Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-29T17:38:39.108Z Has data issue: false hasContentIssue false
  • English
  • Français

Shear flow over a surface containing a groove covered by an incompressible surfactant phase

Published online by Cambridge University Press:  30 September 2022

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, 64287 Darmstadt, Germany
Steffen Hardt
Affiliation:
Fachbereich Maschinenbau, Technische Universität Darmstadt, 64287 Darmstadt, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study shear-driven liquid flow over a planar surface with an embedded gas-filled groove, with the gas–liquid interface protruding slightly above or below the planar surface. The flow direction is along the groove, which is taken to be much longer than it is wide, and the gas–liquid interface is assumed to be covered by an incompressible surface fluid, representing a surfactant phase. Using the incompressibility condition for the surface fluid, the equations of motion and corresponding boundary conditions for the liquid phase are obtained by minimizing the dissipation rate. Assuming a moderate deformation of the interface, a domain perturbation technique with the maximal deformation as the small parameter is employed. The Stokes equation in the liquid phase under corresponding boundary conditions is solved to second order in the deformation using the Keldysh–Sedov formalism. The obtained analytical results are compared with numerical calculations of the same problem, allowing an assessment of the limits of validity of the expansion. While on a planar gas–liquid interface no flow is induced, a recirculating flow is observed on an interface protruding slightly above or below the planar surface. The study sheds light onto the mobility of curved gas–liquid interfaces in the presence of surfactants acting as an incompressible surface fluid.

JFM classification

Flow Control: Drag reduction Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 949 , 25 October 2022 , A34
Check for updates
Copyright
© The Author(s), 2022. 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

Ageev, A., Golubkina, I. & Osiptsov, A. 2018 Application of boundary element method to stokes flows over a striped superhydrophobic surface with trapped gas bubbles. Phys. Fluids 30 (1), 012102.CrossRefGoogle Scholar
Alinovi, E. & Bottaro, A. 2018 Apparent slip and drag reduction for the flow over superhydrophobic and lubricant-impregnated surfaces. Phys. Rev. Fluids 3, 124002.CrossRefGoogle Scholar
Asmolov, E.S., Nizkaya, T.V. & Vinogradova, O.I. 2018 Enhanced slip properties of lubricant-infused grooves. Phys. Rev. E 98 (3), 033103.CrossRefGoogle Scholar
Baier, T. & Hardt, S. 2021 Influence of insoluble surfactants on shear flow over a surface in Cassie state at large Péclet numbers. J. Fluid Mech. 907, A3.CrossRefGoogle Scholar
Baier, T., Steffes, C. & Hardt, S. 2010 Thermocapillary flow on superhydrophobic surfaces. Phys. Rev. E 82 (3), 037301.CrossRefGoogle ScholarPubMed
Barentin, C., Muller, P., Ybert, C., Joanny, J.-F. & Di Meglio, J.-M. 2000 Shear viscosity of polymer and surfactant monolayers. Eur. Phys. J. E 2 (2), 153–159.CrossRefGoogle Scholar
Batchelor, G. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Bolognesi, G., Cottin-Bizonne, C. & Pirat, C. 2014 Evidence of slippage breakdown for a superhydrophobic microchannel. Phys. Fluids 26 (8), 082004.CrossRefGoogle Scholar
Crowdy, D. 2010 Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles. Phys. Fluids 22 (12), 121703.CrossRefGoogle Scholar
Crowdy, D. 2015 Effective slip lengths for longitudinal shear flow over partial-slip circular bubble mattresses. Fluid Dyn. Res. 47 (6), 065507.CrossRefGoogle Scholar
Crowdy, D.G. 2016 Analytical formulae for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci. J. Fluid Mech. 791, R7.CrossRefGoogle Scholar
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
Edwards, D., Brenner, H. & Wasan, D. 1991 Interfacial Transport Processes and Rheology. Bufferworth-Heinemann.Google Scholar
Elfring, G.J., Leal, L.G. & Squires, T.M. 2016 Surface viscosity and marangoni stresses at surfactant laden interfaces. J. Fluid Mech. 792, 712–739.CrossRefGoogle Scholar
England, A.H. 2003 Complex Variable Methods in Elasticity. Dover Publications.Google Scholar
Gaddam, A., Agrawal, A., Joshi, S.S. & Thompson, M.C. 2018 Slippage on a particle-laden liquid-gas interface in textured microchannels. Phys. Fluids 30 (3), 032101.CrossRefGoogle Scholar
Gakhov, F.D. 1966 Boundary Value Problems. Pergamon Press.CrossRefGoogle Scholar
Gogolin, A.O. 2014 Lectures on Complex Integration. Springer.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
Honerkamp-Smith, A.R., Woodhouse, F.G., Kantsler, V. & Goldstein, R.E. 2013 Membrane viscosity determined from shear-driven flow in giant vesicles. Phys. Rev. Lett. 111 (3), 038103.CrossRefGoogle ScholarPubMed
Karatay, E., Haase, A.S., Visser, C.W., Sun, C., Lohse, D., Tsai, P.A. & Lammertink, R.G. 2013 Control of slippage with tunable bubble mattresses. Proc. Natl Acad. Sci. USA 110 (21), 8422–8426.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
Kim, S. & Karrila, S. 2005 Microhydrodynamics: Principles and Selected Applications. Dover Publications.Google Scholar
Kim, H. & Park, H. 2019 Diffusion characteristics of air pockets on hydrophobic surfaces in channel flow: three-dimensional measurement of air–water interface. Phys. Rev. Fluids 4 (7), 074001.CrossRefGoogle Scholar
Kirk, T.L. 2018 Asymptotic formulae for flow in superhydrophobic channels with longitudinal ridges and protruding menisci. J. Fluid Mech. 839, R3.CrossRefGoogle Scholar
Kirk, T.L., Karamanis, G., Crowdy, D.G. & Hodes, M. 2020 Thermocapillary stress and meniscus curvature effects on slip lengths in ridged microchannels. J. Fluid Mech. 894, A15.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
Lawrentjew, M. & Schabat, B. 1967 Methoden der komplexen Funktionentheorie. Deutscher Verlag der Wissenschaften.Google Scholar
Leal, L.G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Lee, C., Choi, C.-H. & Kim, C.-J. 2016 Superhydrophobic drag reduction in laminar flows: a critical review. Exp. Fluids 57 (12), 176.CrossRefGoogle Scholar
Levich, V.G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.Google Scholar
Li, Y., Alame, K. & Mahesh, K. 2017 Feature-resolved computational and analytical study of laminar drag reduction by superhydrophobic surfaces. Phys. Rev. Fluids 2, 054002.CrossRefGoogle Scholar
Li, H., Li, Z., Tan, X., Wang, X., Huang, S., Xiang, Y., Lv, P. & Duan, H. 2020 Three-dimensional backflow at liquid–gas interface induced by surfactant. J. Fluid Mech. 899, A8.CrossRefGoogle Scholar
Manikantan, H. & Squires, T.M. 2020 Surfactant dynamics: hidden variables controlling fluid flows. J. Fluid Mech. 892, P1.CrossRefGoogle ScholarPubMed
Merson, R. & Quinn, J. 1965 Stagnation in a fluid interface: properties of the stagnant film. AIChE J. 11 (3), 391–395.CrossRefGoogle Scholar
Muskhelishvili, N.I. 2008 Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics. Dover Publications.Google Scholar
Muskhelishvili, N.I. 2013 Some Basic Problems of the Mathematical Theory of Elasticity. Springer Science & Business Media.Google Scholar
Ng, C.-O. & Wang, C. 2011 Effective slip for stokes flow over a surface patterned with two-or three-dimensional protrusions. Fluid Dyn. Res. 43 (6), 065504.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
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
Sbragaglia, M. & Prosperetti, A. 2007 A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19 (4), 043603.CrossRefGoogle 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
Scott, J.C. 1982 Flow beneath a stagnant film on water: the Reynolds ridge. J. Fluid Mech. 116, 283–296.CrossRefGoogle Scholar
Slattery, J.C., Sagis, L. & Oh, E.-S. 2007 Interfacial Transport Phenomena. Springer Science & Business Media.Google 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
Steinberger, A., Cottin-Bizonne, C., Kleimann, P. & Charlaix, E. 2007 High friction on a bubble mattress. Nat. Mater. 6 (9), 665–668.CrossRefGoogle ScholarPubMed
Temprano-Coleto, F., Smith, S.M., Peaudecerf, F.J., Landel, J.R., Gibou, F. & Luzzatto-Fegiz, P. 2021 Slip on three-dimensional surfactant-contaminated superhydrophobic gratings. arXiv:2103.16945.Google Scholar
Woodhouse, F.G. & Goldstein, R.E. 2012 Shear-driven circulation patterns in lipid membrane vesicles. J. Fluid Mech. 705, 165–175.CrossRefGoogle Scholar
Yariv, E. & Crowdy, D. 2020 Longitudinal thermocapillary flow over a dense bubble mattress. SIAM J. Appl. Maths 80 (1), 1–19.CrossRefGoogle Scholar
Yariv, E. & Kirk, T.L. 2021 Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles. IMA J. Appl. Maths 86 (3), 490–501.CrossRefGoogle Scholar