Skip to main content
SpringerLink
Log in

Apparent slip arising from Stokes shear flow over a bidimensional patterned surface

  • Research Paper
  • Published:
Microfluidics and Nanofluidics Aims and scope Submit manuscript

Abstract

A mathematical model is presented for the problem of apparent slip arising from Stokes shear flow over a composite surface featuring mixed boundary conditions on the microscale. The surface can be composed of a bidimensional array of solid areas placed on an otherwise no-shear surface corresponding to an envelope over the tops of posts, or no-shear areas placed on an otherwise solid surface corresponding to an envelope over the tops of holes. Posts and holes of circular or square cross section, and solid areas of no-slip or partial-slip types are studied. Following some previously proposed scaling laws, the effective slip length is expressed as a certain function of the solid fraction for some specific cases. More refined equations based on linear regression of the computed results are obtained for these cases. Amounts of slippage arising from these bidimensional patterns are compared with those from the one-dimensional patterns of grooves/grates. It is also shown that a larger slip length can result from an arrangement where the pitch is larger in the spanwise direction than in the streamwise direction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Choi CH, Westin KJA, Breuer KS (2003) Apparent slip flows in hydrophilic and hydrophobic microchannels. Phys Fluids 15: 2897–2902

    Article  Google Scholar 

  • Choi CH, Ulmanella U, Kim J, Ho CM, Kim CJ (2006) Effective slip and friction reduction in nanograted superhydrophobic microchannels. Phys Fluids 18:087105

    Article  Google Scholar 

  • Cottin-Bizonne C, Jurine S, Baudry J, Crassous J, Restagno F, Charlaix E (2002) Nanorheology: an investigation of the boundary condition at hydrophobic and hydrophilic interfaces. Eur Phys J E 9:47–53

    Google Scholar 

  • Cottin-Bizonne C, Barrat JL, Bocquet L, Charlaix E (2003) Low-friction flows of liquid at nanopatterned interfaces. Nat Mater 2:237–240

    Article  Google Scholar 

  • Cottin-Bizonne C, Barentin C, Charlaix E, Bocquet L, Barrat JL (2004) Dynamics of simple liquids at heterogeneous surfaces: molecular-dynamics simulations and hydrodynamic description. Eur Phys J E 15:427–438

    Article  Google Scholar 

  • Davies J, Maynes D, Webb BW, Woolford B (2006) Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys Fluids 18:087110

    Article  Google Scholar 

  • Feuillebois F, Bazant MZ, Vinogradova OI (2009) Effective slip over superhydrophobic surfaces in thin channels. Phys Rev Lett 102:026001

    Article  Google Scholar 

  • Hendy SC, Lund NJ (2007) Effect slip boundary conditions for flows over nanoscale chemical heterogeneities. Phys Rev E 76:066313

    Article  Google Scholar 

  • Hendy SC, Jasperse M, Burnell J (2005) Effect of patterned slip on micro- and nanofluidic flows. Phys Rev E 72:016303

    Article  Google Scholar 

  • Joseph P, Cottin-Bizonne C, Benoit JM, Ybert C, Journet C, Tabeling P, Bocquet L (2006) Slippage of water past superhydrophobic carbon nanotube forests in microchannels. Phys Rev Lett 97:156104

    Article  Google Scholar 

  • Lauga E, Stone HA (2003) Effective slip in pressure-driven Stokes flow. J Fluid Mech 489:55–77

    Article  MATH  MathSciNet  Google Scholar 

  • Lauga E, Brenner MP, Stone HA (2007) Microfluidics: the no-slip boundary condition. In: Foss J et al (eds) Springer handbook of experimental fluid mechanics. Springer, Heidelberg, pp 1219–1240

  • Lee C, Choi CH, Kim CJ (2008) Structured surfaces for a giant liquid slip. Phys Rev Lett 101:064501

    Article  Google Scholar 

  • Navier CLMH (1823) Memoire sur les lois du movement des fluids. Memoires de l’Academie Royale des Sciences de l’Institut de France VI:389–440

    Google Scholar 

  • Neto C, Evans DR, Bonaccurso E, Butt HJ, Craig VSJ (2005) Boundary slip in Newtonian liquids: a review of experimental studies. Rep Prog Phys 68:2859–2897

    Article  Google Scholar 

  • Ng CO, Wang CY (2009) Stokes shear flow over a grating: implications for superhydrophobic slip. Phys Fluids 21:013602

    Article  Google Scholar 

  • Ou J, Rothstein JP (2005) Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys Fluids 17:103606

    Article  Google Scholar 

  • Philip JR (1972) Flows satisfying mixed no-slip and no-shear conditions. Z Angew Math Phys 23:353–372

    Article  MATH  MathSciNet  Google Scholar 

  • Priezjev NV, Troian SM (2006) Influence of periodic wall roughness on the slip behavior at liquid/solid interfaces: molecular-scale simulations versus continuum predictions. J Fluid Mech 554:25–46

    Article  MATH  Google Scholar 

  • Priezjev NV, Darhuber AA, Troian SM (2005) Slip behavior in liquid films on surfaces of patterned wettability: comparison between continuum and molecular dynamics simulations. Phys Rev E 71:041608

    Article  Google Scholar 

  • Sbragaglia M, Prosperetti A (2007a) A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys Fluids 19:043603

    Article  Google Scholar 

  • Sbragaglia M, Prosperetti A (2007b) Effective velocity boundary condition at a mixed slip surface. J Fluid Mech 578:435–451

    Article  MATH  MathSciNet  Google Scholar 

  • Tretheway DC, Meinhart CD (2002) Apparent fluid slip at hydrophobic microchannel walls. Phys Fluids 14:L9–L12

    Article  Google Scholar 

  • Wang CY (2003) Flow over a surface with parallel grooves. Phys Fluids 15:1114–1121

    Article  Google 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

    Article  Google Scholar 

  • Zhang J, Kwok DY (2004) Apparent slip over a solid-liquid interface with a no-slip boundary condition. Phys Rev E 70:056701

    Article  Google Scholar 

  • Zhang X, Shi F, Niu J, Jiang Y, Wang Z (2008) Superhydrophobic surfaces: from structural control to functional application. J Mater Chem 18:621–633

    Article  Google Scholar 

  • Zhu Y, Granick S (2001) Rate-dependent slip of Newtonian liquid at smooth surfaces. Phys Rev Lett 87:096105

    Article  Google Scholar 

  • Zhu Y, Granick S (2002) Limits of the hydrodynamic no-slip boundary condition. Phys Rev Lett 88(10):106102

    Article  Google Scholar 

Download references

Acknowledgments

The authors have benefitted from personal communication with Cécile Cottin-Bizonne. The work was initiated by the second author when he was a William Mong Visiting Research Fellow associating with the first author in May, 2008. The financial support by the William M.W. Mong Engineering Research Fund of the University of Hong Kong is gratefully acknowledged. The work was also supported by the University of Hong Kong through the Small Project Funding Scheme under Project Code 200807176081.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong

    Chiu-On Ng

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    C. Y. Wang

Authors
  1. Chiu-On Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Y. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chiu-On Ng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ng, CO., Wang, C.Y. Apparent slip arising from Stokes shear flow over a bidimensional patterned surface. Microfluid Nanofluid 8, 361–371 (2010). https://doi.org/10.1007/s10404-009-0466-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10404-009-0466-x

Keywords

Search

Navigation