Effects of slowly varying meniscus curvature on internal flows in the Cassie state

Abstract

The flow rate of a pressure-driven liquid through a microchannel may be enhanced by texturing its no-slip boundaries with grooves aligned with the flow. In such cases, the grooves may contain vapour and/or an inert gas and the liquid is trapped in the Cassie state, resulting in (apparent) slip. The flow-rate enhancement is of benefit to different applications including the increase of throughput of a liquid in a lab-on-a-chip, and the reduction of thermal resistance associated with liquid metal cooling of microelectronics. At any given cross-section, the meniscus takes the approximate shape of a circular arc whose curvature is determined by the pressure difference across it. Hence, it typically protrudes into the grooves near the inlet of a microchannel and is gradually drawn into the microchannel as it is traversed and the liquid pressure decreases. For sufficiently large Reynolds numbers, the variation of the meniscus shape and hence the flow geometry necessitates the inclusion of inertial (non-parallel) flow effects. We capture them for a slender microchannel, where our small parameter is the ratio of ridge pitch-to-microchannel height, and order-one Reynolds numbers. This is done by using a hybrid analytical–numerical method to resolve the nonlinear three-dimensional (3-D) problem as a sequence of two-dimensional (2-D) linear ones in the microchannel cross-section, allied with non-local conditions that determine the slowly varying pressure distribution at leading and first orders. When the pressure difference across the microchannel is constrained by the advancing contact angle of the liquid on the ridges and its surface tension (which is high for liquid metals), inertial effects can significantly reduce the flow rate for realistic parameter values. For example, when the solid fraction of the ridges is 0.1, the microchannel height-to-(half) ridge pitch ratio is 6, the Reynolds number of the flow is 1 and the small parameter is 0.1, they reduce the flow rate of a liquid metal (Galinstan) by approximately 50 %. Conversely, for sufficiently large microchannel heights, they enhance it. Physical explanations of both of these phenomena are given.