Skip to main content Accessibility help

Slip flow past a gas–liquid interface with embedded solid particles

  • A. Vidal (a1) and L. Botto (a1)


We simulate shear flow past a stationary monolayer of spherical particles embedded in a flat gas–liquid interface. This problem is relevant to the understanding of the microhydrodynamics of particle-laden interfacial structures, including particle-laden drops, bubbles and foams. The combination of the free-shear condition at the gas–liquid interface and the no-slip condition at the particle surfaces gives rise to a velocity slip at the particle-laden interface. We study the characteristics of the flow near the monolayer, focusing on slip velocity, slip length and interfacial shear stress. Two microstructures are compared: a square array, and a reticulated array mimicking a percolating network of aggregated particles. We demonstrate that the scaling laws for the dependence of the slip length on solid area fraction developed for flow past superhydrophobic microstructured surfaces apply to the case of interfacial particles. The calculated slip lengths are in general smaller that those reported for microstructured superhydrophobic surfaces. This difference, which is due to the significant protrusion of the spherical particles in the liquid, can be accounted for in the case of the square array by an approximate argument. For a given area fraction, the reticulated array yields a larger slip length than the square array. We analyse the hydrodynamic forces acting on the particles, and the corresponding tangential stress exerted by the bulk ‘subphase’.


Corresponding author

Email address for correspondence:


Hide All
Aveyard, R., Clint, J. H., Nees, D. & Paunov, V. N. 2000 Compression and structure of monolayers of charged latex particles at air/water and octane/water interfaces. Langmuir 16 (4), 19691979.
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.
Binks, B. P. 2002 Particles as surfactants: similarities and differences. Curr. Opin. Colloid Interface Sci. 7 (1), 2141.
Binks, B. P. & Horozov, T. S. 2006 Colloidal Particles at Liquid Interfaces. Cambridge University Press.
Bluemink, J. J., Lohse, D., Prosperetti, A. & Van Wijngaarden, L. 2008 A sphere in a uniformly rotating or shearing flow. J. Fluid Mech. 600, 201233.
Bluemink, J. J., Lohse, D., Prosperetti, A. & Van Wijngaarden, L. 2010 Drag and lift forces on particles in a rotating flow. J. Fluid Mech. 643, 131.
Boniello, G., Blanc, C., Fedorenko, D., Medfai, M., Mbarek, N. B., In, M., Gross, M., Stocco, A. & Nobili, M. 2015 Brownian diffusion of a partially wetted colloid. Nat. Mater. 14, 908911.
Botto, L. & Prosperetti, A. 2012 A fully resolved numerical simulation of turbulent flow past one or several spherical particles. Phys. Fluids 24 (1), 013303.
Bournival, G., Ata, S. & Wanless, E. J. 2015 The roles of particles in multiphase processes: particles on bubble surfaces. Adv. Colloid Interface Sci. 225, 114133.
Brenner, H. 2013 Interfacial Transport Processes and Rheology. Elsevier.
Buttinoni, I., Zell, Z. A., Squires, T. M. & Isa, L. 2015 Colloidal binary mixtures at fluid–fluid interfaces under steady shear: structural, dynamical and mechanical response. Soft Matt. 11 (42), 83138321.
Dani, A., Keiser, G., Yeganeh, M. S. & Maldarelli, C. 2015 Hydrodynamics of particles at an oil–water interface. Langmuir 31 (49), 1329013302.
Danov, K., Aust, R., Durst, F. & Lange, U. 1995 Influence of the surface viscosity on the hydrodynamic resistance and surface diffusivity of a large Brownian particle. J. Colloid Interface Sci. 175 (1), 3645.
Danov, K. D., Dimova, R. & Pouligny, B. 2000 Viscous drag of a solid sphere straddling a spherical or flat surface. Phys. Fluids 12 (11), 27112722.
Davis, A. M. J. & Lauga, E. 2009 The friction of a mesh-like super-hydrophobic surface. Phys. Fluids 21 (11), 113101.
Davis, A. M. J. & Lauga, E. 2010 Hydrodynamic friction of fakir-like superhydrophobic surfaces. J. Fluid Mech. 661, 402411.
De Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827863.
Deemer, A. R. & Slattery, J. C. 1978 Balance equations and structural models for phase interfaces. Intl J. Multiphase Flow 4 (2), 171192.
Dörr, A. & Hardt, S. 2015 Driven particles at fluid interfaces acting as capillary dipoles. J. Fluid Mech. 770, 526.
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, 607618.
Edwards, D. A. & Wasan, D. T. 1991 A micromechanical model of linear surface rheological behavior. Chem. Engng Sci. 46 (5), 12471257.
Fischer, Th. M., Dhar, P. & Heinig, P. 2006 The viscous drag of spheres and filaments moving in membranes or monolayers. J. Fluid Mech. 558, 451475.
Frijters, S., Günther, F. & Harting, J. 2012 Effects of nanoparticles and surfactant on droplets in shear flow. Soft Matt. 8 (24), 65426556.
Gu, C. & Botto, L. 2016 Direct calculation of anisotropic surface stresses during deformation of a particle-covered drop. Soft Matt. 12 (3), 705716.
Happel, J. & Brenner, H. 2012 Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, vol. 1. Springer Science and Business Media.
Horozov, T. S. 2008 Foams and foam films stabilised by solid particles. Curr. Opin. Colloid Interface Sci. 13 (3), 134140.
Hunter, T. N., Pugh, R. J., Franks, G. V. & Jameson, G. J. 2008 The role of particles in stabilising foams and emulsions. Adv. Colloid Interface Sci. 137 (2), 5781.
Irvine, W. T. M., Vitelli, V. & Chaikin, P. M. 2010 Pleats in crystals on curved surfaces. Nature 468 (7326), 947951.
Kotula, A. P. & Anna, S. L. 2012 Probing timescales for colloidal particle adsorption using slug bubbles in rectangular microchannels. Soft Matt. 8 (41), 1075910772.
Kumar, A., Datta, S. & Kalyanasundaram, D. 2016 Permeability and effective slip in confined flows transverse to wall slippage patterns. Phys. Fluids 28 (8), 082002.
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 5577.
Lewandowski, E. P., Cavallaro, M. Jr, Botto, L., Bernate, J. C., Garbin, V. & Stebe, K. J. 2010 Orientation and self-assembly of cylindrical particles by anisotropic capillary interactions. Langmuir 26 (19), 1514215154.
Lishchuk, S. V. & Halliday, I. 2009 Effective surface viscosities of a particle-laden fluid interface. Phys. Rev. E 80 (1), 016306.
Liu, Q. & Prosperetti, A. 2011 Pressure-driven flow in a channel with porous walls. J. Fluid Mech. 679, 77100.
Luo, H. & Pozrikidis, C. 2008 Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall. J. Engng Maths 62 (1), 121.
Martinez, A. C., Rio, E., Delon, G., Saint-Jalmes, A., Langevin, D. & Binks, B. P. 2008 On the origin of the remarkable stability of aqueous foams stabilised by nanoparticles: link with microscopic surface properties. Soft Matt. 4 (7), 15311535.
Ng, C.-O. & Wang, C. Y. 2009 Stokes shear flow over a grating: implications for superhydrophobic slip. Phys. Fluids 21 (1), 013602.
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), 147154.
Poulichet, V. & Garbin, V. 2015 Ultrafast desorption of colloidal particles from fluid interfaces. Proc. Natl Acad. Sci. USA 112 (19), 59325937.
Pozrikidis, C. 2007 Particle motion near and inside an interface. J. Fluid Mech. 575, 333357.
Rapacchietta, A. V. & Neumann, A. W. 1977 Force and free-energy analyses of small particles at fluid interfaces: II. Spheres. J. Colloid Interface Sci. 59 (3), 555567.
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89109.
Sbragaglia, M. & Prosperetti, A. 2007 A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19 (4), 043603.
Shang, J., Flury, M. & Deng, Y. 2009 Force measurements between particles and the air–water interface: implications for particle mobilization in unsaturated porous media. Water Resour. Res. 45, W06420.
Shelley, S. R., Smith, J. D., Hibbins, A. P., Sambles, J. R. & Horsley, S. A. R. 2016 Fluid mobility over corrugated surfaces in the Stokes regime. Phys. Fluids 28 (8), 083101.
Sierakowski, A. J. 2016 GPU-centric resolved-particle disperse two-phase flow simulation using the Physalis method. Comput. Phys. Commun 207, 2434.
Singh, P. & Joseph, D. D. 2005 Fluid dynamics of floating particles. J. Fluid Mech. 530, 3180.
Stancik, E. J., Kouhkan, M. & Fuller, G. G. 2004 Coalescence of particle-laden fluid interfaces. Langmuir 20 (1), 9094.
Subramaniam, A. B., Abkarian, M. & Stone, H. A. 2005 Controlled assembly of jammed colloidal shells on fluid droplets. Nat. Mater. 4 (7), 553556.
Subrahmanyam, T. V. & Forssberg, E. 1988 Froth stability, particle entrainment and drainage in flotation – a review. Intl J. Miner. Process. 23 (1), 3353.
Tambe, D. E. & Sharma, M. M. 1994 The effect of colloidal particles on fluid–fluid interfacial properties and emulsion stability. Adv. Colloid Interface Sci. 52, 163.
Tsapis, N., Dufresne, E. R., Sinha, S. S., Riera, C. S., Hutchinson, J. W., Mahadevan, L. & Weitz, D. A. 2005 Onset of buckling in drying droplets of colloidal suspensions. Phys. Rev. Lett. 94 (1), 018302.
Weber, M. E., Blanchard, D. C. & Syzdek, L. D. 1983 The mechanism of scavenging of waterborne bacteria by a rising bubble. Limnol. Oceanogr. 28 (1), 101105.
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.
Yunker, P. J., Still, T., Lohr, M. A. & Yodh, A. G. 2011 Suppression of the coffee-ring effect by shape-dependent capillary interactions. Nature 476 (7360), 308311.
Zhang, Z. & Prosperetti, A. 2005 A second-order method for three-dimensional particle simulation. J. Comput. Phys. 210 (1), 292324.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Slip flow past a gas–liquid interface with embedded solid particles

  • A. Vidal (a1) and L. Botto (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed