Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T18:11:54.269Z Has data issue: false hasContentIssue false
  • English
  • Français

The effective shear and dilatational viscosities of a particle-laden interface in the dilute limit

Published online by Cambridge University Press:  28 September 2020

Michael Eigenbrod  and
Steffen Hardt Open the ORCID record for Steffen Hardt [Opens in a new window]
Michael Eigenbrod
Affiliation:
Institute for Nano- and Microfluidics, Technische Universität Darmstadt, 64287Darmstadt, Germany
Steffen Hardt*
Affiliation:
Institute for Nano- and Microfluidics, Technische Universität Darmstadt, 64287Darmstadt, Germany
*
†Email address for correspondence: hardt@nmf.tu-darmstadt.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The effective dilatational and shear viscosities of a particle-laden fluid interface are computed in the dilute limit under the assumption of an asymptotically vanishing viscosity ratio between the two fluids. Spherical particles with a given contact angle of the fluid interface at the particle surface are considered. A planar fluid interface and a small Reynolds number are assumed. The theoretical analysis is based on a domain perturbation expansion in the deviation of the contact angle from $90^{\circ }$ up to the second order. The resulting effective dilatational viscosity shows a stronger dependence on the contact angle than the effective shear viscosity, and its magnitude is larger for all contact angles. The limits of validity of the theory are discussed.

JFM classification

Complex Fluids: Suspensions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A26
Check for updates
Copyright
© The Author(s), 2020. 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

Aussillous, P. & Quéré, D. 2001 Liquid marbles. Nature 411 (6840), 924–927.CrossRefGoogle ScholarPubMed
Aveyard, R., Binks, B. P. & Clint, J. H. 2003 Emulsions stabilised solely by colloidal particles. Adv. Colloid Interface Sci. 100, 503–546.CrossRefGoogle Scholar
Avramescu, R.-E., Ghica, M.-V., Dinu-Pîrvu, C., Udeanu, D. I. & Popa, L. 2018 Liquid marbles: from industrial to medical applications. Molecules 23 (5), 1120.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (3), 545–570.CrossRefGoogle Scholar
Batchelor, G. K. 1976 Brownian diffusion with hydrodynamic interaction. J. Fluid Mech. 74 (1), 1–29.CrossRefGoogle Scholar
Batchelor, G. K. 1977 The effect of Brownian motion on the bulk stress in a suspension of spherical particles. J. Fluid Mech. 83 (1), 97–117.CrossRefGoogle Scholar
Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order $c^2$. J. Fluid Mech. 56 (3), 401–427.CrossRefGoogle Scholar
Binks, B. P. 2002 Particles as surfactants – similarities and differences. Curr. Opin. Colloid Interface Sci. 7 (1–2), 21–41.CrossRefGoogle Scholar
Bormashenko, E., Pogreb, R., Bormashenko, Y., Musin, A. & Stein, T. 2008 New investigations on ferrofluidics: ferrofluidic marbles and magnetic-field-driven drops on superhydrophobic surfaces. Langmuir 24 (21), 12119–12122.CrossRefGoogle ScholarPubMed
Boussinesq, J. V. 1913 Sur l'existence d'une viscosité superficielle, dans la mince couche de transition séparant un liquide d'un autre fluide contigu. J. Ann. Chim. Phys. 29, 349–357.Google Scholar
Brenner, H. 1964 The Stokes resistance of a slightly deformed sphere. Chem. Engng Sci. 19 (8), 519–539.CrossRefGoogle Scholar
Brenner, H. 1991 Interfacial Transport Processes and Rheology. Elsevier.Google Scholar
Byerly, W. E. 1893 An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. Dover.Google Scholar
De Corato, M. & Garbin, V. 2018 Capillary interactions between dynamically forced particles adsorbed at a planar interface and on a bubble. J. Fluid Mech. 847, 71–92.CrossRefGoogle Scholar
Dickinson, E. 2010 Food emulsions and foams: stabilization by particles. Curr. Opin. Colloid Interface Sci. 15, 40–49.CrossRefGoogle Scholar
Dörr, A. & Hardt, S. 2015 Driven particles at fluid interfaces acting as capillary dipoles. J. Fluid Mech. 770, 5–26. arXiv:1411.1183v3.CrossRefGoogle Scholar
Dörr, A., Hardt, S., Masoud, H. & Stone, H. A. 2016 Drag and diffusion coefficients of a spherical particle attached to a fluid–fluid interface. J. Fluid Mech. 790, 607–618. arXiv:1502.05488.CrossRefGoogle Scholar
Eigenbrod, M., Bihler, F. & Hardt, S. 2018 Electrokinetics of a particle attached to a fluid interface: electrophoretic mobility and interfacial deformation. Phys. Rev. Fluids 3 (10), 103701.CrossRefGoogle Scholar
Einstein, A. 1906 Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 324 (2), 289–306.CrossRefGoogle Scholar
Eshtiaghi, N., Liu, J. J. S. & Hapgood, K. P. 2010 Formation of hollow granules from liquid marbles: small scale experiments. Powder Technol. 197 (3), 184–195.CrossRefGoogle Scholar
Foss, D. R. & Brady, J. F. 2000 Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulation. J. Fluid Mech. 407, 167–200.CrossRefGoogle Scholar
Fox, F. E. & Rock, G. D. 1946 Compressional viscosity and sound absorption in water at different temperatures. Phys. Rev. 70 (1–2), 68–73.CrossRefGoogle Scholar
Galatola, P. & Fournier, J.-B. 2014 Capillary force acting on a colloidal particle floating on a deformed interface. Soft Matter 10 (13), 2197–2212.CrossRefGoogle ScholarPubMed
Gatignol, R. & Prud'homme, R. 2001 Mechanical and Thermodynamical Modeling of Fluid Interfaces. World Scientific.CrossRefGoogle Scholar
Guazzelli, E. & Morris, J. F. 2011 A Physical Introduction to Suspension Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 2012 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media. Springer Science & Business Media.Google Scholar
Kim, S. & Karrila, S. J. 2013 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.Google Scholar
Léandri, J. & Würger, A. 2013 Trapping energy of a spherical particle on a curved liquid interface. J. Colloid Interface Sci. 405, 249–255.CrossRefGoogle ScholarPubMed
Lishchuk, S. V. 2014 Effective surface dilatational viscosity of highly concentrated particle-laden interfaces. Phys. Rev. E 90 (5), 053005.CrossRefGoogle ScholarPubMed
Lishchuk, S. V. 2016 Dilatational viscosity of dilute particle-laden fluid interface at different contact angles. Phys. Rev. E 94 (6), 063111.CrossRefGoogle ScholarPubMed
Lishchuk, S. V. & Halliday, I. 2009 Effective surface viscosities of a particle-laden fluid interface. Phys. Rev. E 80 (June), 1–7.CrossRefGoogle ScholarPubMed
Lorentz, H. A. 1896 A general theorem concerning the motion of a viscous fluid and a few consequences derived from it. Z. Akad. Wet. Amsterdam 5, 168–175.Google Scholar
MacRobert, T. M. 1947 Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications. Dover.Google Scholar
Maru, H. C. & Wasan, D. T. 1979 Dilatational viscoelastic properties of fluid interfaces—II: experimental study. Chem. Engng Sci. 34 (11), 1295–1307.CrossRefGoogle Scholar
McHale, G. & Newton, M. I. 2011 Liquid marbles: principles and applications. Soft Matter 7 (12), 5473–5481.CrossRefGoogle Scholar
Mewis, J. & Wagner, N. J. 2012 Colloidal Suspension Rheology. Cambridge University Press.Google Scholar
Newton, M. I., Herbertson, D. L., Elliott, S. J., Shirtcliffe, N. J. & McHale, G. 2007 Electrowetting of liquid marbles. J. Phys. D: Appl. Phys. 40 (1), 20–24.CrossRefGoogle Scholar
Park, J. Y. & Advincula, R. C. 2011 Nanostructuring polymers, colloids, and nanomaterials at the air–water interface through Langmuir and Langmuir–Blodgett techniques. Soft Matter 7 (21), 9829–9843.CrossRefGoogle Scholar
Park, B. J. & Furst, E. M. 2011 Attractive interactions between colloids at the oil–water interface. Soft Matter 7 (17), 7676.CrossRefGoogle Scholar
Peaudecerf, J. R., 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. 114 (28), 7254–7259.CrossRefGoogle ScholarPubMed
Petkov, J. T., Denkov, N. D., Danov, K. D., Velev, O. D., Aust, R. & Durst, F. 1995 Measurement of the drag coefficient of spherical particles attached to fluid interfaces. J. Colloid Interface Sci. 172 (1), 147–154.CrossRefGoogle Scholar
Scriven, L. E. 1960 Dynamics of a fluid interface equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12 (2), 98–108.CrossRefGoogle Scholar
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115–146.CrossRefGoogle Scholar
Slattery, J. C., Sagis, L. & Oh, E.-S. 2007 Interfacial Transport Phenomena. Springer Science & Business Media.Google Scholar
Stamou, D., Duschl, C. & Johannsmann, D. 2000 Long-range attraction between colloidal spheres at the air–water interface: the consequence of an irregular meniscus. Phys. Rev. E 62 (4), 5263–5272.CrossRefGoogle ScholarPubMed
Toro-Mendoza, J., Rodriguez-Lopez, G. & Paredes-Altuve, O. 2017 Brownian diffusion of a particle at an air/liquid interface: the elastic (not viscous) response of the surface. Phys. Chem. Chem. Phys. 19 (13), 9092–9095.CrossRefGoogle ScholarPubMed
Wang, Y. & Oberlack, M. 2011 A thermodynamic model of multiphase flows with moving interfaces and contact line. Contin. Mech. Thermodyn. 23 (5), 409–433.CrossRefGoogle Scholar
Wu, J. & Ma, G.-H. 2016 Recent studies of pickering emulsions: particles make the difference. Small 12 (34), 4633–4648.CrossRefGoogle ScholarPubMed