Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T15:18:39.341Z Has data issue: false hasContentIssue false
  • English
  • Français

Slip length formulas for longitudinal shear flow over a superhydrophobic grating with partially filled cavities

Published online by Cambridge University Press:  01 September 2021

Darren G. Crowdy Open the ORCID record for Darren G. Crowdy [Opens in a new window]
Darren G. Crowdy*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, LondonSW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Explicit formulas are given for the hydrodynamic slip lengths associated with longitudinal shear flow over a superhydrophobic grating where the menisci have partially invaded the cavities and are only weakly curved. For flat menisci that have depinned from the top of the grating and have displaced downwards into the cavities, the axial velocity is determined analytically and the slip length extracted from it. This solution is then combined with an integral identity to determine the first-order correction to the slip length when the displaced menisci bow weakly into the cavity. It is argued that the new formulas provide useful upper bounds for quantifying slip in microchannel flows involving partially filled cavities. The new solutions are natural extensions of prior results due to Philip (Z. Angew. Math. Phys., vol. 23, 1972, pp. 353–372) for shear flow over mixed no-slip/no-shear surfaces and due to Bechert & Bartenwerfer (J. Fluid Mech., vol. 206, 1989, pp. 105–129) for shear flow over blade-shaped riblets.

JFM classification

Flow Control: Drag reduction Interfacial Flows (free surface): Capillary flows
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 925 , 25 October 2021 , R2
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ahuja, A., Taylor, J.A., Lifton, V., Sidorenko, A.A., Salamon, T.R., Lobaton, E.J., Kolodner, P. & Krupenkin, T.N. 2008 Nanonails: a simple geometrical approach to electrically tunable superlyophobic surfaces. Langmuir 24 (1), 9–14.CrossRefGoogle ScholarPubMed
Bechert, D.W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105–129.CrossRefGoogle Scholar
Biben, T. & Joly, L. 2008 Wetting on nanorough surfaces. Phys. Rev. Lett. 100, 186103.CrossRefGoogle ScholarPubMed
Crowdy, D.G. 2011 a Frictional slip lengths and blockage coefficients. Phys. Fluids 23 (9), 091703.CrossRefGoogle Scholar
Crowdy, D.G. 2011 b Frictional slip lengths for unidirectional superhydrophobic grooved surfaces. Phys. Fluids 23 (7), 072001.CrossRefGoogle Scholar
Crowdy, D.G. 2017 Perturbation analysis of subphase gas and meniscus curvature effects for longitudinal flows over superhydrophobic surfaces. J. Fluid Mech. 822, 307–326.CrossRefGoogle Scholar
Ge, Z., Holmgren, H., Kronbichler, M., Brandt, L. & Kreiss, G. 2018 Effective slip over partially filled microcavities and its possible failure. Phys. Rev. Fluids 3, 054201.CrossRefGoogle Scholar
Hensel, R., Helbig, R., Aland, S., Braun, H.-G., Voigt, A., Neinhuis, C. & Werner, C. 2013 Wetting resistance at its topographical limit: the benefit of mushroom and Serif T structures. Langmuir 29 (4), 1100–1112.CrossRefGoogle ScholarPubMed
Kirk, T.L. 2018 Asymptotic formulae for flow in superhydrophobic channels with longitudinal ridges and protruding menisci. J. Fluid Mech. 839, R3.CrossRefGoogle Scholar
Lauga, E. & Stone, H.A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.CrossRefGoogle Scholar
Lee, C., Choi, C.H. & Kim, C.J. 2016 Superhydrophobic drag reduction in laminar flows: a critical review. Exp. Fluids 57, 176.CrossRefGoogle Scholar
Lee, C. & Kim, C.J 2009 Maximizing the giant liquid slip on superhydrophobic microstructures by nanostructuring their sidewalls. Langmuir 25 (21), 12812–12818.CrossRefGoogle ScholarPubMed
Ng, C.-O. & Wang, C.Y. 2009 Stokes shear flow over a grating: implications for superhydrophobic slip. Phys. Fluids 21, 013602.CrossRefGoogle Scholar
Philip, J.R. 1972 Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23, 353–372.CrossRefGoogle Scholar
Rothstein, J.P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89–109.CrossRefGoogle Scholar
Sbragaglia, M. & Prosperetti, A. 2007 A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19, 043603.CrossRefGoogle Scholar
Schnitzer, O. 2016 Singular effective slip length for longitudinal flow over a dense bubble mattress. Phys. Rev. Fluids 1, 052101(R).CrossRefGoogle Scholar
Teo, C.J. & Khoo, B.C. 2010 Flow past superhydrophobic surfaces containing longitudinal grooves: effects of interface curvature. Microfluid Nanofluid 9, 499–511.CrossRefGoogle Scholar
Tuteja, A., Choi, W., Mabry, J.M., McKinley, G.H. & Cohen, R.E. 2008 Robust omniphobic surfaces. Proc. Natl Acad. Sci. USA 105 (47), 18200–18205.CrossRefGoogle ScholarPubMed