Skip to main content
×
×
Home

Small-solid-fraction approximations for the slip-length tensor of micropillared superhydrophobic surfaces

  • Ory Schnitzer (a1) and Ehud Yariv (a2)
Abstract

Fakir-like superhydrophobic surfaces, formed by doubly periodic arrays of thin pillars that sustain a lubricating gas layer, exhibit giant liquid-slip lengths that scale as $\unicode[STIX]{x1D719}^{-1/2}$ relative to the periodicity, $\unicode[STIX]{x1D719}$ being the solid fraction (Ybert et al., Phys. Fluids, vol. 19, 2007, 123601). Considering arbitrarily shaped pillars distributed over an arbitrary Bravais lattice, we employ matched asymptotic expansions to calculate the slip-length tensor in the limit $\unicode[STIX]{x1D719}\rightarrow 0$ . The leading $O(\unicode[STIX]{x1D719}^{-1/2})$ slip length is determined from a local analysis of an ‘inner’ region close to a single pillar, in conjunction with a global force balance. This leading term, which is independent of the lattice geometry, is related to the drag due to pure translation of a flattened disk shaped like the pillar cross-section; its calculation is illustrated for the case of elliptical pillars. The $O(1)$ slip length is associated with the excess velocity induced about a given pillar by all the others. Since the field induced by each pillar corresponds on the ‘outer’ lattice scale to a Stokeslet whose strength is fixed by the shear rate, the $O(1)$ slip length depends upon the lattice geometry but is independent of the cross-sectional shape. Its calculation entails asymptotic evaluation of singular lattice sums. Our approximations are in excellent agreement with existing numerical computations for both circular and square pillars.

Copyright
Corresponding author
Email address for correspondence: o.schnitzer@imperial.ac.uk
References
Hide All
Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions, 3rd edn. Dover.
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419441.
Cottin-Bizonne, C., Barentin, C., Charlaix, É., Bocquet, L. & Barrat, J.-L. 2004 Dynamics of simple liquids at heterogeneous surfaces: molecular-dynamics simulations and hydrodynamic description. Eur. Phys. J. 15 (4), 427438.
Davis, A. M. J. & Lauga, E. 2010 Hydrodynamic friction of fakir-like superhydrophobic surfaces. J. Fluid Mech. 661, 402411.
Extrand, C. W. 2004 Criteria for ultralyophobic surfaces. Langmuir 20 (12), 50135018.
Hasimoto, H. 1959 On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (2), 317328.
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.
Kittel, C. 2005 Introduction to Solid State Physics. Wiley.
Lamb, H. 1945 Hydrodynamics, 6th edn. Cambridge University Press.
Lee, C., Choi, C.-H. & Kim, C.-J. 2008 Structured surfaces for a giant liquid slip. Phys. Rev. Lett. 101 (6), 064501.
Ng, C.-O. & Wang, C. 2010 Apparent slip arising from Stokes shear flow over a bidimensional patterned surface. Microfluid. Nanofluid. 8 (3), 361371.
Nijboer, B. R. A. & De Wette, F. W. 1957 On the calculation of lattice sums. Physica 23, 309321.
Pozrikidis, C. 1996 Computation of periodic Green’s functions of Stokes flow. J. Engng Maths 30 (1–2), 7996.
Quéré, D. 2008 Wetting and roughness. Annu. Rev. Mater. Res. 38 (1), 7199.
Ybert, C., Barentin, C., Cottin-Bizonne, C., Joseph, P. & Bocquet, L. 2007 Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries. Phys. Fluids 19 (12), 123601.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 5
Total number of PDF views: 146 *
Loading metrics...

Abstract views

Total abstract views: 258 *
Loading metrics...

* Views captured on Cambridge Core between 26th March 2018 - 14th August 2018. This data will be updated every 24 hours.