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Change in drag, apparent slip and optimum air layer thickness for laminar flow over an idealised superhydrophobic surface

Published online by Cambridge University Press:  28 June 2013

A. Busse ,
N. D. Sandham ,
G. McHale  and
M. I. Newton
A. Busse*
Affiliation:
Faculty of Engineering and the Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK
N. D. Sandham
Affiliation:
Faculty of Engineering and the Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK
G. McHale
Affiliation:
Faculty of Engineering and Environment, Northumbria University, Newcastle upon Tyne NE1 8ST, UK
M. I. Newton
Affiliation:
School of Science and Technology, Nottingham Trent University, Clifton Lane, Nottingham NG11 8NS, UK
*
†Email address for correspondence: a.busse@soton.ac.uk
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Abstract

Analytic results are derived for the apparent slip length, the change in drag and the optimum air layer thickness of laminar channel and pipe flow over an idealised superhydrophobic surface, i.e. a gas layer of constant thickness retained on a wall. For a simple Couette flow the gas layer always has a drag reducing effect, and the apparent slip length is positive, assuming that there is a favourable viscosity contrast between liquid and gas. In pressure-driven pipe and channel flow blockage limits the drag reduction caused by the lubricating effects of the gas layer; thus an optimum gas layer thickness can be derived. The values for the change in drag and the apparent slip length are strongly affected by the assumptions made for the flow in the gas phase. The standard assumptions of a constant shear rate in the gas layer or an equal pressure gradient in the gas layer and liquid layer give considerably higher values for the drag reduction and the apparent slip length than an alternative assumption of a vanishing mass flow rate in the gas layer. Similarly, a minimum viscosity contrast of four must be exceeded to achieve drag reduction under the zero mass flow rate assumption whereas the drag can be reduced for a viscosity contrast greater than unity under the conventional assumptions. Thus, traditional formulae from lubrication theory lead to an overestimation of the optimum slip length and drag reduction when applied to superhydrophobic surfaces, where the gas is trapped.

JFM classification

Multiphase and Particle-laden Flows: Core-annular flow Low-Reynolds-number flows: Low-Reynolds-number flows Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 727 , 25 July 2013 , pp. 488 - 508
Copyright
©2013 Cambridge University Press 

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References

Bhushan, B. 2011 Biomimetics inspired surfaces for drag reduction and oleophobicity/philicity. Beilstein J. Nanotechnology 2, 66–84.Google Scholar
Busse, A. & Sandham, N. D. 2012 Influence of an anisotropic slip-length boundary condition on turbulent channel flow. Phys. Fluids 24, 055111.CrossRefGoogle Scholar
Bye, J. A. T. 1966 Numerical solutions of the steady-state vorticity equation in rectangular basins. J. Fluid Mech. 26, 577–598.Google Scholar
Choi, C.-H., Westin, K. J. A. & Breuer, K. S. 2003 Apparent slip flows in hydrophilic and hydrophobic microchannels. Phys. Fluids 15 (10), 2897–2902.Google Scholar
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. & Knuth, D. E. 1996 On the Lambert W function. Adv. Comput. Math. 5, 329–359.Google Scholar
Daniello, R. J., Waterhouse, N. E. & Rothstein, J. P. 2009 Drag reduction in turbulent flows over superhydrophobic surfaces. Phys. Fluids 21, 085103.Google Scholar
Ditsche-Kuru, P., Schneider, E. S., Melskotte, J.-E., Brede, M., Leder, A. & Barthlott, W. 2011 Superhydrophobic surfaces of the water bug Notonecta glauca: a model for friction reduction and air retention. Beilstein J. Nanotechnology 2, 137–144.Google Scholar
Elbing, B. R., Winkel, E. S., Lay, K. A., Ceccio, S. L., Dowling, D. R. & Perlin, M. 2008 Bubble-induced skin-friction drag reduction and the abrupt transition to air-layer drag reduction. J. Fluid Mech. 612, 201–236.Google Scholar
Elboth, T., Reif, B. A. P., Andreassen, O. & Martell, M. B. 2012 Flow noise reduction from superhydrophobic surfaces. Geophysics 77, P1–P10.Google Scholar
Flynn, M. R. & Bush, J. W. M. 2008 Underwater breathing: the mechanics of plastron respiration. J. Fluid Mech. 608, 275–296.Google Scholar
Garcia-Mayoral, Ricardo & Jimenez, Javier 2011 Drag reduction by riblets. Phil. Trans. R. Soc. A 369, 1412.Google Scholar
Ghosh, S., Mandal, T. K., Das, G. & Das, P. K. 2009 Review of oil water core annular flow. Renewable and Sustainable Energy Reviews 13, 1957–1965.Google Scholar
Gotge, S., Vorobieff, P., Truesdell, R., Mammoli, A. & van Swol, F. 2005 Effective slip on textured superhydrophobic surfaces. Phys. Fluids 17, 051701.Google Scholar
Govardhan, R. N., Srinivas, F. S., Asthana, A. & Bobji, M. S. 2009 Time dependence of effective slip on textured hydrophobic surfaces. Phys. Fluids 21, 052001.CrossRefGoogle Scholar
Greidanus, A. J., Delfos, R. & Westerweel, J 2011 Drag reduction by surface treatment in turbulent Taylor–Couette flow. J. Phys.: Conf. Ser. 318, 082016.Google Scholar
Gruncell, B. R. K., Sandham, N. D. & McHale, G. 2012a Simulations of laminar flow past a superhydrophobic sphere with drag reduction and separation delay. Phys. Fluids 25, 043601.Google Scholar
Gruncell, B. R. K., Sandham, N. D. & Prince, M. P. 2012b Experimental and numerical investigation of the drag on superhydrophobic surfaces. In Proceedings of the 9th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements. ETMM Symposium, June 6–8 2012, Thessaloniki, Greece.Google Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 65–90.Google Scholar
Joseph, D. D., Nguyen, K. & Beavers, G. S. 1984 Non-uniqueness and stability of the configuration of flow of immiscible fluids with different viscosities. J. Fluid Mech. 141, 319–345.Google Scholar
Joseph, D. D. & Renardy, Y. Y. 1992 Fundamentals of Two-Fluid Dynamics. Springer.Google Scholar
Joseph, P., Cottin-Bizonne, C., Benoît, 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.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon Press.Google Scholar
Leidenfrost, J. G. 1966 On fixation of water in diverse fire. Intl J. Heat Mass Transfer 9, 1153.Google Scholar
Lockerby, D. A., Reese, J. M., Emerson, D. R. & Barber, R. W. 2004 Velocity boundary condition at solid walls in rarefied gas calculations. Phys. Rev. E 70, 017303.CrossRefGoogle ScholarPubMed
Looman, M. D. 1916 US Patent No. 1,192,438.Google Scholar
Maali, A. & Bhushan, B. 2012 Measurement of slip length on superhydrophobic surfaces. Proc. R. Soc. A 370, 2304–2320.Google ScholarPubMed
McHale, G., Flynn, M. R. & Newton, M. I. 2011 Plastron induced drag reduction and increased slip on a superhydrophobic sphere. Soft Matt. 7 (21), 10100–10107.Google Scholar
McHale, G., Newton, M. I. & Shirtcliffe, N. J. 2010 Immersed superhydrophobic surfaces: gas exchange, slip and drag reduction properties. Soft Matt. 6, 714–719.CrossRefGoogle Scholar
McHale, G., Shirtcliffe, N. J., Evans, C. R. & Newton, M. I. 2009 Terminal velocity and drag reduction measurements on superhydrophobic spheres. Appl. Phys. Lett. 94, 064104.CrossRefGoogle Scholar
Min, T. & Kim, J. 2004 Effects of hydrophobic surface on skin-friction drag. Phys. Fluids 16 (7), L55–58.Google Scholar
Muralidhar, P., Ferrer, N., Daniello, R. & Rothstein, J. P. 2011 Influence of slip on the flow past superhydrophobic circular cylinders. J. Fluid Mech. 680, 459–476.Google 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.Google Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16, 4635.Google Scholar
Ou, J. & Rothstein, J. P. 2005 Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys. Fluids 17, 103606.Google Scholar
Quéré, D. 2008 Wetting and roughness. Annu. Rev. Mater. Res. 42, 89–109.Google Scholar
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89–109.CrossRefGoogle Scholar
Sadhal, S. S., Ayyaswamy, P. S. & Chung, J. N. 1996 Transport Phenomena with Drops and Bubbles. Springer.Google Scholar
Shirtcliffe, N. J., McHale, G., Newton, M. I., Perry, C. C. & Pyatt, F. B. 2006 Plastron properties of a superhydrophobic surface. Appl. Phys. Lett. 89, 104106.Google Scholar
Than, P. T., Rosso, F. & Joseph, D. D. 1987 Instability of Poiseuille flow of two immiscible liquids with different viscosities in a channel. Intl J. Engng Sci. 25, 189–204.Google Scholar
Thorpe, W. H. & Crisp, D. J. 1947 Plastron respiration in the Coleoptera. J. Expl Biol. 26, 219–260.Google 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.Google Scholar
Tsai, P., 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.Google Scholar
Tuteja, A., Choi, W., Ma, M., Mabry, J. M., Mazzella, S. A., Rutledge, G. C., McKinley, G. H. & Cohen, R. E. 2007 Designing superoleophobic surfaces. Science 318, 1618.CrossRefGoogle ScholarPubMed
Tuteja, A., Choi, W., Mabry, J. M., McKinley, G. H. & Cohen, R. E. 2008 Robust omniphobic surfaces. Proc. Natl Acad. Sci. USA 105, 18200–18205.Google Scholar
Vakareslki, I., Marston, J., Chan, D. & Thoroddsen, S. 2011 Drag reduction by Leidenfrost vapor layers. Phys. Rev. Lett. 106.Google Scholar
Vinogradova, O. I. 1999 Slippage of water over hydrophobic surfaces. Intl J. Miner. Process. 56, 31–60.Google Scholar
Vinogradova, O. I. & Dubov, A. L. 2012 Superhydrophobic textures for microfluidics. Mendeleev Commun. 22 (5), 229–236.Google Scholar
Voronov, R. S., Papavassiliou, D. V. & Lee, L. L. 2008 Review of fluid slip over superhydrophobic surfaces and its dependence on the contact angle. Ind. Engng Chem. Res. 47, 2455–2477.Google Scholar
Wong, T.-S., 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, 443–447.CrossRefGoogle ScholarPubMed
Yang, Y., Straatman, A. G., Martinuzzi, R. J. & Yanful, E. K. 2002 A study of laminar flow in low aspect ratio lid-driven cavities. Can. J. Civil Engng 29, 436–447.Google Scholar