Longitudinal thermocapillary flow over a dense bubble mattress

A common form of superhydrophobic surface is made out of a periodically grooved solid substrate, wherein cylindrical bubbles are trapped in a Cassie state. When a macroscopic temperature gradient is externally applied, Marangoni forces generate thermocapillary flow of characteristic magnitude μ = −aG\sigma _T /\mu , in which 2a is the groove width, G the applied-gradient magnitude, \mu the liquid viscosity, and \sigma _T the derivative of interfacial-tension coefficient with respect to the temperature. We consider the case of a gradient which is applied parallel to the grooves. Assuming a highly conducting solid substrate, we seek to calculate the longitudinal velocity component, antiparallel to the applied gradient, and in particular its ``slip"" value, attained at large distances away from the surface. Normalized by μ, this value depends only upon the bubble protrusion angle and the solid fraction \epsilon . In this paper we consider the small solid-fraction limit \epsilon ≪ 1 and focus upon the case of a 90^\circ protrusion angle, for which this limit is known to be highly singular in the comparable problem of shear-driven flow [O. Schnitzer, Phys. Rev. Fluids, 1 (2016), 052101]._ Using matched asymptotic expansions, we find that the dimensionless slip velocity is given by \pi ² /√8\epsilon + ln \epsilon + I − ln(8\pi ² ) + o(1). The first two terms in this expansion follow from a lubrication-type analysis of the narrow gap region separating two neighboring bubbles. The subsequent O(1) terms follow from asymptotic matching with the bubble-scale region, where the bubbles appear to be touching. The solution in that region is obtained using conformal mapping techniques, with the constant I given as an explicit integral, evaluated numerically to be 2.27.