Abstract
Integral expressions for the first-order correction to the effective slip length for longitudinal flows over a unidirectional superhydrophobic surface with rectangular grooves are determined under the assumptions that the meniscus curvature is small and the viscosity contrast between the groove-trapped subphase gas and the working fluid is significant. Both pressure-driven channel flows and semi-infinite shear flows are considered. Reciprocity ideas, based on use of Green’s second identity, provide the integral expressions with integrands dependent on known flat-meniscus solutions found by Philip (Z. Angew. Math. Phys., vol. 23, 1972, pp. 353–372). The results extend earlier work by Sbragaglia & Prosperetti (Phys. Fluids, vol. 19, 2007, 043603) on how weak meniscus curvature affects hydrodynamic slip. In particular, we derive a new integral expression for the first-order slip length correction due to weak meniscus curvature.
Highlights
The study of superhydrophobic surfaces is an active area of research owing to its relevance in a rich variety of micro- and nano-fluidics applications
This is natural since we expect that additional dissipation associated with the working fluid having to drag the viscous subphase along will lead to a decrease in the effective slip for a given pressure gradient
If θ < 0, so that the meniscus bows into the groove the term θ λ(2θ) is positive – as is evident from the expression (1.6) – since it is associated with the increase in cross-sectional area of the longitudinal flow
Summary
The study of superhydrophobic surfaces is an active area of research owing to its relevance in a rich variety of micro- and nano-fluidics applications. Sbragaglia & Prosperetti (2007) calculated the first-order correction to the effective longitudinal slip length for flows in channels assuming that the curvature of the meniscus is small Their analysis places no restriction on the no-shear fraction of the surface. While Sbragaglia & Prosperetti (2007) did not use any such reciprocity arguments ours is not the first study to exploit the latter to understand superhydrophobic surfaces: Squires (2008) employed reciprocal theorems to examine electrokinetic effects on flat slipping surfaces; Baier, Steffes & Hardt (2010) used them to study thermal Marangoni flow over a superhydrophobic array of fins oriented parallel or perpendicular to an applied temperature gradient None of these prior works using reciprocity ideas, take non-zero meniscus curvature into account
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