Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T07:13:03.271Z Has data issue: false hasContentIssue false
  • English
  • Français

Onset of motion of a three-dimensional droplet on a wall in shear flow at moderate Reynolds numbers

Published online by Cambridge University Press:  06 March 2008

HANG DING  and
PETER D. M. SPELT
HANG DING
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
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the critical conditions for the onset of motion of a three-dimensional droplet on a wall in shear flows at moderate Reynolds number. A diffuse-interface method is used for this purpose, which also circumvents the stress singularity at the moving contact line, and the method allows for a density and viscosity contrast between the fluids. Contact-angle hysteresis is represented by the prescription of a receding contact angle θR and an advancing contact angle value θA. Critical conditions are determined by tracking the motion and deformation of a droplet (initially a spherical cap with a uniform contact angle θ0). At sufficiently low values of a Weber number, We (based on the applied shear rate and the drop volume), the drop deforms and translates for some time, but subsequently reaches a stationary position and attains a steady-state shape. At sufficiently large values of We no such steady state is found. We present results for the critical value of We as a function of Reynolds number Re for cases with the initial value of the contact angle θ0=θR as well as for θ0=θA. A scaling argument based on a force balance on the drop is shown to represent the results very accurately. Results are also presented for the static shape, transient motion and flow structure at criticality. It is shown that at low Re our results agree (with some qualifications) with those of Dimitrakopoulos & Higdon (1998, J. Fluid Mech. vol. 377, p. 189). Overall, the results indicate that the critical value of We is affected significantly by inertial effects at moderate Reynolds numbers, whereas the steady shape of droplets still shows some resemblance to that obtained previously for creeping flow conditions. The paper concludes with an investigation into the complex structure of a steady wake behind the droplet and the occurrence of a stagnation point at the upstream side of the droplet.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 599 , 25 March 2008 , pp. 341 - 362
Copyright
Copyright © Cambridge University Press 2008

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

Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165.CrossRefGoogle Scholar
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
Berejnov, V. & Thorne, R. E. 2007 Effect of transient pinning on stability of drops sitting on an inclined plane. Phys. Rev.E 75, 066308.Google Scholar
Cao, J., Donell, B., Deaver, D. R., Lawrence, M. B. & Dong, C. 1998 In vitro side-view imaging technique and analysis of human t-leukemic cell adhesion to ICAM-1 in Microvasc. Res. 55 124–137.CrossRefGoogle ScholarPubMed
Dimitrakopoulos, P. 2007 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. & 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
Dimitrakopoulos, P. & Higdon, J. J. L. 1999 On the gravitational displacement of three-dimensional fluid droplets from inclined solid surfaces in low-Reynolds-number shear flows. J. Fluid Mech. 395, 181–209.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.Google ScholarPubMed
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.Google Scholar
Dupont, J.-B. Legendre, D. & Morgante, A. M. 2007 Numerical simulation of static or sliding drop with contact-line hysteresis. In Proc. 6th Intl Conf. Multiphase Flow, Leipzig, Germany, paper S4 Wed C37.Google Scholar
Durbin, P. A. 1988 On the wind force needed to dislodge a drop adhered to a surface J. Fluid Mech. 196, 205–222.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
Feng, J. Q. & Basaran, O. A. 1994 Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall. J. Fluid Mech. 275, 351–378.CrossRefGoogle Scholar
de Gennes, P. G. 1985 Wetting: Statistics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
Ha, J. W., Park, I. J. & Lee, S. B. 2005 Hydrophobicity and sliding behavior of liquid droplets on the fluorinated latex films. Macromolecules 38, 736–744.CrossRefGoogle Scholar
Hocking, L. M. 1981 Sliding and spreading of thin two-dimensional drops. Q. J. Mech. Appl. Maths 24, 37–55.Google 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
Hohenberg, P. C. & Halperin, B. I. 1977 Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479CrossRefGoogle Scholar
Hwang, J. Y. & Yang, K. S. 2004 Numerical study of vortical structures around a wall-mounted cubic obstacle in channel flow. Phys. Fluids 16, 2382–2394.CrossRefGoogle Scholar
Hyväluoma, J., Koponen, A., Raiskinmäki, P. & Timonen, J. 2007 Droplets on inclined rough surfaces. Eur. Phys. J. E 23, 289–293.CrossRefGoogle ScholarPubMed
Jacqmin, D. 1999 Calculation of two-phase Navier-Stokes flows using phase-field modelling. J. Comput. Phys. 155, 96–127.CrossRefGoogle Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57.CrossRefGoogle Scholar
Joanny, J. F. & Robbins, M. O. 1990 Motion of a contact line on a heterogeneous surface. J. Chem. Phys. 92, 3206–3212.Google Scholar
Jones, J. L., Lal, M., Ruddock, J. N. & Spenley, N. A. 1999 Dynamics of a drop at a liquid/solid interface in simple shear fields: A mesoscopic simulation study. Faraday Discuss. 112, 129–142.CrossRefGoogle Scholar
Li, X. & Pozrikidis, C. 1996 Shear flow over a liquid drop adhering to a solid surface. J. Fluid Mech. 307, 167–190.CrossRefGoogle Scholar
Liu, X. D., Osher, S. & Chan, T. 1994 Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212.CrossRefGoogle Scholar
Mahé, M., Vignes-Adler, M., Rousseau, A., Jacquin, C. G. & Adler, P. M. 1988 Adhesion of droplets on a solid wall and detachment by a shear flow. J. Colloid Interface Sci. 126, 314–328.CrossRefGoogle Scholar
Marsh, J. A., Garoff, S. & Dussan, V. E. B. 1993 Dynamic contact angles and hydrodynamics near a moving contact line. Phys. Rev. Lett. 70, 2778–2781.CrossRefGoogle Scholar
Nikolayev, V. S. 2005 Dynamics and depinning of the triple contact linein the presence of surface defects. J. Phys.: Condens. Matt. 17, 2111–2117.Google Scholar
Quéré, D., Azzopardi, M. J. & Delattre, L. 1998 Drops at rest on a tilted plane. Langmuir 14, 2213–2216.CrossRefGoogle Scholar
Seppecher, P. 1996 Moving contact lines in the Cahn-Hilliard theory. Intl J. Engng Sci. 34, 977–992.CrossRefGoogle Scholar
Spelt, P. D. M. 2005 A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. J. Comput. Phys. 207, 389–404.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. 2007 Linear shear flow past a hemispherical droplet adhering to a solid surface. J. Comput. Phys. 207, 389–404.Google Scholar
Theodorakakos, A., Ous, T., Gavaises, M., Nouri, J. M., Nikolopoulos, N. & Yanagihara, H. 2006 Dynamics of water droplets detached from porous surfaces of relevance to PEM fuel cell. J. Colloid Interface Sci. 300, 673–687.CrossRefGoogle Scholar
Thiele, U. & Knobloch, E. 2006 On the depinning of a driven drop on a heterogeneous substrate. New J. Phys. 8, 1–37.Google Scholar
Thoreau, V., Malki, B., Berthome, G., Boulange-Petermann, L. & Joud, J. C. 2006 Physico-chemical and dynamic study of oil-drop removal from bare and coated stainless-steel surfaces. J. Adhesion Sci. Technol. 20, 1819–1831.Google Scholar
Zhang, J., Miksis, M. J. & Bankoff, S. G. 2006 Nonlinear dynamics of a two-dimensional viscous drop under shear flow. Phys. Fluids 18, 072106.CrossRefGoogle Scholar