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

Published online by Cambridge University Press:  06 January 2014

Clarissa Schönecker ,
Tobias Baier  and
Steffen Hardt
Clarissa Schönecker*
Affiliation:
Institute for Nano- and Microfluidics, Center of Smart Interfaces, Technische Universität Darmstadt, 64287 Darmstadt, Germany
Tobias Baier
Affiliation:
Institute for Nano- and Microfluidics, Center of Smart Interfaces, Technische Universität Darmstadt, 64287 Darmstadt, Germany
Steffen Hardt
Affiliation:
Institute for Nano- and Microfluidics, Center of Smart Interfaces, Technische Universität Darmstadt, 64287 Darmstadt, Germany
*
†Email address for correspondence: [email protected]
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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.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 740 , 10 February 2014 , pp. 168 - 195
Copyright
© 2014 Cambridge University Press 

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References

öeferences

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.Google Scholar
Asmolov, E. S. & Vinogradova, O. I. 2012 Effective slip boundary conditions for arbitrary one-dimensional surfaces. J. Fluid Mech. 706, 108–117.Google Scholar
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.Google Scholar
Bechert, D. W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105–129.Google Scholar
Belyaev, A. V. & Vinogradova, O. I. 2010 Effective slip in pressure-driven flow past super-hydrophobic stripes. J. Fluid Mech. 652, 489–499.Google Scholar
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, 488–508.CrossRefGoogle Scholar
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, 427–438.CrossRefGoogle ScholarPubMed
Crowdy, D. 2010 Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles. Phys. Fluids 22 (12), 121703.Google Scholar
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.Google Scholar
Davis, A. M. J. & Lauga, E. 2009 Geometric transition in friction for flow over a bubble mattress. Phys. Fluids 21 (1), 011701.CrossRefGoogle Scholar
Davis, A. M. J. & Lauga, E. 2010 Hydrodynamic friction of fakir-like superhydrophobic surfaces. J. Fluid Mech. 661, 402–411.Google Scholar
Eijkel, J. 2007 Liquid slip in micro- and nanofluidics: recent research and its possible implications. Lab on a Chip 7 (3), 299–301.Google Scholar
Feuillebois, F., Bazant, M. Z. & Vinogradova, O. I. 2010 Transverse flow in thin superhydrophobic channels. Phys. Rev. E 82, 055301(R).Google Scholar
Garabedian, P. R. 1966 Free boundary flows of a viscous liquid. Commun. Pure Appl. Maths 19 (4), 421–434.CrossRefGoogle Scholar
de Gennes, P. G. 2002 On fluid/wall slippage. Langmuir 18, 3413–3414.Google Scholar
Higdon, J. J. L. 1985 Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. J. Fluid Mech. 159, 195–226.Google Scholar
Hocking, L. M. 1976 A moving fluid interface on a rough surface. J. Fluid Mech. 76 (4), 801–817.Google Scholar
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), 7–26.CrossRefGoogle Scholar
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, 409–437.Google Scholar
Kamrin, K. & Stone, H. 2011 The symmetry of mobility laws for viscous flow along arbitrarily patterned surfaces. Phys. Fluids 23, 031701.Google Scholar
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.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven stokes flow. J. Fluid Mech. 489, 55–77.Google Scholar
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87–109.Google Scholar
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.Google Scholar
Mongruel, A., Chastel, T., Asmolov, E. S. & Vinogradova, O. I. 2013 Effective hydrodynamic boundary conditions for microtextured surfaces. Phys. Rev. E 87, 011002.Google Scholar
Muskhelishvili, N. I. 1975 Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff.Google 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.Google Scholar
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.Google Scholar
Pan, F. & Acrivos, A. 1967 Steady flows in rectangular cavities. J. Fluid Mech. 28 (4), 643–655.CrossRefGoogle Scholar
Philip, J. R. 1972a Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (3), 353–372.Google Scholar
Philip, J. R. 1972b Integral properties of flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (6), 960–968.Google Scholar
Richardson, S. 1971 A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49, 327–336.CrossRefGoogle Scholar
Sbragaglia, M. & Prosperetti, A. 2007a Effective velocity boundary condition at a mixed slip surface. J. Fluid Mech. 578, 435–451.Google Scholar
Sbragaglia, M. & Prosperetti, A. 2007b A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19, 043603.Google Scholar
Schmieschek, S., Belyaev, A. V., Harting, J. & Vinogradova, O. I. 2012 Tensorial slip of superhydrophobic channels. Phys. Rev. E 85, 016324.Google Scholar
Schönecker, C. & Hardt, S. 2013 Longitudinal and transverse flow over a cavity containing a second immiscible fluid. J. Fluid Mech. 717, 376–394.Google Scholar
Shankar, P. N. 1993 The eddy structure in Stokes flow in a cavity. J. Fluid Mech. 250, 371–383.CrossRefGoogle Scholar
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. North Holland.Google Scholar
Squires, T. M. 2008 Electrokinetic flows over inhomogeneously slipping surfaces. Phys. Fluids 20 (9), 092105.Google Scholar
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), 85–88.Google Scholar
Vinogradova, O. I. 1995 Drainage of a thin liquid film confined between hydrophobic surfaces. Langmuir 11, 2213–2220.CrossRefGoogle Scholar
Wang, C. Y. 2003 Flow over a surface with parallel grooves. Phys. Fluids 15 (5), 1114–1121.Google Scholar
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), 443–447.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 (12), 123601.CrossRefGoogle Scholar
Zhou, J., Belyaev, A. V., Schmid, F. & Vinogradova, O. I. 2012 Anisotropic flow in striped superhydrophobic channels. J. Chem. Phys. 136, 194706.CrossRefGoogle ScholarPubMed