Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-23T10:50:07.626Z Has data issue: false hasContentIssue false

Local interfacial stability near a zero vorticity point

Published online by Cambridge University Press:  30 June 2015

Yu-Hau Tseng
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
Andrea Prosperetti*
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA Faculty of Science and Technology and Burgerscentrum, University of Twente, AE 7500 Enschede, The Netherlands
Email address for correspondence:


It is often observed that small drops or bubbles detach from the interface separating two co-flowing immiscible fluids. The size of these drops or bubbles can be orders of magnitude smaller than the length scales of the parent fluid mass. Examples are tip-streaming from drops or coaxial jets in microfluidics, selective withdrawal, ‘skirt’ formation around bubbles or drops, and others. It is argued that these phenomena are all reducible to a common instability that can occur due to a local convergence of streamlines in the neighbourhood of a zero-vorticity point or line on the interface. When surfactants are present, this converging flow tends to concentrate them in these regions weakening the effect of surface tension, which is the only mechanism opposing the instability. Several analytical and numerical calculations are presented to substantiate this interpretation of the phenomenon. In addition to some idealized cases, the results of two-dimensional simulations of co-flowing jets and a rising drop are presented.

© 2015 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.)


Present address: Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung City 81148, Taiwan, ROC.


Adami, S., Hu, X. Y. & Adams, N. A.2010 Tipstreaming of a drop in simple shear flow in the presence of surfactant. arXiv:1010.3646.Google Scholar
Anna, S. L., Bontoux, N. & Stone, H. A. 2003 Formation of dispersions using flow focusing in microchannels. Appl. Phys. Lett. 82, 364366.Google Scholar
Barrero, A. & Loscertales, I. G. 2007 Micro- and nanoparticles via capillary flows. Annu. Rev. Fluid Mech. 39, 89106.Google Scholar
Basaran, O. A. 2002 Small-scale free surface flows with breakup: drop formation and emerging applications. AIChE J. 48, 18421848.Google Scholar
Bhaga, D. & Weber, M. E. 1981 Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105, 6185.Google Scholar
Blanchette, F. & Zhang, W. W. 2009 Force balance at the transition from selective withdrawal to viscous entrainment. Phys. Rev. Lett. 102, 144501.Google Scholar
de Bruijn, R. A. 1993 Tipstreaming of drops in simple shear flows. Chem. Engng Sci. 48, 277284.Google Scholar
Cohen, I. & Nagel, S. R. 2002 Scaling at the selective withdrawal transition through a tube suspended above the fluid surface. Phys. Rev. Lett. 88, 074501.CrossRefGoogle ScholarPubMed
Collins, R. T., Krishnaraj, S., Harris, M. T. & Basaran, O. A. 2013 Universal scaling laws for the disintegration of electrified drops. Proc. Natl Acad. Sci. USA 110, 49054910.Google Scholar
Eggleton, C. D. & Stebe, K. J. 1998 An adsorption–desorption-controlled surfactant on a deforming droplet. J. Colloid Interface Sci. 208, 6880.CrossRefGoogle ScholarPubMed
Eggleton, C. D., Tsai, T.-M. & Stebe, K. J. 2001 Tip streaming from a drop in the presence of surfactants. Phys. Rev. Lett. 87, 048302.Google Scholar
Frankel, I. & Weihs, D. 1985 Stability of a capillary jet with linearly increasing axial velocity (with application to shaped charges). J. Fluid Mech. 155, 289307.CrossRefGoogle Scholar
Gopalan, B. & Katz, J. 2010 Turbulent shearing of crude oil mixed with dispersants generates long microthreads and microdroplets. Phys. Rev. Lett. 104, 054501.Google Scholar
Gordillo, J. M., Sevilla, A. & Campo-Cortes, F. 2014 Global stability of stretched jets: conditions for the generation of monodisperse micro-emulsions using coflows. J. Fluid Mech. 738, 335357.Google Scholar
Hao, Y. & Prosperetti, A. 1999 The effect of viscosity on the spherical stability of oscillating gas bubbles. Phys. Fluids 11, 13091317.CrossRefGoogle Scholar
Jeong, W. C., Lim, J. M., Choi, J. H., Kim, J. H., Lee, Y. J., Kim, S. H., Lee, G., Kim, J. D., Yi, G. R. & Yang, S. M. 2012 Controlled generation of submicron emulsion droplets via highly stable tip-streaming mode in microfluidic devices. Lab on a Chip 12, 14461453.Google Scholar
Lai, M.-C., Tseng, Y.-H. & Huang, H. 2008 An immersed boundary method for interfacial flows with insoluble surfactant. J. Comput. Phys. 227, 72797293.Google Scholar
Lu, X. Z. & Prosperetti, A. 2009 Numerical study of Taylor bubbles. Ind. Engng Chem. Res. 48, 242252.Google Scholar
Marín, A. G., Campo-Cortés, F. & Gordillo, J. M. 2009 Generation of micron-size drops and bubbles through viscous coflows. Colloids Surf. A 344, 27.Google Scholar
Mulligan, M. K. & Rothstein, J. P. 2011 The effect of confinement-induced shear on drop deformation and breakup in microfluidic extensional flows. Phys. Fluids 23, 022004.CrossRefGoogle Scholar
OSCA, Oil Spill Commission Action, 2011 The use of surface and subsea dispersants during the BP Deepwater Horizon oil spill. Available at: Scholar
Peskin, C. S. 2002 The immersed boundary method. Acta Numerica 11, 479517.Google Scholar
Plesset, M. S. 1954 On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25, 9698.CrossRefGoogle Scholar
Sherwood, J. D. 1984 Tip streaming from slender drops in a nonlinear extensional flow. J. Fluid Mech. 144, 281295.Google Scholar
Stone, H. A. 1990 A simple derivation of the time-dependent convective diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2, 111112.Google Scholar
Suryo, R. & Basaran, O. A. 2006 Tip streaming from a liquid drop forming from a tube in a co-flowing outer fluid. Phys. Fluids 18, 082102.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146, 501523.Google Scholar
Thomson, J. J. & Newall, H. F. 1885 On the formation of vortex rings by drops falling into liquids, and some allied phenomena. Proc. R. Soc. Lond. A 29, 417436.Google Scholar
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulations of Gas–Liquid Multiphase Flows. Cambridge University Press.Google Scholar