Slip length estimation for flow over lubricant-impregnated surface

TL;DR

The paper tackles drag reduction and anisotropic flow control over lubricant-impregnated ribbed surfaces by formulating a two-fluid Stokes problem with a flat fluid-fluid interface and solving it via domain decomposition and eigenfunction expansion. It yields semi-analytical solutions for flows along and across the ribs, providing explicit expressions for slip velocity and slip length in terms of geometric parameters ($d$, $a$, $L$) and fluid properties ($\mu^A$, $\mu^B$). The work demonstrates high accuracy through numerical validation and reveals how lubricant fraction $a$, rib height $d$, and viscosity ratio $\mu=\mu^A/\mu^B$ govern slip and anisotropy, offering design guidance for maximizing drag reduction or achieving tailored anisotropic responses. Overall, the approach enables robust LIS design across broad parameter ranges with practical implications for microfluidic control and drag reduction in engineering systems, with all key quantities expressed in $...$-style mathematics.$b = \frac{u}{du/dy}$ at the reference plane and related surface-averaged definitions are employed to quantify slip.

Abstract

Lubricant-impregnated surfaces (LIS) and superhydrophobic surfaces (SHSs) are known to passively reduce drag over a surface, which, with a suitable design such as the ribbed texture, can also steer flows anisotropically. Analytical predictions are developed for ribbed textures using an eigenfunction expansion approach. Compared to currently available analytical predictions, these predictions demonstrate superior numerical accuracy and avoid restrictive assumptions on rib geometry, working fluid and lubricant properties. The predictions provide contrasting design prescriptions depending on whether a lower drag or a larger degree of anisotropic flow deflection is desired.

Figures (6)

• Figure 1: Schematic diagram of (a) surface having ribs, with driving force applied along two principal directions, and (b) details of the corrugation geometry. As the corrugations are periodically repeating along $x-$direction, hence only one cell is shown in Figure (a). Due to the symmetric nature of the problem along the mid-plane, figure (b) shows only one-half of the cell along with the boundary condition used to solve the problem.
• Figure 2: Comparison between analytical and numerical results for shear-driven longitudinal flow on rectangular shaped LIS for the case of (a) $\mu = 50$, and (b) $\mu=0.02$.
• Figure 3: Comparison between analytical and numerical results for transverse flow on rectangular shaped LIS for the case of (a) $\mu=55$ and (b) $\mu=0.02$.
• Figure 4: Variation of slip length for the flow along $z-$ direction with the lubricant fraction ($a$) at the constant lubricant volume ($v=2da$) for the case of (a) $\mu=50$, (b) $\mu=10$, (c) $\mu=1$, and (d) $\mu=0.02$.
• Figure 5: Variation of effective slip length for the flow along $x-$ direction with the lubricant fraction ($a$) at the constant lubricant volume ($v=2da$) for the case of (a) $\mu=50$, and (b) $\mu=0.02$.