in Wexler et al. [8], showing (top) the isometric overview with steady-state length of lubricant, $L_{\infty }$, and the upstream and downstream cross sections highlighted, with all interfacial curvature confined to the upstream end; and (bottom) the cross-sectional views, showing the upstream radius of curvature, $R_r$, receding contact angle, ${\theta }_r$, and groove half width $a_0$ and depth $h_0$. The interface and lubricant in the groove is colored gray.

Figure 3

The steady-state pressure gradient as a function of aspect ratio, $a/h$, and contact angle, $\theta$, calculated numerically over the exact domain (black solid lines); by domain perturbation (green dash-dotted); for a flat interface (red dotted); and using the heuristic scaling of Eq. (20) (blue dashed).

Figure 4

The steady-state retention length correction factor for a uniform groove, determined by integrating the steady momentum equation (solid black) or approximating the variation of the maximum deformation according to Liu et al. [27] (dashed line).

Figure 5

A trapezoidally shaped groove, with linearly varying half-width, $a\left(x\right).$

Figure 6

The ratio of retained fluid in a groove with linearly varying half-width, $L_{\infty }^{\text{linear}}$, vs a uniform groove of the same average aspect ratio, $L_{\infty }^{\text{uniform}}$, as a function of downstream width, $a_0$, and wall slope, $a^{\prime }$. Calculations were performed for ${\theta }_r=\pi /3$.

Figure 7

The ratio of retained fluid in a groove with linearly varying depth, $L_{\infty }^{\text{linear}}$, vs a uniform groove of the same average aspect ratio, $L_{\infty }^{\text{uniform}}$, as a function of downstream depth, $h_0$, and floor slope, $h^{\prime }$. Calculations were performed for ${\theta }_r=\pi /3$.

Figure 8

The nondimensionalized variation of interfacial shear stress for a flat interface at the centerline, ${\tau }_s(z=0,y=\eta )$, with respect to variations the aspect ratio of the groove, $h/a$ for $\lambda =2$. Theory from Schönecker and Hardt [23] in solid black line; numerical calculations of the theory for flat interface in blue dashed line; numerics for curved interface in red dash-dotted line.