Singular effective slip length for longitudinal flow over a dense bubble mattress

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Title: Singular effective slip length for longitudinal flow over a dense bubble mattress
Authors: Schnitzer, O
Item Type: Journal Article
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 π/2 contact angle. Specifically, we carry out a singular-perturbation analysis in the limit ǫ ≪ 1 where the bubbles are closely spaced, finding the effective slip length (normalised by the bubble radius) for longitudinal flow along the the ridges as π/√ 2ǫ − (12/π) ln 2 + (13π/24)√ 2ǫ + o( √ ǫ), the small parameter ǫ 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 O(1) 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 (D. Crowdy, IMA J. Appl. Math., 80 1902, 2015). The comparison demonstrates that our asymptotic formula, together with the diametric “dilute-limit” approximation (D. Crowdy, J. Fluid Mech., 791 R7, 2016), provides an elementary analytical description for essentially arbitrary no-slip fractions.
Issue Date: 13-Sep-2016
Date of Acceptance: 26-Aug-2016
ISSN: 2469-990X
Publisher: American Physical Society
Journal / Book Title: Physical Review Fluids
Volume: 1
Copyright Statement: © 2016 American Physical Society
Keywords: Science & Technology
Physical Sciences
Physics, Fluids & Plasmas
Publication Status: Published
Open Access location:
Article Number: 052101
Appears in Collections:Mathematics
Faculty of Natural Sciences

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