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Global stability of stretched jets: conditions for the generation of monodisperse micro-emulsions using coflows

Published online by Cambridge University Press:  05 December 2013

J. M. Gordillo ,
A. Sevilla  and
F. Campo-Cortés
J. M. Gordillo*
Affiliation:
Área de Mecánica de Fluidos, Departamento de Ingenería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos s/n, 41092 Sevilla, Spain
A. Sevilla*
Affiliation:
Área de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos. Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
F. Campo-Cortés
Affiliation:
Área de Mecánica de Fluidos, Departamento de Ingenería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos s/n, 41092 Sevilla, Spain
*
†Email addresses for correspondence: jgordill@us.es, asevilla@ing.uc3m.es
†Email addresses for correspondence: jgordill@us.es, asevilla@ing.uc3m.es
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Abstract

In this paper we reveal the physics underlying the conditions needed for the generation of emulsions composed of uniformly sized drops of micrometric or submicrometric diameters when two immiscible streams flow in parallel under the so-called tip streaming regime after Suryo and Basaran (Phys. Fluids, vol. 18, 2006, 082102). Indeed, when inertial effects in both liquid streams are negligible, the inner to outer flow-rate and viscosity ratios are small enough and the capillary number is above an experimentally determined threshold which is predicted by our theoretical results with small relative errors, a steady micrometre-sized jet is issued from the apex of a conical drop. Under these conditions, the jet disintegrates into drops with a very well-defined mean diameter, giving rise to a monodisperse microemulsion. Here, we demonstrate that the regime in which uniformly sized drops are produced corresponds to values of the capillary number for which the cone-jet system is globally stable. Interestingly enough, our general stability theory reveals that liquid jets with a cone-jet structure are much more stable than their cylindrical counterparts thanks, mostly, to a capillary stabilization mechanism described here for the first time. Our findings also limit the validity of the type of stability analysis based on the common parallel flow assumption to only those situations in which the liquid jet diameter is almost constant.

JFM classification

Instability: Absolute/convective instability Drops and Bubbles: Drops Low-Reynolds-number flows: Slender-body theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 738 , 10 January 2014 , pp. 335 - 357
Copyright
©2013 Cambridge University Press 

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References

Acrivos, A. & Lo, T. S. 1977 Deformation and breakup of a single slender drop in an extensional flow. J. Fluid Mech. 86, 641–672.CrossRefGoogle Scholar
Ambravaneswaran, B., Subramani, H. J., Phillips, S. D. & Basaran, O. A. 2004 Dripping–jetting transitions in a dripping faucet. Phys. Rev. Lett. 93, 034501.CrossRefGoogle Scholar
Ambravaneswaran, B., Wilkes, E. D. & Basaran, O. A. 2002 Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops. Phys. Fluids 14, 2606–2621.CrossRefGoogle Scholar
Anna, S. L., Bontoux, N. & Stone, H. A. 2003 Formation of dispersions using flow focusing in microchannels. Appl. Phys. Lett. 82, 364–366.CrossRefGoogle Scholar
Ashley, H. & Landahl, M. 1965 Aerodynamics of Wings and Bodies. Adison-Wesley.Google Scholar
Barrero, A. & Loscertales, I. G. 2007 Micro- and nanoparticles via capillary flows. Annu. Rev. Fluid Mech. 39, 89–106.CrossRefGoogle Scholar
Basaran, O. A. 2002 Small-scale free surface flows with breakup: drop formation and emerging applications. AIChE J. 48, 1842–1848.CrossRefGoogle Scholar
Buckmaster, J. D. 1971 Pointed bubbles in slow viscous flow. J. Fluid Mech. 55, 385–400.CrossRefGoogle Scholar
Castro-Hernández, E., Campo-Cortés, F. & Gordillo, J. M. 2012 Slender-body theory for the generation of micrometre-sized emulsions through tip streaming. J. Fluid Mech. 698, 423–445.CrossRefGoogle Scholar
Christodoulou, K. N. & Scriven, L. N. 1992 Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. 99 (4), 39–55.CrossRefGoogle Scholar
Clanet, C. & Lasheras, J. C. 1999 Transition from dripping to jetting. J. Fluid Mech. 383, 307–326.CrossRefGoogle Scholar
Collins, R. T., Jones, J. J., Harris, M. T. & Basaran, O. A. 2008 Electrohydrodynamic tip streaming and emission of charged drops from liquid cones. Nat. Phys. 4 (4), 149–154.CrossRefGoogle Scholar
Collins, R. T., Sambath, K., Harris, M. T. & Basaran, O. A. 2013 Universal scaling laws for the didintegration of electrified drops. Proc. Acad. Sci. Amer. 110 (13), 4905–4910.CrossRefGoogle ScholarPubMed
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–222.CrossRefGoogle Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.CrossRefGoogle Scholar
Fernandez de la Mora, J. & Loscertales, I. G. 1994 The current emitted by highly conducting Taylor cones. J. Fluid Mech. 260, 155–184.CrossRefGoogle 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, 289–307.CrossRefGoogle Scholar
Gañán Calvo, A. M. 2008 Unconditional jetting. Phys. Rev. E 78, 026304.CrossRefGoogle ScholarPubMed
García, F. J. & Castellanos, A. 1994 One-dimensional models for slender axisymmetric viscous liquid jets. Phys. Fluids 6 (8), 2676–2689.CrossRefGoogle Scholar
Gordillo, J. M., Gañán Calvo, A. M. & Pérez-Saborid, M. 2001 Monodisperse microbubbling: absolute instabilities in coflowing gas–liquid jets. Phys. Fluids 13, 3839–3842.CrossRefGoogle Scholar
Guillot, P., Colin, A., Utada, A. S. & Ajdari, A. 2007 Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number. Phys. Rev. Lett. 99, 104502.CrossRefGoogle Scholar
Hinch, E. J. & Acrivos, A. 1979 Steady long slender droplets in two-dimensional straining motion. J. Fluid Mech. 91, 401–414.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flows. Gordon & Breach.Google Scholar
Marín, A. G., Campo-Cortés, F. & Gordillo, J. M. 2009 Generation of micron-sized drops and bubbles through viscous coflows. Colloids Surf. A: Physicochem. Eng. Aspects 344, 2–7.CrossRefGoogle Scholar
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323–344.CrossRefGoogle Scholar
Powers, T. R. & Goldstein, R. E. 1997 Pearling and pinching: propagation of Rayleigh instabilities. Phys. Rev. Lett. 78, 2555–2558.CrossRefGoogle Scholar
Powers, T. R., Zhang, D. F., Goldstein, R. E. & Stone, H. A. 1998 Propagation of a topological transition: the Rayleigh instability. Phys. Fluids 10, 1052–1057.CrossRefGoogle Scholar
Rubio-Rubio, M., Sevilla, A. & Gordillo, J. M. 2013 On the thinnest steady threads obtained by gravitational stretching of capillary jets. J. Fluid Mech. 729, 471–483.CrossRefGoogle Scholar
Sevilla, A., Gordillo, J. M. & Martínez-Bazán, C. 2005 Transition from bubbling to jetting in a coaxial air–water jet. Phys. Fluids 17, 018105.CrossRefGoogle Scholar
Sherwood, J. D. 1983 Tip streaming from slender drops in a nonlinear extensional flow. J. Fluid Mech. 144, 281–295.CrossRefGoogle Scholar
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26, 65–102.CrossRefGoogle Scholar
Stone, H. A., Strook, A. D. & Adjari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.CrossRefGoogle 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.CrossRefGoogle Scholar
Taylor, G. I. 1964 Conical free surfaces and fluid interfaces. In Proceedings of the 11th International Congress of Applied Mechanics (Munich), pp. 790–796.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.CrossRefGoogle Scholar
Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. 150, 322–337.Google Scholar
Tomotika, S. 1936 Breaking up of a drop of viscous liquid immersed in another viscous fluid which is extending at a uniform rate. Proc. R. Soc. 153, 302–318.Google Scholar
Utada, A. S., Fernández-Nieves, A., Gordillo, J. M. & Weitz, D. 2008 Absolute instability of a liquid jet in a coflowing stream. Phys. Rev. Lett. 100, 014502.CrossRefGoogle Scholar
Utada, A. S., Fernández-Nieves, A., Stone, H. A. & Weitz, D. 2007 Dripping to jetting transitions in co-flowing liquid streams. Phys. Rev. Lett. 99, 094502.CrossRefGoogle Scholar
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21, 155–165.CrossRefGoogle Scholar
Zhang, W. W. 2004 Viscous entrainment from a nozzle: singular liquid spouts. Phys. Rev. Lett. 93, 184502.CrossRefGoogle ScholarPubMed

Gordillo Arias de Saavedra supplementary movie

Non regular drop formation process for $\lambda=0.01$ and Ca=0.9, with Ca

Download Gordillo Arias de Saavedra supplementary movie(Video)
Video 1.7 MB

Gordillo Arias de Saavedra supplementary movie

Drop formation process for $\lambda=0.01$ and Ca\simeq Ca^*. Observe that there are two well defined frequencies of drop formation.

Download Gordillo Arias de Saavedra supplementary movie(Video)
Video 1.9 MB

Gordillo Arias de Saavedra supplementary movie

Drop formation process for $\lambda=0.01$ and $Ca=1.12$, with $Ca>Ca^*$. Observe that the drops ejected are virtually identical.

Download Gordillo Arias de Saavedra supplementary movie(Video)
Video 1.6 MB