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Capillary waves with surface viscosity

  • Li Shen (a1), Fabian Denner (a1), Neal Morgan (a2), Berend van Wachem (a1) (a3) and Daniele Dini (a1)...
Abstract

Experiments over the last 50 years have suggested a tentative correlation between the surface (shear) viscosity and the stability of a foam or emulsion. We examine this link theoretically using small-amplitude capillary waves in the presence of a surfactant solution of dilute concentration, where the associated Marangoni and surface viscosity effects are modelled via the Boussinesq–Scriven formulation. The resulting integro-differential initial value problem is solved analytically, and surface viscosity is found to contribute an overall damping effect to the amplitude of the capillary wave with varying degree depending on the length scale of the system. Numerically, we find that the critical damping wavelength increases for increasing surface concentration but the rate of increase remains different for both the surface viscosity and the Marangoni effect.

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Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
Email address for correspondence: l.shen14@imperial.ac.uk
References
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Aarts, D. G. A. L., Schmidt, M. & Lekkerkerker, H. N. W. 2004 Direct visual observation of thermal capillary waves. Science 304 (5672), 847850.
Aris, R. 1963 Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover.
Batchelor, G. K., Moffatt, H. K., Worster, M. G. & Osborn, T. R. 2003 Perspectives in Fluid Dynamics: A Collective Introduction to Current Research. Cambridge University Press.
Blanchette, F. & Bigioni, T. P. 2006 Partial coalescence of drops at liquid interfaces. Nat. Phys. 2 (4), 254257.
Brown, A. G., Thuman, W. C. & McBain, J. W. 1953 The surface viscosity of detergent solutions as a factor in foam stability. J. Colloid Interface Sci. 8 (5), 491553.
Denner, F. 2016 Frequency dispersion of small-amplitude capillary waves in viscous fluids. Phys. Rev. E 94, 023110.
Djabbarah, N. F. & Wasan, D. T. 1982 Dilational viscoelastic properties of fluid interfaces-III Mixed surfactant systems. Chem. Engng Sci. 37 (2), 175184.
Edwards, D. A., Brenner, H. & Wasan, D. T. 1991 Interfacial Transport Processes and Rheology. Butterworth-Heinemann.
Gavranovic, G. T., Kurtz, R. E., Golemanov, K., Lange, A. & Fuller, G. G. 2006 Interfacial rheology and structure of straight-chain and branched hexadecanol mixtures. Ind. Engng Chem. Res. 45 (21), 68806884.
Gounley, J., Boedec, G., Jaeger, M. & Leonetti, M. 2016 Influence of surface viscosity on droplets in shear flow. J. Fluid Mech. 791, 464494.
Kanner, B. & Glass, J. E. 1969 Surface viscosity and elasticity: significant parameters in industrial processes. Ind. Engng Chem. 61 (5), 3141.
Lamb, S. H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Langevin, D. 2014 Rheology of adsorbed surfactant monolayers at fluid surfaces. Annu. Rev. Fluid Mech. 46 (1), 4765.
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.
Levich, V. G. & Krylov, V. S. 1969 Surface-tension-driven phenomena. Annu. Rev. Fluid Mech. 1, 293316.
Lopez, J. M. & Hirsa, A. 1998 Direct determination of the dependence of the surface shear and dilatational viscosities on the thermodynamic state of the interface: theoretical foundations. J. Colloid Interface Sci. 206, 231239.
Lucassen, J. & Hansen, R. S. 1966 Damping of waves on monolayer-covered surfaces. J. Colloid Interface Sci. 22 (1), 3244.
Ponce-Torres, A., Montanero, J. M., Herrada, M. A., J., Vega, E., M. & Vega, J. 2017 Influence of the surface viscosity on the breakup of a surfactant-laden drop. Phys. Rev. Lett. 118, 024501.
Prosperetti, A. 1976 Viscous effects on small-amplitude surface waves. Phys. Fluids 19 (2), 195203.
Scheid, B., Delacotte, J., Dollet, B., Rio, E., Restagno, F., van Nierop, E. A., Cantat, I., Langevin, D. & Stone, H. A. 2010 The role of surface rheology in liquid film formation. Europhys. Lett. 90 (2), 24002.
Scheludko, A. 1967 Thin liquid films. Adv. Colloid Interface Sci. 1 (1), 391464.
Scriven, L. E. 1960 Dynamics of a fluid interface: equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12 (2), 98108.
Shen, L., Denner, F., Morgan, N., van Wachem, B. G. M. & Dini, D. 2017 The Marangoni effect on small-amplitude capillary waves in viscous fluids. Phys. Rev. E 96, 053110.
Sinclair, D., Levy, R. & Daniels, K. E. 2018 Simulating surfactant spreading: influence of a physically motivated equation of state. Eur. J. Appl. Maths 29 (1), 3054.
Slattery, J. C., Sagis, L. & Oh, E.-S. 2007 Interfacial Transport Phenomena. Springer.
Stevenson, P. 2005 Remarks on the shear viscosity of surfaces stabilised with soluble surfactants. J. Colloid Interface Sci. 290 (2), 603606.
Stone, H. A. 1990 A simple derivation of the time-dependent convective–diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2 (1), 111112.
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), 36773682.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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