Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T20:43:35.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic formulae for flow in superhydrophobic channels with longitudinal ridges and protruding menisci

Published online by Cambridge University Press:  01 February 2018

Toby L. Kirk Open the ORCID record for Toby L. Kirk [Opens in a new window]
Toby L. Kirk*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents new asymptotic formulae for flow in a channel with one or both walls patterned with a longitudinal array of ridges and arbitrarily protruding menisci. Derived from a matched asymptotic expansion, they extend results by Crowdy (J. Fluid Mech., vol. 791, 2016, R7) for shear flow, and thus make no restriction on the protrusion into or out of the liquid. The slip length formula is compared against full numerical solutions and, despite the assumption of small ridge period in its derivation, is found to have a very large range of validity; relative errors are small even for periods large enough for the protruding menisci to degrade the flow and touch the opposing wall.

JFM classification

Flow Control: Drag reduction Drops and Bubbles: Drops and Bubbles Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 839 , 25 March 2018 , R3
Copyright
© 2018 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

Cottin-Bizonne, C., Barentin, C., Charlaix, E., 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
Crowdy, D. G. 2010 Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles. Phys. Fluids 22 (12), 121703.Google Scholar
Crowdy, D. G. 2016 Analytical formulae for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci. J. Fluid Mech. 791, R7.Google Scholar
Davis, A. M. J. & Lauga, E. 2009 Geometric transition in friction for flow over a bubble mattress. Phys. Fluids 21 (1), 011701.Google Scholar
Enright, R., Hodes, M., Salamon, T. & Muzychka, Y. 2014 Isoflux Nusselt number and slip length formulae for superhydrophobic microchannels. Trans. ASME J. Heat Transfer 136 (1), 012402.Google Scholar
Hodes, M., Kirk, T. L., Karamanis, G. & MacLachlan, S. 2017 Effect of thermocapillary stress on slip length for a channel textured with parallel ridges. J. Fluid Mech. 814, 301–324.Google Scholar
Karamanis, G., Hodes, M., Kirk, T. & Papageorgiou, D. 2017 Solution of the Graetz–Nusselt problem for liquid flow over isothermal parallel ridges. Trans. ASME J. Heat Transfer 139 (9), 091702.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.CrossRefGoogle Scholar
Lam, L. S., Hodes, M. & Enright, R. 2015 Analysis of Galinstan-based microgap cooling enhancement using structured surfaces. Trans. ASME J. Heat Transfer 137 (9), 091003.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.Google Scholar
Maynes, D. & Crockett, J. 2014 Apparent temperature jump and thermal transport in channels with streamwise rib and cavity featured superhydrophobic walls at constant heat flux. Trans. ASME J. Heat Transfer 136 (1), 011701.Google Scholar
Maynes, D., Webb, B. W., Crockett, J. & Solovjov, V. 2013 Analysis of laminar slip-flow thermal transport in microchannels with transverse rib and cavity structured superhydrophobic walls at constant heat flux. Trans. ASME J. Heat Transfer 135 (2), 021701.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
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
Schnitzer, O. 2017 Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: the small-solid-fraction singularity. J. Fluid Mech. 820, 580–603.Google Scholar
Steinberger, A., Cottin-Bizonne, C., Kleimann, P. & Charlaix, E. 2007 High friction on a bubble mattress. Nat. Mater. 6 (9), 665–668.Google Scholar
Sullivan, J. M., Paterson, C., Wilson, S. K. & Duffy, B. R. 2012 A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow. Phys. Fluids 24 (8), 082109.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
Teo, C. J. & Khoo, B. C. 2010 Flow past superhydrophobic surfaces containing longitudinal grooves: effects of interface curvature. Microfluid. Nanofluid. 9 (2–3), 499–511.Google Scholar
Wilson, S. K. & Duffy, B. R. 1998 On the gravity-driven draining of a rivulet of viscous fluid down a slowly varying substrate with variation transverse to the direction of flow. Phys. Fluids 10 (1), 13–22.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
Supplementary material: File

Kirk supplementary material

Kirk supplementary material 1

Download Kirk supplementary material(File)
File 183.7 KB