Skip to main content Accesibility Help
×
×
Home

Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state

  • Clarissa Schönecker (a1), Tobias Baier (a1) and Steffen Hardt (a1)
Abstract

Analytical expressions for the flow field as well as for the effective slip length of a shear flow over a surface with periodic rectangular grooves are derived. The primary fluid is in the Cassie state with the grooves being filled with a secondary immiscible fluid. The coupling of the two fluids is reflected in a locally varying slip distribution along the fluid–fluid interface, which models the effect of the secondary fluid on the outer flow. The obtained closed-form analytical expressions for the flow field and effective slip length of the primary fluid explicitly contain the influence of the viscosities of the two fluids as well as the magnitude of the local slip, which is a function of the surface geometry. They agree well with results from numerical computations of the full geometry. The analytical expressions allow an investigation of the influence of the viscous stresses inside the secondary fluid for arbitrary geometries of the rectangular grooves. For classic superhydrophobic surfaces, the deviations in the effective slip length compared to the case of inviscid gas flow are pointed out. Another important finding with respect to an accurate modelling of flow over microstructured surfaces is that not only the effective slip length, but also the local slip length of a grooved surface, is anisotropic.

Copyright
Corresponding author
Email address for correspondence: schoenecker@csi.tu-darmstadt.de
References
Hide All
Asmolov, E. S., Schmieschek, S., Harting, J. & Vinogradova, O. I. 2013 Flow past superhydrophobic surfaces with cosine variation in local slip length. Phys. Rev. E 87, 023005.
Asmolov, E. S. & Vinogradova, O. I. 2012 Effective slip boundary conditions for arbitrary one-dimensional surfaces. J. Fluid Mech. 706, 108117.
Asmolov, E. S., Zhou, J., Schmid, F. & Vinogradova, O. I. 2013 Effective slip-length tensor for a flow over weakly slipping stripes. Phys. Rev. E 88, 023004.
Bechert, D. W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105129.
Belyaev, A. V. & Vinogradova, O. I. 2010 Effective slip in pressure-driven flow past super-hydrophobic stripes. J. Fluid Mech. 652, 489499.
Busse, A., Sandham, N. D., McHale, Glen & Newton, M. I. 2013 Change in drag, apparent slip and optimum air layer thickness for laminar flow over an idealized superhydrophobic surface. J. Fluid Mech. 727, 488508.
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. E 15, 427438.
Crowdy, D. 2010 Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles. Phys. Fluids 22 (12), 121703.
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.
Davis, A. M. J. & Lauga, E. 2009 Geometric transition in friction for flow over a bubble mattress. Phys. Fluids 21 (1), 011701.
Davis, A. M. J. & Lauga, E. 2010 Hydrodynamic friction of fakir-like superhydrophobic surfaces. J. Fluid Mech. 661, 402411.
Eijkel, J. 2007 Liquid slip in micro- and nanofluidics: recent research and its possible implications. Lab on a Chip 7 (3), 299301.
Feuillebois, F., Bazant, M. Z. & Vinogradova, O. I. 2010 Transverse flow in thin superhydrophobic channels. Phys. Rev. E 82, 055301(R).
Garabedian, P. R. 1966 Free boundary flows of a viscous liquid. Commun. Pure Appl. Maths 19 (4), 421434.
de Gennes, P. G. 2002 On fluid/wall slippage. Langmuir 18, 34133414.
Higdon, J. J. L. 1985 Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. J. Fluid Mech. 159, 195226.
Hocking, L. M. 1976 A moving fluid interface on a rough surface. J. Fluid Mech. 76 (4), 801817.
Joseph, D. D. & Sturges, L. 1978 The convergence of biorthogonal series for biharmonic and stokes flow edge problems: Part ii. SIAM J. Appl. Maths 34 (1), 726.
Kamrin, K., Bazant, M. Z. & Stone, H. A. 2010 Effective slip boundary conditions for arbitrary periodic surfaces: the surface mobility tensor. J. Fluid Mech. 658, 409437.
Kamrin, K. & Stone, H. 2011 The symmetry of mobility laws for viscous flow along arbitrarily patterned surfaces. Phys. Fluids 23, 031701.
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.
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven stokes flow. J. Fluid Mech. 489, 5577.
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87109.
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.
Mongruel, A., Chastel, T., Asmolov, E. S. & Vinogradova, O. I. 2013 Effective hydrodynamic boundary conditions for microtextured surfaces. Phys. Rev. E 87, 011002.
Muskhelishvili, N. I. 1975 Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff.
Ng, C.-O. & Wang, C. Y. 2010 Apparent slip arising from Stokes shear flow over a bidimensional patterned surface. Microfluid Nanofluid 8, 361371.
Ng, C.-O. & Wang, C. Y. 2011 Effective slip for Stokes flow over a surface patterned with two- or three-dimensional protrusions. Fluid Dyn. Res. 43, 065504.
Pan, F. & Acrivos, A. 1967 Steady flows in rectangular cavities. J. Fluid Mech. 28 (4), 643655.
Philip, J. R. 1972a Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (3), 353372.
Philip, J. R. 1972b Integral properties of flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (6), 960968.
Richardson, S. 1971 A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49, 327336.
Sbragaglia, M. & Prosperetti, A. 2007a Effective velocity boundary condition at a mixed slip surface. J. Fluid Mech. 578, 435451.
Sbragaglia, M. & Prosperetti, A. 2007b A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19, 043603.
Schmieschek, S., Belyaev, A. V., Harting, J. & Vinogradova, O. I. 2012 Tensorial slip of superhydrophobic channels. Phys. Rev. E 85, 016324.
Schönecker, C. & Hardt, S. 2013 Longitudinal and transverse flow over a cavity containing a second immiscible fluid. J. Fluid Mech. 717, 376394.
Shankar, P. N. 1993 The eddy structure in Stokes flow in a cavity. J. Fluid Mech. 250, 371383.
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. North Holland.
Squires, T. M. 2008 Electrokinetic flows over inhomogeneously slipping surfaces. Phys. Fluids 20 (9), 092105.
Steffes, C., Baier, T. & Hardt, S. 2011 Enabling the enhancement of electroosmotic flow over superhydrophobic surfaces by induced charges. Colloids Surf. A 376 (1–3), 8588.
Vinogradova, O. I. 1995 Drainage of a thin liquid film confined between hydrophobic surfaces. Langmuir 11, 22132220.
Wang, C. Y. 2003 Flow over a surface with parallel grooves. Phys. Fluids 15 (5), 11141121.
Wong, T.-S., Kang, S. H., Tang, S. K. Y., Smythe, E. J., Hatton, B. D., Grinthal, A. & Aizenberg, J. 2011 Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity. Nature 477 (7365), 443447.
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.
Zhou, J., Belyaev, A. V., Schmid, F. & Vinogradova, O. I. 2012 Anisotropic flow in striped superhydrophobic channels. J. Chem. Phys. 136, 194706.
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: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed