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Stokes resistance of a solid cylinder near a superhydrophobic surface. Part 1. Grooves perpendicular to cylinder axis

Published online by Cambridge University Press:  10 April 2019

Ory Schnitzer Open the ORCID record for Ory Schnitzer [Opens in a new window]  and
Ehud Yariv Open the ORCID record for Ehud Yariv [Opens in a new window]
Ory Schnitzer*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Ehud Yariv
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
*
†Email address for correspondence: [email protected]
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Abstract

An important class of canonical problems that is employed in quantifying the slipperiness of microstructured superhydrophobic surfaces is concerned with the calculation of the hydrodynamic loads on adjacent solid bodies whose size is large relative to the microstructure period. The effect of superhydrophobicity is most pronounced when the latter period is comparable to the separation between the solid probe and the superhydrophobic surface. We address the above distinguished limit, considering a simple configuration where the superhydrophobic surface is formed by a periodically grooved array, in which air bubbles are trapped in a Cassie state, and the solid body is an infinite cylinder. In the present part, we consider the case where the grooves are aligned perpendicular to the cylinder and allow for three modes of rigid-body motion: rectilinear motion perpendicular to the surface; rectilinear motion parallel to the surface, in the groove direction; and angular rotation about the cylinder axis. In this scenario, the flow is periodic in the direction parallel to the axis. Averaging over the small-scale periodicity yields a modified lubrication description where the small-scale details are encapsulated in two auxiliary two-dimensional cell problems which respectively describe pressure- and boundary-driven longitudinal flow through an asymmetric rectangular domain, bounded by a compound surface from the bottom and a no-slip surface from the top. Once the integral flux and averaged shear stress associated with each of these cell problems are calculated as a function of the slowly varying cell geometry, the hydrodynamic loads experienced by the cylinder are provided as quadratures of nonlinear functions of the latter distributions over a continuous sequence of cells.

JFM classification

Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 868 , 10 June 2019 , pp. 212 - 243
Copyright
© 2019 Cambridge University Press 

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References

Asmolov, E. S., Belyaev, A. V. & Vinogradova, O. I. 2011 Drag force on a sphere moving toward an anisotropic superhydrophobic plane. Phys. Rev. E 84 (2), 026330.Google Scholar
Belyaev, A. V. & Vinogradova, O. I. 2010a Effective slip in pressure-driven flow past super-hydrophobic stripes. J. Fluid Mech. 652, 489–499.Google Scholar
Belyaev, A. V. & Vinogradova, O. I. 2010b Hydrodynamic interaction with super-hydrophobic surfaces. Soft Matt. 6 (18), 4563–4570.Google Scholar
Chastel, T. & Mongruel, A. 2016 Squeeze flow between a sphere and a textured wall. Phys. Fluids 28 (2), 023301.Google Scholar
Choi, C.-H. & Kim, C.-J. 2006 Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett. 96 (6), 066001.Google 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 (4), 427–438.Google Scholar
Davis, A. M. J., Kezirian, M. T. & Brenner, H. 1994 On the Stokes–Einstein model of surface diffusion along solid surfaces: slip boundary conditions. J. Colloid Interface Sci. 165 (1), 129–140.Google Scholar
Davis, S. H. 2017 The importance of being thin. J. Engng Math. 105 (1), 3–30.Google Scholar
Feuillebois, F., Bazant, M. Z. & Vinogradova, O. I. 2009 Effective slip over superhydrophobic surfaces in thin channels. Phys. Rev. Lett. 102 (2), 026001.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1981 The slow motion of a cylinder next to a plane wall. Q. J. Mech. Appl. Math. 34 (2), 129–137.Google Scholar
Kaynan, U. & Yariv, E. 2017 Stokes resistance of a cylinder near a slippery wall. Phys. Rev. Fluids 2 (10), 104103.Google Scholar
Kirk, T. L., Hodes, M. & Papageorgiou, D. T. 2017 Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature. J. Fluid Mech. 811, 315–349.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.Google Scholar
Lee, C., Choi, C.-H. & Kim, C.-J. 2008 Structured surfaces for a giant liquid slip. Phys. Rev. Lett. 101 (6), 064501.Google Scholar
Lee, C., Choi, C.-H. & Kim, C.-J. 2016 Superhydrophobic drag reduction in laminar flows: a critical review. Exp. Fluids 57 (12), 1–20.Google Scholar
Maali, A., Pan, Y., Bhushan, B. & Charlaix, E. 2012 Hydrodynamic drag-force measurement and slip length on microstructured surfaces. Phys. Rev. E 85 (6), 066310.Google Scholar
Marshall, J. S. 2017 Exact formulae for the effective slip length of a symmetric superhydrophobic channel with flat or weakly curved menisci. SIAM J. Appl. Math. 77 (5), 1606–1630.Google Scholar
Mongruel, A., Chastel, T., Asmolov, E. S. & Vinogradova, O. I. 2013 Effective hydrodynamic boundary conditions for microtextured surfaces. Phys. Rev. E 87 (1), 011002.Google Scholar
Nizkaya, T. V., Dubov, A. L., Mourran, A. & Vinogradova, O. I. 2016 Probing effective slippage on superhydrophobic stripes by atomic force microscopy. Soft Matt. 12 (33), 6910–6917.Google Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16 (12), 4635–4643.Google Scholar
Ou, J. & Rothstein, J. P. 2005 Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys. Fluids 17 (10), 103606.Google 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
Quéré, D. 2008 Wetting and roughness. Annu. Rev. Mater. Res. 38 (1), 71–99.Google Scholar
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42 (1), 89–109.Google Scholar
Sbragaglia, M. & Prosperetti, A. 2007 A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19 (4), 043603.Google Scholar
Schmieschek, S., Belyaev, A. V., Harting, J. & Vinogradova, O. I. 2012 Tensorial slip of superhydrophobic channels. Phys. Rev. E 85 (1), 016324.Google Scholar
Schnitzer, O. & Yariv, E. 2017 Longitudinal pressure-driven flows between superhydrophobic grooved surfaces: large effective slip in the narrow-channel limit. Phys. Rev. Fluids 2 (7), 072101.Google Scholar
Seo, J. & Mani, A. 2016 On the scaling of the slip velocity in turbulent flows over superhydrophobic surfaces. Phys. Fluids 28 (2), 025110.Google Scholar
Teo, C. J. & Khoo, B. C. 2009 Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves. Microfluid. Nanofluid. 7 (3), 353–382.Google Scholar
Yariv, E. 2017 Velocity amplification in pressure-driven flows between superhydrophobic gratings of small solid fraction. Soft Matt. 13, 6287–6292.Google Scholar
Yariv, E. & Schnitzer, O. 2018 Pressure-driven plug flows between superhydrophobic surfaces of closely spaced circular bubbles. J. Engng Math. 111, 15–22.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.Google Scholar