The use of superhydrophobic surfaces has attracted significant research in recent years, due to their ability to reduce flow resistance, giving a range of applications in microfluidics. They are partially non-wetting surfaces consisting of microscale roughness that resists a liquid fully wetting the solid substrate due to surface tension. A natural example is the lotus leaf, but artificial examples can be made, such as arrays of microscale grooves, ridges or pillars. The focus of this talk is on the application of such surfaces to thermal transport, which has received relatively little attention but has consequences for, e.g., liquid cooling of microelectronics. In particular, we consider forced convection from a heated substrate textured with parallel ridges aligned with a pressure-driven flow. The reduced contact area between the solid and liquid enhances the flow rate (measured by an effective slip length) but reduces the convective heat transfer (quantified by a Nusselt number). These competing effects both contribute to the total heat transfer, which can be enhanced or diminished depending on the ridge geometry and liquid properties. One of the most important geometric effects is the curvature of the liquid-gas interfaces (menisci) that span the ridge tips.

In the first part of the talk, we analyse the periodic flow and thermal problems accounting for curved menisci using boundary perturbation methods, and provide semi-analytical series solutions for the slip length and Nusselt number. In the second part, we consider the additional impact of thermal Marangoni stresses due to temperature variations along the menisci, and derive asymptotic expressions for the slip length in a suite of limits using matched asymptotic expansions and conformal mapping techniques. These formulae are compared against full numerical simulations, and found to have a wide range of validity, spanning much of the relevant parameter space.

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