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, Phys. Rev. Lett. 114, 168301 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.168301] 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.