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

  • HANG DING (a1), MOHAMMAD N. H. GILANI (a1) and PETER D. M. SPELT (a1)

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.

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Corresponding author

Email address for correspondence: p.spelt@imperial.ac.uk

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

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References

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Bagchi, P. & Balachandar, S. 2002 Shear versus vortex-induced lift force on a rigid sphere at moderate Re. J. Fluid Mech. 473, 379388.
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, 243275.
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, 249263.
Cristini, V., Guido, S., Alfani, A., Blawzdziewicz, J. & Loewenberg, M. 2003 Drop breakup and fragment size distribution in shear flow. J. Rheol. 47, 12831298.
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169194.
Dimitrakopoulos, P 2007 a Deformation of a droplet adhering to a solid surface in shear flow: onset of interfacial sliding. J. Fluid Mech. 435, 327350.
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.
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, 189222.
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, 287296.
Ding, H. & Spelt, P. D. M. 2007 b Wetting condition in diffuse interface simulation of contact line motion. Phys. Rev. E 75, 046708.
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, 341362.
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, 20782095.
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, 381397.
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, 129.
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, 261275.
Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865929.
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, 8595.
Hodges, S. R. & Jensen, O. E. 2002 Spreading and peeling dynamics in a model of cell adhesion. J. Fluid Mech. 460, 381409.
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 5788.
Kang, Q., Zhang, D. & Chen, S. 2005 Displacement of a three-dimensional immiscible droplet in a duct. J. Fluid Mech. 545, 4166.
Khismatullin, et al. 2003 Inertia-induced breakup of highly viscous drops subjected to simple shear. Phys. Fluids 15, 13511354.
Le Grand, N., Daerr, A. & Limat, L. 2005 Shape and motion of drops sliding down an inclined plane. J. Fluid Mech. 541, 293315.
Lister, J. R. & Stone, H. A. 1998 Capillary breakup of a viscous thread surrounded by another viscous fluid. Phys. Fluids 10, 27582764.
Marmottant, P. & Villermaux, E. 2004 a On spray formation. J. Fluid Mech. 498, 73112.
Marmottant, P. & Villermaux, E. 2004 b Fragmentation of stretched liquid ligaments. Phys. Fluids 16, 27322741 and Erratum: 18, 059901-1.
Notz, P. K. & Basaran, O. A. 2004 Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 512, 223256.
Pan, L. & Hanratty, T. J. 2002 Correlation of entrainment for annular flow in horizontal pipes. Intl J. Multiph. Flow 28, 385408.
Rallison, J. M. 1984 The deformation of small drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16, 4566.
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.
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, 2954.
Sierou, A. & Lister, J. R. 2003 Self-similar solutions for the viscous capillary pinch-off. J. Fluid Mech. 497, 381403.
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, 439463.
Sugiyama, K. & Sbragaglia, M. 2008 Linear shear flow past a hemispherical droplet adhering to a solid surface. J. Engng Math. 62, 3555.
Thiele, U. & Knobloch, E. 2006 On the depinning of a driven drop on a heterogeneous substrate. New J. Phys. 8, 137.
Thoroddsen, S. T. & Takehara, K. 2000 The coalescence cascade of a drop. Phys. Fluids 12, 12651267.
Tjahjadi, M., Stone, H. A. & Ottino, J. M. 1992 Satellite and subsatellite formation in capillary breakup. J. Fluid Mech. 243, 297317.
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Sliding, pinch-off and detachment of a droplet on a wall in shear flow

  • HANG DING (a1), MOHAMMAD N. H. GILANI (a1) and PETER D. M. SPELT (a1)

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