Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T11:59:53.845Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Asmolov, E. S. & Vinogradova, O. I. 2012 Effective slip boundary conditions for arbitrary one-dimensional surfaces. J. Fluid Mech. 706, 108–117.CrossRefGoogle Scholar
Bhushan, B. 2018 Plant leaf surfaces in living nature. In Biomimetics: Bioinspired Hierarchical-Structured (ed. Bhushan, B.), pp. 81–107. Springer.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
Bond, W. & Newton, D. A. 1928 Bubbles, drops and Stokes’ law. Phil. Mag. 5, 794–800.CrossRefGoogle Scholar
Cartagena, E. J. G., Arenas, I., Bernardini, M. & Leonardi, S. 2018 Dependence of the drag over super hydrophobic and liquid infused surfaces on the textured surface and Weber number. Flow Turbul. Combust. 100 (4), 945–960.CrossRefGoogle Scholar
Chang, C. H. & Franses, E. I. 1995 Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms. Colloids Surf. A 100, 1–45.CrossRefGoogle Scholar
Cheng, Y., Xu, J. & Sui, Y. 2015 Numerical study on drag reduction and heat transfer enhancement in microchannels with superhydrophobic surfaces for electronic cooling. Appl. Therm. Engng 88, 71–81; Special Issue for International Heat Transfer Symposium 2014.CrossRefGoogle Scholar
Cottin-Bizonne, C., Barentin, C., Charlaix, 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
Crowdy, D. G. 2016 Analytical formulae for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci. J. Fluid Mech. 791, R7.CrossRefGoogle Scholar
Crowdy, D. G. 2017a Effective slip lengths for immobilized superhydrophobic surfaces. J. Fluid Mech. 825, R2.CrossRefGoogle Scholar
Crowdy, D. G. 2017b Perturbation analysis of subphase gas and meniscus curvature effects for longitudinal flows over superhydrophobic surfaces. J. Fluid Mech. 822, 307–326.CrossRefGoogle Scholar
Davis, A. M. J. & Lauga, E. 2009 Geometric transition in friction for flow over a bubble mattress. Phys. Fluids 21 (1), 011701.CrossRefGoogle 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
Facchini, M. C., Decesari, S., Mircea, M., Fuzzi, S. & Loglio, G. 2000 Surface tension of atmospheric wet aerosol and cloud/fog droplets in relation to their organic carbon content and chemical composition. Atmos. Environ. 34 (28), 4853–4857.CrossRefGoogle Scholar
Frumkin, A. N. & Levich, V. G. 1947 Effect of surface-active substances on movements at the boundaries of liquid phases. Zhur. Fiz. Khim. 21, 1183–1204; (in Russian). This work is summarized in the textbook by Levich (1962), also translated from Russian.Google Scholar
Game, S. E., Hodes, M. & Papageorgiou, D. T. 2019 Effects of slowly varying meniscus curvature on internal flows in the cassie state. J. Fluid Mech. 872, 272–307.CrossRefGoogle Scholar
Gose, J. W., Golovin, K., Boban, M., Mabry, J. M., Tuteja, A., Perlin, M. & Ceccio, S. L. 2018 Characterization of superhydrophobic surfaces for drag reduction in turbulent flow. J. Fluid Mech. 845, 560–580.CrossRefGoogle Scholar
Gruncell, B.2014 Superhydrophobic surfaces and their potential application to hydrodynamic drag reduction. PhD thesis, University of Southampton.Google Scholar
Harper, J. F. 2004 Stagnant-cap bubbles with both diffusion and adsorption rate-determining. J. Fluid Mech. 521, 115–123.CrossRefGoogle Scholar
He, Z., Maldarelli, C. & Dagan, Z. 1991 The size of stagnant caps of bulk soluble surfactant on the interfaces of translating fluid droplets. J. Colloid Interface Sci. 146, 442–451.CrossRefGoogle Scholar
Hourlier-Fargette, A., Dervaux, J., Antkowiak, A. & Neukirch, S. 2018 Extraction of silicone uncrosslinked chains at air–water-polydimethylsiloxane triple lines. Langmuir 34, 12244–12250.CrossRefGoogle ScholarPubMed
Jones, E., Oliphant, T. E. & Peterson, P.2001 SciPy: open source scientific tools for Python. Available at: https://www.scipy.org.Google Scholar
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
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
Kotula, A. P. & Anna, S. L. 2015 Regular perturbation analysis of small amplitude oscillatory dilatation of an interface in a capillary pressure tensiometer. J. Rheol. 59 (1), 85–117.CrossRefGoogle Scholar
Kropfli, R. A., Ostrovski, L. A., Stanton, T. P., Skirta, E. A., Keane, A. N. & Irisov, V. 1999 Relationships between strong internal waves in the coastal zone and their radar and radiometric signatures. J. Geophys. Res. 104 (C2), 3133–3148.CrossRefGoogle Scholar
Lam, L. S., Hodes, M. & Enright, R. 2015 Analysis of Galinstan-based microgap cooling enhancement using structured surfaces. Trans. ASME J. Heat Mass Transfer 137, 091003.Google Scholar
Landel, J. R., Thomas, A. L., McEvoy, H. & Dalziel, S. B. 2016 Convective mass transfer from a submerged drop in a thin falling film. J. Fluid Mech. 789, 630–668.CrossRefGoogle Scholar
Langevin, D. 2014 Rheology of adsorbed surfactant monolayers at fluid surfaces. Annu. Rev. Fluid Mech. 46 (1), 47–65.CrossRefGoogle Scholar
Langevin, D. & Monroy, F. 2014 Marangoni stresses and surface compression rheology of surfactant solutions. Achievements and problems. Adv. Colloid Interface Sci. 206, 141–149.CrossRefGoogle ScholarPubMed
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.CrossRefGoogle 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. 2008 Structured surfaces for a giant liquid slip. Phys. Rev. Lett. 101 (6), 064501.CrossRefGoogle ScholarPubMed
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
Lévêque, A. M. 1928 Les lois de la transmission de chaleur par convection. In Annales des Mines, ou Recueil de Mémoires sur l’Exploitation des Mines et sur les Sciences et les Arts qui s’y Rapportent, Tome XIII, pp. 201–299, 305–362, 381–415. Dunod.Google Scholar
Levich, V. 1962 Physicochemical Hydrodynamics. Prentice Hall.Google Scholar
Lewis, M. A. 1991 Chronic and sublethal toxicities of surfactants to aquatic animals: A review and risk assessment. Water Res. 25 (1), 101–113.CrossRefGoogle Scholar
Ling, H., Srinivasan, S., Golovin, K., McKinley, G. H., Tuteja, A. & Katz, J. 2016 High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces. J. Fluid Mech. 801, 670–703.CrossRefGoogle Scholar
Luzzatto-Fegiz, P. & Helfrich, K. R. 2014 Laboratory experiments and simulations for solitary internal waves with trapped cores. J. Fluid Mech. 757, 354–380.CrossRefGoogle Scholar
Maali, A., Boisgard, R., Chraibi, H., Zhang, Z., Kellay, H. & Würger, A. 2017 Viscoelastic drag forces and crossover from no-slip to slip boundary conditions for flow near air–water interfaces. Phys. Rev. Lett. 118, 084501.CrossRefGoogle ScholarPubMed
Manor, O., Vakarelski, I. U., Tang, X., O’Shea, S. J., Stevens, G. W., Grieser, F., Dagastine, R. R. & Chan, D. Y. C. 2008 Hydrodynamic boundary conditions and dynamic forces between bubbles and surfaces. Phys. Rev. Lett. 101, 024501.CrossRefGoogle ScholarPubMed
Mayer, H. C. & Krechetnikov, R. 2012 Landau-Levich flow visualization: revealing the flow topology responsible for the film thickening phenomena. Phys. Fluids 24 (5), 052103.CrossRefGoogle Scholar
Nayar, K. G., Panchanathan, D., McKinley, G. H. & Lienhard, J. H. 2014 Surface tension of seawater. J. Phys. Chem. Ref. Data 43 (4), 043103.CrossRefGoogle Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16 (12), 4635–4643.CrossRefGoogle Scholar
Ou, J. & Rothstein, J. P. 2005 Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys. Fluids 17 (10), 103606.CrossRefGoogle Scholar
Palaparthi, R., Papageorgiou, D. T. & Maldarelli, C. 2006 Theory and experiments on the stagnant cap regime in the motion of spherical surfactant-laden bubbles. J. Fluid Mech. 559, 1–44.CrossRefGoogle Scholar
Park, H., Park, H. & Kim, J. 2013 A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow. Phys. Fluids 25, 110815.CrossRefGoogle Scholar
Park, H., Sun, G. & Kim, C.-J. 2014 Superhydrophobic turbulent drag reduction as a function of surface grating parameters. J. Fluid Mech. 747, 722–734.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, 7254–7259.CrossRefGoogle ScholarPubMed
Pereira, R., Ashton, I., Sabbaghzadeh, B., Shutler, J. D. & Upstill-Goddard, R. C. 2018 Reduced air–sea CO2 exchange in the Atlantic Ocean due to biological surfactants. Nat. Geosci. 11 (7), 492–496.CrossRefGoogle Scholar
Philip, J. R. 1972a Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23, 353–372.CrossRefGoogle Scholar
Philip, J. R. 1972b Integral properties of flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23, 960–968.CrossRefGoogle Scholar
Pogorzelski, S. & Kogut, A. D. 2001 Kinetics of marine surfactant adsorption at an air/water interface. Baltic sea studies. Oceanologia 43 (4), 389–404.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Prosser, A. J. & Franses, E. I. 2001 Adsorption and surface tension of ionic surfactants at the air–water interface: review and evaluation of equilibrium models. Colloids Surf. A 178 (1), 1–40.CrossRefGoogle Scholar
Quéré, D. 2008 Wetting and roughness. Annu. Rev. Mater. Res. 38, 71–99.CrossRefGoogle Scholar
Rastegari, A. & Akhavan, R. 2018 The common mechanism of turbulent skin-friction drag reduction with superhydrophobic longitudinal microgrooves and riblets. J. Fluid Mech. 838, 68–104.CrossRefGoogle Scholar
Rosen, M. J. & Kunjappu, J. T. 2012 Surfactants and Interfacial Phenomena, 4th edn. Wiley.CrossRefGoogle Scholar
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89–109.CrossRefGoogle Scholar
Samaha, M. A., Tafreshi, H. V. & Gad-el Hak, M. 2012 Superhydrophobic surfaces: from the lotus leaf to the submarine. C. R. Méc. 340, 18–34.CrossRefGoogle 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, 134501.CrossRefGoogle ScholarPubMed
Schmidt, R. & Schneider, B. 2011 The effect of surface films on the air–sea gas exchange in the Baltic sea. Mar. Chem. 126 (1), 56–62.CrossRefGoogle Scholar
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. 2013 Longitudinal and transverse flow over a cavity containing a second immiscible fluid. J. Fluid Mech. 717, 376–394.CrossRefGoogle Scholar
Shirtcliffe, N. J., McHale, G., Newton, M. I., Perry, C. C. & Pyatt, F. B. 2006 Plastron properties of a superhydrophobic surface. Appl. Phys. Lett. 89 (10), 104106.CrossRefGoogle Scholar
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. North-Holland Pub. Co.; Wiley.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
Temprano-Coleto, F., Peaudecerf, F. J., Landel, J. R., Gibou, F. & Luzzatto-Fegiz, P. 2018 Soap opera in the maze: geometry matters in Marangoni flows. Phys. Rev. Fluids 3 (10), 100507.CrossRefGoogle 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, 353–382.CrossRefGoogle Scholar
Truesdell, R., Mammoli, A., Vorobieff, P., van Swol, F. & Brinker, C. J. 2006 Drag reduction on a patterned superhydrophobic surface. Phys. Rev. Lett. 97 (4), 044504.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Zell, Z. A., Nowbahar, A., Mansard, V., Leal, L. G., Deshmukh, S. S., Mecca, J. M., Tucker, C. J. & Squires, T. M. 2014 Surface shear inviscidity of soluble surfactants. Proc. Natl Acad. Sci. USA 111 (10), 3677–3682.CrossRefGoogle ScholarPubMed
Supplementary material: File

Landel et al. supplementary material

Landel et al. supplementary material

Download Landel et al. supplementary material(File)
File 135.3 KB