Hostname: page-component-5d84bcc8dc-whph2 Total loading time: 0 Render date: 2026-08-11T06:38:50.236Z Has data issue: false hasContentIssue false
  • English
  • Français

Sliding, pinch-off and detachment of a droplet on a wall in shear flow

Published online by Cambridge University Press:  28 January 2010

HANG DING ,
MOHAMMAD N. H. GILANI  and
PETER D. M. SPELT
HANG DING
Affiliation:
Department of Chemical Engineering, Imperial College London SW7 2AZ, UK
MOHAMMAD N. H. GILANI
Affiliation:
Department of Chemical Engineering, Imperial College London SW7 2AZ, UK
PETER D. M. SPELT*
Affiliation:
Department of Chemical Engineering, Imperial College London SW7 2AZ, UK
*
‡Email address for correspondence: p.spelt@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate here what happens beyond the onset of motion of a droplet on a wall by the action of an imposed shear flow, accounting for inertial effects and contact-angle hysteresis. A diffuse-interface method is used for this purpose, which alleviates the shear stress singularity at a moving contact line, resulting in an effective slip length. Various flow regimes are investigated, including steadily moving drops, and partial or entire droplet entrainment. In the regime of quasi-steadily moving drops, the drop speed is found to be linear in the imposed shear rate, but to exhibit an apparent discontinuity at the onset of motion. The results also include the relation between a local maximum angle between the interface and the wall and the instantaneous value of the contact-line speed. The critical conditions for the onset of entrainment are determined for pinned as well as for moving drops. The corresponding critical capillary numbers are found to be in a rather narrow range, even for quite substantial values of a Reynolds number. The approach to breakup is then investigated in detail, including the growth of a ligament on a drop, and the reduction of the radius of a pinching neck. A model based on an energy argument is proposed to explain the results for the rate of elongation of ligaments. The paper concludes with an investigation of detachment of a hydrophobic droplet from the solid wall.

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 644 , 10 February 2010 , pp. 217 - 244
Copyright
Copyright © Cambridge University Press 2010

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.)

Article purchase

Temporarily unavailable

Footnotes

†

Present address: Department of Chemical Engineering, University of California, Santa Barbara, CA 93106-5080, USA

References

REFERENCES

Bagchi, P. & Balachandar, S. 2002 Shear versus vortex-induced lift force on a rigid sphere at moderate Re. J. Fluid Mech. 473, 379–388.CrossRefGoogle Scholar
Booty, M. R. & Siegel, M. 2005 Steady deformation and tip-streaming of a slender bubble with surfactant in an extensional flow. J. Fluid Mech. 544, 243–275.CrossRefGoogle Scholar
Chatterjee, J. 2001 A criterion for buoyancy induced drop detachment based on an analytical approximation of the drop shape. Coll. Surf. A: Physiochem. Engng Asp. 178, 249–263.CrossRefGoogle Scholar
Cristini, V., Guido, S., Alfani, A., Blawzdziewicz, J. & Loewenberg, M. 2003 Drop breakup and fragment size distribution in shear flow. J. Rheol. 47, 1283–1298.CrossRefGoogle Scholar
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.CrossRefGoogle Scholar
Dimitrakopoulos, P 2007 a Deformation of a droplet adhering to a solid surface in shear flow: onset of interfacial sliding. J. Fluid Mech. 435, 327–350.CrossRefGoogle Scholar
Dimitrakopoulos, P 2007 b Gravitational effects on the deformation of a droplet adhering to a horizontal solid surface in shear flow. Phys. Fluids 19, 122105.CrossRefGoogle Scholar
Dimitrakopoulos, P. & Higdon, J. J. L. 1998 On the displacement of three-dimensional fluid droplets from solid surfaces in low-Reynolds-number shear flows. J. Fluid Mech. 377, 189–222.CrossRefGoogle Scholar
Ding, H. & Spelt, P. D. M. 2007 a Inertial effects in droplet spreading: a comparison between diffuse interface and level-set simulations. J. Fluid Mech. 576, 287–296.CrossRefGoogle Scholar
Ding, H. & Spelt, P. D. M. 2007 b Wetting condition in diffuse interface simulation of contact line motion. Phys. Rev. E 75, 046708.CrossRefGoogle ScholarPubMed
Ding, H. & Spelt, P. D. M. 2008 Onset of motion of a three-dimensional droplet on a wall in shear flow at moderate Reynolds numbers. J. Fluid Mech. 599, 341–362.CrossRefGoogle Scholar
Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226, 2078–2095.CrossRefGoogle Scholar
Dussan, V. E. B., 1987 On the ability of drops to stick to surfaces of solids. Part 3. The influences of the motion of the surrounding fluid on dislodging drops. J. Fluid Mech. 174, 381–397.CrossRefGoogle Scholar
Dussan, V., E. B. & Chow, R. T. P. 1983 On the ability of drops or bubbles to stick to non-horizontal surfaces of solids. J. Fluid Mech. 137, 1–29.CrossRefGoogle Scholar
Eames, I., Gilbertson, M. A. & Landeryou, M. 2005 The effect of an ambient flow on the spreading of a viscous gravity current. J. Fluid Mech. 523, 261–275.CrossRefGoogle Scholar
Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865–929.CrossRefGoogle Scholar
Golpaygan, A. & Ashgriz, N. 2008 Multiphase flow model to study channel flow dynamics of PEM fuel cells: deformation and detachment of water droplets. Intl J. Comput. Fluid Dyn. 22, 85–95.CrossRefGoogle Scholar
Hodges, S. R. & Jensen, O. E. 2002 Spreading and peeling dynamics in a model of cell adhesion. J. Fluid Mech. 460, 381–409.CrossRefGoogle Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.CrossRefGoogle Scholar
Kang, Q., Zhang, D. & Chen, S. 2005 Displacement of a three-dimensional immiscible droplet in a duct. J. Fluid Mech. 545, 41–66.CrossRefGoogle Scholar
Khismatullin, et al. 2003 Inertia-induced breakup of highly viscous drops subjected to simple shear. Phys. Fluids 15, 1351–1354.CrossRefGoogle Scholar
Le Grand, N., Daerr, A. & Limat, L. 2005 Shape and motion of drops sliding down an inclined plane. J. Fluid Mech. 541, 293–315.CrossRefGoogle Scholar
Lister, J. R. & Stone, H. A. 1998 Capillary breakup of a viscous thread surrounded by another viscous fluid. Phys. Fluids 10, 2758–2764.CrossRefGoogle Scholar
Marmottant, P. & Villermaux, E. 2004 a On spray formation. J. Fluid Mech. 498, 73–112.CrossRefGoogle Scholar
Marmottant, P. & Villermaux, E. 2004 b Fragmentation of stretched liquid ligaments. Phys. Fluids 16, 2732–2741 and Erratum: 18, 059901-1.CrossRefGoogle Scholar
Notz, P. K. & Basaran, O. A. 2004 Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 512, 223–256.CrossRefGoogle Scholar
Pan, L. & Hanratty, T. J. 2002 Correlation of entrainment for annular flow in horizontal pipes. Intl J. Multiph. Flow 28, 385–408.CrossRefGoogle Scholar
Rallison, J. M. 1984 The deformation of small drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16, 45–66.CrossRefGoogle Scholar
Rio, E., Daerr, A., Andreotti, B. & Limat, L. 2005 Boundary conditions in the vicinity of a dynamic contact line: experimental investigation of viscous drops sliding down an inclined plane. Phys. Rev. Lett. 94, 024503.CrossRefGoogle ScholarPubMed
Schleizer, A. D. & Bonnecaze, R. T. 1999 Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows. J. Fluid Mech. 383, 29–54.CrossRefGoogle Scholar
Sierou, A. & Lister, J. R. 2003 Self-similar solutions for the viscous capillary pinch-off. J. Fluid Mech. 497, 381–403.CrossRefGoogle Scholar
Spelt, P. D. M. 2006 Shear flow past two-dimensional droplets pinned or moving on an adhering channel wall at moderate Reynolds numbers: a numerical study. J. Fluid Mech. 561, 439–463.CrossRefGoogle Scholar
Sugiyama, K. & Sbragaglia, M. 2008 Linear shear flow past a hemispherical droplet adhering to a solid surface. J. Engng Math. 62, 35–55.CrossRefGoogle Scholar
Thiele, U. & Knobloch, E. 2006 On the depinning of a driven drop on a heterogeneous substrate. New J. Phys. 8, 1–37.CrossRefGoogle Scholar
Thoroddsen, S. T. & Takehara, K. 2000 The coalescence cascade of a drop. Phys. Fluids 12, 1265–1267.CrossRefGoogle Scholar
Tjahjadi, M., Stone, H. A. & Ottino, J. M. 1992 Satellite and subsatellite formation in capillary breakup. J. Fluid Mech. 243, 297–317.CrossRefGoogle Scholar