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Singular effective slip length for longitudinal flow over a dense bubble mattress
Phys. Rev. Fluids 1, 052101(R) – Published 13 September, 2016
DOI: https://doi.org/10.1103/PhysRevFluids.1.052101
Abstract
We consider the effective hydrophobicity of a periodically grooved surface immersed in liquid, with trapped shear-free bubbles protruding between the no-slip ridges at a contact angle. Specifically, we carry out a singular-perturbation analysis in the limit where the bubbles are closely spaced, finding the effective slip length (normalized by the bubble radius) for longitudinal flow along the ridges as , the small parameter being the planform solid fraction. The square-root divergence highlights the strong hydrophobic character of this configuration; this leading singular term (along with the third term) follows from a local lubrication-like analysis of the gap regions between the bubbles, together with general matching considerations and a global conservation relation. The constant term is found by matching with a leading-order solution in the outer region, where the bubbles appear to be touching. We find excellent agreement between our slip-length formula and a numerical scheme recently derived using a unified-transform method [Crowdy, IMA J. Appl. Math. 80, 1902 (2015)]. The comparison demonstrates that our asymptotic formula, together with the diametric dilute-limit approximation [Crowdy, J. Fluid Mech. 791, R7 (2016)], provides an elementary analytical description for essentially arbitrary no-slip fractions.
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