Skip to main content Accessibility help

Global stability of axisymmetric flow focusing

  • F. Cruz-Mazo (a1), M. A. Herrada (a1), A. M. Gañán-Calvo (a1) and J. M. Montanero (a2)


In this paper, we analyse numerically the stability of the steady jetting regime of gaseous flow focusing. The base flows are calculated by solving the full Navier–Stokes equations and boundary conditions for a wide range of liquid viscosities and gas speeds. The axisymmetric modes characterizing the asymptotic stability of those flows are obtained from the linearized Navier–Stokes equations and boundary conditions. We determine the flow rates leading to marginally stable states, and compare them with the experimental values at the jetting-to-dripping transition. The theoretical predictions satisfactorily agree with the experimental results for large gas speeds. However, they do not capture the trend of the jetting-to-dripping transition curve for small gas velocities, and considerably underestimate the minimum flow rate in this case. To explain this discrepancy, the Navier–Stokes equations are integrated over time after introducing a small perturbation in the tapering liquid meniscus. There is a transient growth of the perturbation before the asymptotic exponential regime is reached, which leads to dripping. Our work shows that flow focusing stability can be explained in terms of the combination of asymptotic global stability and the system short-term response for large and small gas velocities, respectively.


Corresponding author

Email address for correspondence:


Hide All
Acero, A. J., Ferrera, C., Montanero, J. M. & Gañán-Calvo, A. M. 2012 Focusing liquid microjets with nozzles. J. Micromech. Microengng 22, 065011.
Chapman, H. N., Fromme, P., Barty, A., White, T. A., Kirian, R. A., Aquila, A., Hunter, M. S., Schulz, J., DePonte, D. P., Weierstall, U. et al. 2011 Femtosecond x-ray protein nanocrystallography. Nature 470, 7379.
Chomaz, J. 2005 Global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 37, 357392.
DePonte, D. P., Weierstall, U., Schmidt, K., Warner, J., Starodub, D., Spence, J. C. H. & Doak, R. B. 2008 Gas dynamic virtual nozzle for generation of microscopic droplet streams. J. Phys. D: Appl. Phys. 41, 195505.
Dizes, S. L. 1997 Global modes in falling capillary jets. Eur. J. Mech. (B/Fluids) 16, 761778.
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.
Forbes, T. P. & Sisco, E. 2014 Chemical imaging of artificial fingerprints by desorption electro-flow focusing ionization mass spectrometry. Analyst 139, 2982.
Gañán-Calvo, A. M. 1998 Generation of steady liquid microthreads and micron-sized monodisperse sprays in gas streams. Phys. Rev. Lett. 80, 285288.
Gañán-Calvo, A. M., Montanero, J. M., Martín-Banderas, L. & Flores-Mosquera, M. 2013 Building functional materials for health care and pharmacy from microfluidic principles and Flow Focusing. Adv. Drug Deliv. Rev. 65, 14471469.
García, F. & González, H. 2008 Normal-mode linear analysis and initial conditions of capillary jets. J. Fluid Mech. 602, 81117.
González, H. & García, F. 2009 The measurement of growth rates in capillary jets. J. Fluid Mech. 619, 179212.
Gordillo, J. M., Sevilla, A. & Campo-Cortés, F. 2014 Global stability of stretched jets: conditions for the generation of monodisperse micro-emulsions using coflows. J. Fluid Mech. 738, 335357.
Herrada, M. A., Gañán-Calvo, A. M., Ojeda-Monge, A., Bluth, B. & Riesco-Chueca, P. 2008 Liquid flow focused by a gas: Jetting, dripping, and recirculation. Phys. Rev. E 78, 036323.
Herrada, M. A. & Montanero, J. M. 2016 A numerical method to study the dynamics of capillary fluid systems. J. Comput. Phys. 306, 137147.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilites in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Korczyk, P. M., Cybulski, O., Makulskaa, S. & Garstecki, P. 2011 Effects of unsteadiness of the rates of flow on the dynamics of formation of droplets in microfluidic systems. Lab on a Chip 11, 173175.
Lakkis, J. M. 2016 Encapsulation and Controlled Release Technologies in Food Systems. Wiley.
Leib, S. J. & Goldstein, M. E. 1986 Convective and absolute instability of a viscous liquid jet. Phys. Fluids 29, 952954.
de Luca, L. 1999 Experimental investigation of the global instability of plane sheet flows. J. Fluid Mech. 399, 355376.
de Luca, L., Costa, M. & Caramiello, C. 2002 Energy growth of initial perturbations in two-dimensional gravitational jets. Phys. Fluids 14, 289299.
Montanero, J. M., Rebollo-Muñoz, N., Herrada, M. A. & Gañán-Calvo, A. M. 2011 Global stability of the focusing effect of fluid jet flows. Phys. Rev. E 83, 036309.
Rayleigh, J. W. S. 1892 On the instability of a cylinder of viscous liquid under capillary force. Phil. Mag. 35, 145155.
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, 471483.
Sauter, U. S. & Buggisch, H. W. 2005 Stability of initially slow viscous jets driven by gravity. J. Fluid Mech. 533, 237257.
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.
Si, T., Feng, H., Luo, X. & Xu, R. X. 2014 Formation of steady compound cone-jet modes and multilayered droplets in a tri-axial capillary flow focusing device. Microfluid Nanofluid 1, 111.
Si, T., Li, F., Yin, X.-Y. & Yin, X.-Z. 2009 Modes in flow focusing and instability of coaxial liquid–gas jets. J. Fluid Mech. 629, 123.
Tammisola, O., Lundell, F. & Soderberg, L. D. 2012 Surface tension-induced global instability of planar jets and wakes. J. Fluid Mech. 713, 632658.
Tammisola, O., Sasaki, A., Lundell, F., Matsubara, M. & Soderberg, L. D. 2011 Stabilizing effect of surrounding gas flow on a plane liquid sheet. J. Fluid Mech. 672, 532.
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319352.
Trebbin, M., Kruger, K., DePonte, D., Roth, S. V., Chapman, H. N. & Forster, S. 2014 Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions. Lab on a Chip 14, 17331745.
Vega, E. J., Montanero, J. M., Herrada, M. A. & Gañán-Calvo, A. M. 2010 Global and local instability of flow focusing: the influence of the geometry. Phys. Fluids 22, 064105.
Yakubenko, P. A. 1997 Global capillary instability of an inclined jet. J. Fluid Mech. 346, 181200.
Zhuab, P. & Wang, L. 2017 Passive and active droplet generation with microfluidics: a review. Lab on a Chip 17, 3475.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed