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Influence of insoluble surfactants on shear flow over a surface in Cassie state at large Péclet numbers

Published online by Cambridge University Press:  17 November 2020

Tobias Baier Open the ORCID record for Tobias Baier [Opens in a new window]  and
Steffen Hardt Open the ORCID record for Steffen Hardt [Opens in a new window]
Tobias Baier*
Affiliation:
Fachbereich Maschinenbau, Technische Universität Darmstadt, 64287Darmstadt, Germany
Steffen Hardt
Affiliation:
Fachbereich Maschinenbau, Technische Universität Darmstadt, 64287Darmstadt, Germany
*
†Email address for correspondence: baier@nmf.tu-darmstadt.de
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Abstract

Surfactants can immobilize fluid–liquid interfaces under shear stress. We investigate the impact of insoluble surfactants on shear flow along a superhydrophobic surface in Cassie state, with gas trapped in grooves oriented perpendicular to the flow direction. Assuming convection-dominated transport along the gas–liquid interface, analytical results for the surfactant distribution on a groove and the corresponding flow field in its vicinity are derived both for a single groove and for an array of evenly spaced grooves. The results are elaborated for the case where the surface tension depends linearly on the surfactant concentration, which is characteristic for dilute coverage of the gas–liquid interface. For an array of grooves, the relation between the applied shear stress and the effective slip length on the microstructured surface is investigated.

JFM classification

Flow Control: Drag reduction Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 907 , 25 January 2021 , A3
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Baier, T., Steffes, C. & Hardt, S. 2010 Thermocapillary flow on superhydrophobic surfaces. Phys. Rev. E 82 (3), 037301.CrossRefGoogle ScholarPubMed
Belyaev, A. V. & Vinogradova, O. I. 2011 Electro-osmosis on anisotropic superhydrophobic surfaces. Phys. Rev. Lett. 107 (9), 098301.CrossRefGoogle ScholarPubMed
Berg, J. C. & Acrivos, A. 1965 The effect of surface active agents on convection cells induced by surface tension. Chem. Engng Sci. 20 (8), 737–745.CrossRefGoogle Scholar
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2007 Transport Phenomena. Wiley.Google Scholar
Bolognesi, G., Cottin-Bizonne, C. & Pirat, C. 2014 Evidence of slippage breakdown for a superhydrophobic microchannel. Phys. Fluids 26 (8), 082004.CrossRefGoogle Scholar
Carpenter, B. & Homsy, G. M. 1985 The effect of surface contamination on thermocapillary flow in a two-dimensional slot. Part 2. Partially contaminated interfaces. J. Fluid Mech. 155, 429–439.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Davis, R. E. & Acrivos, A. 1966 The influence of surfactants on the creeping motion of bubbles. Chem. Engng Sci. 21 (8), 681–685.CrossRefGoogle Scholar
Harper, J. F. 1992 The leading edge of an oil slick, soap film, or bubble stagnant cap in stokes flow. J. Fluid Mech. 237, 23–32.CrossRefGoogle Scholar
Huang, D. M., Cottin-Bizonne, C., Ybert, C. & Bocquet, L. 2008 Massive amplification of surface-induced transport at superhydrophobic surfaces. Phys. Rev. Lett. 101 (6), 064503.CrossRefGoogle ScholarPubMed
Joly, L., Detcheverry, F. & Biance, A.-L. 2014 Anomalous $\zeta$ potential in foam films. Phys. Rev. Lett. 113 (8), 088301.CrossRefGoogle ScholarPubMed
Kim, T. J. & Hidrovo, C. 2012 Pressure and partial wetting effects on superhydrophobic friction reduction in microchannel flow. Phys. Fluids 24 (11), 112003.CrossRefGoogle Scholar
Landel, J. R., Peaudecerf, F. J., Temprano-Coleto, F., Gibou, F., Goldstein, R. E. & Luzzatto-Fegiz, P. 2020 A theory for the slip and drag of superhydrophobic surfaces with surfactant. J. Fluid Mech. 883, A18.CrossRefGoogle ScholarPubMed
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Lee, K. Y. C. 2008 Collapse mechanisms of Langmuir monolayers. Annu. Rev. Phys. Chem. 59 (1), 771–791.CrossRefGoogle ScholarPubMed
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.Google Scholar
Lyklema, H. 2000 Fundamentals of Interface and Colloid Science: Liquid-Fluid Interfaces. Academic Press.Google Scholar
Merson, R. L. & Quinn, J. A. 1965 Stagnation in a fluid interface: properties of the stagnant film. AIChE J. 11 (3), 391–395.CrossRefGoogle Scholar
Michael, D. H. 1958 The separation of a viscous liquid at a straight edge. Mathematika 5 (1), 82–84.CrossRefGoogle Scholar
Peaudecerf, F. J., Landel, J. R., Goldstein, R. E. & Luzzatto-Fegiz, P. 2017 Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces. Proc. Natl Acad. Sci. USA 114 (28), 7254–7259.CrossRefGoogle ScholarPubMed
Philip, J. R. 1972 Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (3), 353–372.CrossRefGoogle Scholar
Richardson, S. 1970 A ‘stick-slip’ problem related to the motion of a free jet at low Reynolds numbers. Math. Proc. Cambridge 67 (2), 477–489.CrossRefGoogle Scholar
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42 (1), 89–109.CrossRefGoogle Scholar
Sadhal, S. S. & Johnson, R. E. 1983 Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film – exact solution. J. Fluid Mech. 126, 237–250.CrossRefGoogle Scholar
Savic, P. 1953 Circulation and distortion of liquid drops falling through a viscous medium. Rep. MT-22. Nat. Res. Counc. Can., Div. Mech. Engng.Google Scholar
Schäffel, D., Koynov, K., Vollmer, D., Butt, H.-J. & Schönecker, C. 2016 Local flow field and slip length of superhydrophobic surfaces. Phys. Rev. Lett. 116 (13), 134501.CrossRefGoogle ScholarPubMed
Schönecker, C., Baier, T. & Hardt, S. 2014 Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state. J. Fluid Mech. 740, 168–195.CrossRefGoogle Scholar
Schönecker, C. & Hardt, S. 2014 Electro-osmotic flow along superhydrophobic surfaces with embedded electrodes. Phys. Rev. E 89 (6), 063005.CrossRefGoogle ScholarPubMed
Scott, J. C. 1982 Flow beneath a stagnant film on water: the Reynolds ridge. J. Fluid Mech. 116, 283–296.CrossRefGoogle Scholar
Song, D., Song, B., Hu, H., Du, X., Du, P., Choi, C.-H. & Rothstein, J. P. 2018 Effect of a surface tension gradient on the slip flow along a superhydrophobic air-water interface. Phys. Rev. Fluids 3 (3), 033303.CrossRefGoogle Scholar
Squires, T. M. 2008 Electrokinetic flows over inhomogeneously slipping surfaces. Phys. Fluids 20 (9), 092105.CrossRefGoogle Scholar
Steffes, C., Baier, T. & Hardt, S. 2011 Enabling the enhancement of electroosmotic flow over superhydrophobic surfaces by induced charges. Colloids Surf. A 376 (1–3), 85–88.CrossRefGoogle Scholar