Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-18T21:06:20.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydrodynamic friction of fakir-like superhydrophobic surfaces

Published online by Cambridge University Press:  23 August 2010

ANTHONY M. J. DAVIS  and
ERIC LAUGA
ANTHONY M. J. DAVIS
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
ERIC LAUGA*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A fluid droplet located on a superhydrophobic surface makes contact with the surface only at small isolated regions, and is mostly in contact with the surrounding air. As a result, a fluid in motion near such a surface experiences very low friction, and superhydrophobic surfaces display strong drag reduction in the laminar regime. Here we consider theoretically a superhydrophobic surface composed of circular posts (so-called fakir geometry) located on a planar rectangular lattice. Using a superposition of point forces with suitably spatially dependent strength, we derive the effective surface-slip length for a planar shear flow on such a fakir-like surface as the solution to an infinite series of linear equations. In the asymptotic limit of small surface coverage by the posts, the series can be interpreted as Riemann sums, and the slip length can be obtained analytically. For posts on a square lattice, our analytical prediction of the dimensionless slip length, in the low surface coverage limit, is in excellent quantitative agreement with previous numerical computations.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 661 , 25 October 2010 , pp. 402 - 411
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bico, J., Marzolin, C. & Quéré, D. 1999 Pearl drops. Europhys. Lett. 47, 220–226.CrossRefGoogle Scholar
Bocquet, L. & Barrat, J.-L. 2007 Flow boundary conditions: from nano- to micro-scales. Soft Matter 3, 685–693.CrossRefGoogle ScholarPubMed
Choi, C. H. & Kim, C. J. 2006 Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett. 96, 066001.CrossRefGoogle Scholar
Choi, C. H., Ulmanella, U., Kim, J., Ho, C. M. & Kim, C. J. 2006 Effective slip and friction reduction in nanograted superhydrophobic microchannels. Phys. Fluids 18, 087105.CrossRefGoogle Scholar
Cottin-Bizonne, C., Barentin, C., Charlaix, E., Bocquet, L. & Barrat, J. L. 2004 Dynamics of simple liquids at heterogeneous surfaces: molecular-dynamics simulations and hydrodynamic description. Eur. Phys. J. E 15, 427–438.CrossRefGoogle ScholarPubMed
Davies, J., Maynes, D., Webb, B. W. & Woolford, B. 2006 Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys. Fluids 18, 087110.CrossRefGoogle Scholar
Davis, A. M. J. & Lauga, E. 2009 The friction of a mesh-like super-hydrophobic surface. Phys. Fluids 21, 113101.CrossRefGoogle Scholar
Feng, L., Li, S. H., Li, Y. S., Li, H. J., Zhang, L. J., Zhai, J., Song, Y. L., Liu, B. Q., Jiang, L. & Zhu, D. B. 2002 Super-hydrophobic surfaces: from natural to artificial. Adv. Mater. 14, 1857–1860.CrossRefGoogle Scholar
de Gennes, F.Brochard-Wyart, P.-G. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.CrossRefGoogle Scholar
Gogte, S., Vorobieff, P., Truesdell, R., Mammoli, A., van Swol, F., Shah, P. & Brinker, C. J. 2005 Effective slip on textured superhydrophobic surfaces. Phys. Fluids 17, 051701.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice Hall.Google Scholar
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, 317–328.CrossRefGoogle Scholar
Joseph, P., Cottin-Bizonne, C., Benoit, J. M., Ybert, C., Journet, C., Tabeling, P. & Bocquet, L. 2006 Slippage of water past superhydrophobic carbon nanotube forests in microchannels. Phys. Rev. Lett. 97, 156104.CrossRefGoogle ScholarPubMed
Lauga, E., Brenner, M. P. & Stone, H. A. 2007 Microfluidics: the no-slip boundary condition. In Handbook of Experimental Fluid Dynamics (ed. Yarin, A., Tropea, C. & Foss, J. F.), Chap. 19, pp. 1219–1240. Springer.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.CrossRefGoogle Scholar
Lee, C., Choi, C. H. & Kim, C. J. 2008 Structured surfaces for a giant liquid slip. Phys. Rev. Lett. 101, 064501.CrossRefGoogle ScholarPubMed
Maynes, D., Jeffs, K., Woolford, B. & Webb, B. W. 2007 Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction. Phys. Fluids 19, 093603.CrossRefGoogle Scholar
Neto, C., Evans, D. R., Bonaccurso, E., Butt, H.-J. & Craig, V. S. J. 2005 Boundary slip in Newtonian liquids: a review of experimental studies. Rep. Prog. Phys. 68, 2859–2897.CrossRefGoogle Scholar
Ng, C.-O. & Wang, C. Y. 2009 Stokes shear flow over a grating: implications for superhydrophobic slip. Phys. Fluids 21, 013602.CrossRefGoogle Scholar
Ng, C.-O. & Wang, C. Y. 2010 Apparent slip arising from Stokes shear flow over a bidimensional patterned surface. Microfluid Nanofluid 8, 361–371.CrossRefGoogle Scholar
Onda, T., Shibuichi, S., Satoh, N. & Tsujii, K. 1996 Super-water-repellent fractal surfaces. Langmuir 12, 2125–2127.CrossRefGoogle Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16, 4635–4643.CrossRefGoogle Scholar
Ou, J. & Rothstein, J. P. 2005 Drag reduction and μ-PIV measurements of the flow past ultrahydrophobic surfaces. Phys. Fluids 17, 103606.CrossRefGoogle Scholar
Quéré, D. 2008 Wetting and roughness. Annu. Rev. Fluid Mech. 38, 71–99.Google Scholar
Roach, P., Shirtcliffe, N. J. & Newton, M. I. 2008 Progress in superhydrophobic surface development. Soft Matter 4, 224–240.CrossRefGoogle ScholarPubMed
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89–109.CrossRefGoogle Scholar
Teo, C. J. & Khoo, B. C. 2009 Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves. Microfluid Nanofluid 7, 353–382.CrossRefGoogle Scholar
Truesdell, R., Mammoli, A., Vorobieff, P., van Swol, F. & Brinker, C. J. 2006 Drag reduction on a patterned superhydrophobic surface. Phys. Rev. Lett. 97, 044504.CrossRefGoogle ScholarPubMed
Tsai, P. C., Peters, A. M., Pirat, C., Wessling, M., Lammertink, R. G. H. & Lohse, D. 2009 Quantifying effective slip length over micropatterned hydrophobic surfaces. Phys. Fluids 21, 112002.CrossRefGoogle Scholar
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, 123601.CrossRefGoogle Scholar