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A theory for the slip and drag of superhydrophobic surfaces with surfactant

Published online by Cambridge University Press:  25 November 2019

Julien R. Landel Open the ORCID record for Julien R. Landel [Opens in a new window] ,
François J. Peaudecerf Open the ORCID record for François J. Peaudecerf [Opens in a new window] ,
Fernando Temprano-Coleto Open the ORCID record for Fernando Temprano-Coleto [Opens in a new window] ,
Frédéric Gibou Open the ORCID record for Frédéric Gibou [Opens in a new window] ,
Raymond E. Goldstein Open the ORCID record for Raymond E. Goldstein [Opens in a new window]  and
Paolo Luzzatto-Fegiz Open the ORCID record for Paolo Luzzatto-Fegiz [Opens in a new window]
Julien R. Landel
Affiliation:
Department of Mathematics, Alan Turing Building, University of Manchester, Oxford Road,Manchester M139PL, UK
François J. Peaudecerf*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Wilberforce Rd,University of Cambridge, CambridgeCB3 0WA, UK
Fernando Temprano-Coleto
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA93106, USA
Frédéric Gibou
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA93106, USA
Raymond E. Goldstein
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Wilberforce Rd,University of Cambridge, CambridgeCB3 0WA, UK
Paolo Luzzatto-Fegiz*
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA93106, USA
*
‡Present address: Institute of Environmental Engineering, Department of Civil, Environmental and Geomatic Engineering, ETH Zürich, 8093 Zürich, Switzerland.
†Email address for correspondence: pfegiz@ucsb.edu
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Abstract

Superhydrophobic surfaces (SHSs) have the potential to reduce drag at solid boundaries. However, multiple independent studies have recently shown that small amounts of surfactant, naturally present in the environment, can induce Marangoni forces that increase drag, at least in the laminar regime. To obtain accurate drag predictions, one must solve the mass, momentum, bulk surfactant and interfacial surfactant conservation equations. This requires expensive simulations, thus preventing surfactant from being widely considered in SHS studies. To address this issue, we propose a theory for steady, pressure-driven, laminar, two-dimensional flow in a periodic SHS channel with soluble surfactant. We linearize the coupling between flow and surfactant, under the assumption of small concentration, finding a scaling prediction for the local slip length. To obtain the drag reduction and interfacial shear, we find a series solution for the velocity field by assuming Stokes flow in the bulk and uniform interfacial shear. We find how the slip and drag depend on the nine dimensionless groups that together characterize the surfactant transport near SHSs, the gas fraction and the normalized interface length. Our model agrees with numerical simulations spanning orders of magnitude in each dimensionless group. The simulations also provide the constants in the scaling theory. Our model significantly improves predictions relative to a surfactant-free one, which can otherwise overestimate slip and underestimate drag by several orders of magnitude. Our slip length model can provide the boundary condition in other simulations, thereby accounting for surfactant effects without having to solve the full problem.

JFM classification

Flow Control: Drag reduction Micro-/Nano-fluid dynamics: Microfluidics

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A18
Copyright
© 2019 Cambridge University Press

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