Skip to main content
×
Home

Effective velocity boundary condition at a mixed slip surface

  • M. SBRAGAGLIA (a1) and A. PROSPERETTI (a1) (a2)
Abstract

This paper studies the nature of the effective velocity boundary condition for liquid flow over a plane boundary on which small free-slip islands are randomly distributed. It is found that an effective Navier partial-slip condition for the velocity emerges from a statistical analysis valid for arbitrary fractional area coverage β. As an example, the general theory is applied to the low-β limit and this result is extended heuristically to finite β with a resulting slip length proportional to aβ/(1 − β), where a is a characteristic size of the islands. A specification of the nature of the free-slip islands is not required in the analysis. They could be nano-bubbles, as suggested by recent experiments, or hydrophobic surface patches. The results are also relevant for ultra-hydrophobic surfaces exploiting the so-called ‘lotus effect’.

Copyright
References
Hide All
Barthlott W. & Neinhaus C. 1997 Purity of the sacred lotus, or escape from contamination in biological surfaces. Planta 202, 18.
Bunkin N. F., Kochergin A. V., Lobeyev A. V., Ninham B. W. & Vinogradova O. I. 1996 Existence of charged submicrobubble clusters in polar liquids as revealed by correlation between optical cavitation and electrical conductivity. Colloid Interface Sci. A 110, 207212.
Cheng J.-T. & Giordano N. 2002 Fluid flow through nanometer-scale channels. Phys. Rev. Lett. 65, 031206.
Choi C. H. & Kim C. J. 2006 Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett. 96, 066001.
Craig V. S. J., Neto C. & Williams D. R. M. 2001 Shear-dependent boundary slip in an aqueous Newtonian liquid. Phys. Rev. Lett. 87, 054504.
Dammer S. M. & Lohse D. 2006 Gas enrichment at liquid-wall interfaces. Phys. Rev. Lett. 96, 206101.
Davis A. M. J. 1991 Shear flow disturbance due to a hole in the plane. Phys. Fluids A 3, 478480.
Foldy L. 1945 The multiple scattering of waves. Phys. Rev. 67, 107119.
Gradshteyn I. S. & Ryzhik I. M. 2000 Table of Integrals, Series, and Products, 6th edn. Academic.
Holmberg M., Kühle A., Garnaes J., Mörch K. A. & Boisen A. 2003 Nanobubble trouble on gold surfaces. Langmuir 19, 10,51010,513.
Ishida N., Inoue T., Miyahara M. & Higashitani K. 2000 Nano bubbles on a hydrophobic surface in water observed by tapping-mode atomic force microscopy. Langmuir 16, 63776380.
Jäger W. & Mikeli'c A. 2001 On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Diffl Equat. 170, 96122.
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.
Kim S. & Karrila S. 1991 Microhydrodynamics. Butterworth-Heinemann.
Lauga E., Brenner M. P. & Stone H. A. 2005 Microfluidics: the no-slip boundary condition. In Handbook of Experimental Fluid Dynamics (ed. Foss J., Tropea C. & Yarin A.). Springer (to appear).
Lauga E. & Stone H. A. 2003 Effective slip in pressure-driven stokes flow. J. Fluid Mech. 489, 5577.
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, 2859–897.
Ou J., Perot B. & Rothstein P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16, 46354643.
Ou J. & Rothstein P. 2005 Direct velocity measurement of the flow past drag-reducing ultrahydrophobic surfaces. Phys. Fluids 17, 103606.
Philip J. R. 1972 Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23, 353370.
Pit R., Hervet H. & Léger L. 2000 Direct experimental evidence of slip in hexadecane: Solid interfaces. Phys. Rev. Lett. 85, 980983.
Pozrikidis C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Ranger K. B. 1978 The circular disc straddling the interface of a two phase flow. Intl J. Multiphase Flow 4, 263277.
Rubinstein J. & Keller J. 1989 Sedimentation of a dilute suspension. Phys. Fluids A 1, 637643.
Sarkar K. & Prosperetti A. 1995 Effective boundary conditions for the Laplace equation with a rough boundary. Proc. R. Soc. Lond. A 451, 425452.
Sarkar K. & Prosperetti A. 1996 Effective boundary conditions for Stokes flow over a rough surface. J. Fluid Mech. 316, 223240.
Sbragaglia M., Benzi R., Biferale L., Succi S. & Toschi F. 2006 Surface Roughness-Hydrophobicity Coupling in Microchannel and Nanochannel Flows. Phys. Rev. Lett. 97, 204503.
Simonsen A. C., Hansen P. L. & Klosgen B. 2004 Nanobubbles give evidence of incomplete wetting at a hydrophobic interface. J. Colloid Interface Sci. 273, 291299.
Smith S. H. 1987 Stokes flows past slits and holes. Intl J. Multiphase Flow 13, 219231.
Sneddon I. N. 1966 Mixed Boundary Value Problems in Potential Theory. North-Holland.
Steitz R., Gutberlet T., Hauss T., Klösgen B., Krastev R., Schemmel S., Simonsen A. C. & Findenegg G. H. 2003 Nanobubbles and their precursor layer at the interface of water against a hydrophobic substrate. Langmuir 19, 24092418.
Stone H. A. & Ajdari A. 1998 Hydrodynamics of particles embedded in a flat surfactant layer overlying a subphase of finite depth. J. Fluid Mech. 369, 151173.
Tartakovsky D. M. & Xiu D. B. 2006 Stochastic analysis of transport in tubes with rough walls. J. Comput. Phys. 217, 248259.
Tretheway D. & Meinhart C. 2002 Apparent fluid slip at hydrophobic microchannel walls. Phys. Fluids 14, L9L12.
Twersky V. 1957 On scattering and reflection of sound by rough surfaces. J. Acoust. Soc. Am. 29, 209225.
Twersky V. 1983 Reflection and scattering of sound by correlated rough surfaces. J. Acoust. Soc. Am. 73, 8594.
Tyrrell J. W. G. & Attard P. 2001 Images of nanobubbles on hydrophobic surfaces and their interactions. Phys. Rev. Lett. 87, 176104.
Vinogradova O. I. 1999 Slippage of water over hydrophobic surfaces. Intl J. Mineral Proc. 56, 3160.
Watanabe K., Yanuar & Udagawa H. 1999 Drag reduction of Newtonian fluid in a circular pipe with a highly water-repellent wall. J. Fluid Mech. 381, 225238.
Wu Z., Zhang X., Zhang X., Li G., Sun J., Zhang M. & Hu J. 2005 Nanobubbles influence on BSA adsorption on mica surface. Surface Interface Anal. 37, 797801.
Zhu Y. & Granick S. 2001 Rate-dependent slip of newtonian liquid at smooth surfaces. Phys. Rev. Lett. 87, 096105.
Zhu Y. & Granick S. 2002 Limits of the hydrodynamic no-slip boundary condition. Phys. Rev. Lett. 88, 106102.
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

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 58 *
Loading metrics...

Abstract views

Total abstract views: 165 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th November 2017. This data will be updated every 24 hours.