Skip to main content Accessibility help

The steady cone-jet mode of electrospraying close to the minimum volume stability limit

  • A. Ponce-Torres (a1), N. Rebollo-Muñoz (a1), M. A. Herrada (a2), A. M. Gañán-Calvo (a2) and J. M. Montanero (a1)...


We study both numerically and experimentally the steady cone-jet mode of electrospraying close to the stability limit of minimum flow rate. The leaky dielectric model is solved for arbitrary values of the relative permittivity and the electrohydrodynamic Reynolds number. The linear stability analysis of the base flows is conducted by calculating their global eigenmodes. The minimum flow rate is determined as that for which the growth factor of the dominant mode becomes positive. We find a good agreement between this theoretical prediction and experimental values. The analysis of the spatial structure of the dominant perturbation may suggest that instability originates in the cone-jet transition region, which shows the local character of the cone-jet mode. The electric relaxation time is considerably smaller than the residence time of a fluid particle in the cone-jet transition region (defined as the region where the surface and bulk intensities are of the same order of magnitude) except for the high-polarity case, where these characteristic times are commensurate with each other. The superficial charge is not relaxed within the cone-jet transition region except for the high-viscosity case, because significant inner electric fields arise in the cone-jet transition region. However, those electric fields are not large enough to invalidate the scaling laws that do not take them into account. Viscosity and polarization forces compete against the driving electric shear stress in the cone-jet transition region for small Reynolds numbers and large relative permittivities, respectively. Capillary forces may also play a significant role in the minimum flow rate stability limit. The experiments show the noticeable stabilizing effect of the feeding capillary for diameters even two orders of magnitude larger than that of the jet. Stable jets with electrification levels higher than the Rayleigh limit are produced. During the jet break-up, two consecutive liquid blobs may coalesce and form a bigger emitted droplet, probably due to the jet acceleration. The size of droplets exceeds Rayleigh’s prediction owing to the stabilizing effect of both the axial electric field and viscosity.


Corresponding author

Email address for correspondence:


Hide All
Ambravaneswaran, B., Subramani, H. J., Phillips, S. D. & Basaran, O. A. 2004 Dripping-jetting transitions in a dripping faucet. Phys. Rev. Lett. 93, 034501.
Banerjee, S. & Mazumdar, S. 2012 Electrospray ionization mass spectrometry: a technique to access the information beyond the molecular weight of the analyte. Intl J. Anal. Chem. 2012, 140.
Barenblatt, G. I. 2003 Scaling. Cambridge University Press.
Carlier, J., Arscott, S., Camart, J.-C., Cren-Olivé, C. & Gac, S. L. 2005 Integrated microfabricated systems including a purification module and an on-chip nano electrospray ionization interface for biological analysis. J. Chromatogr. A 1071, 213222.
Castrejón-Pita, A. A., Castrejón-Pita, J. R. & Hutchings, I. M. 2012 Breakup of liquid filaments. Phys. Rev. Lett. 108, 074506.
Cherney, L. T. 1999 Structure of Taylor cone-jets: limit of low flow rates. J. Fluid Mech. 378, 167196.
Cloupeau, M. & Prunet-Foch, B. 1989 Electrostatic spraying of liquids in cone-jet mode. J. Electrostat. 22, 135159.
Cruz-Mazo, F., Herrada, M. A., Gañán-Calvo, A. M. & Montanero, J. M. 2017 Global stability of axisymmetric flow focusing. J. Fluid Mech. 832, 329344.
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.
Fernández de la Mora, J. 2007 The fluid dynamics of Taylor cones. Annu. Rev. Fluid Mech. 39, 217243.
Fernandez de la Mora, J. & Loscertales, I. G. 1994 The current transmitted through an electrified conical meniscus. J. Fluid Mech. 260, 155184.
Gamero-Castaño, M. 2010 Energy dissipation in electrosprays and the geometric scaling of the transition region of cone-jets. J. Fluid Mech. 662, 493513.
Gamero-Castaño, M. & Hruby, V. 2001 Electrospray as a source of nanoparticles for efficient colloid thrusters. J. Propul. Power 17, 977987.
Gañán-Calvo, A. M. 1999 The surface charge in electrospraying: its nature and its universal scaling laws. J. Aero. Sci. 30, 863872.
Gañán-Calvo, A. M. 2004 On the general scaling theory for electrospraying. J. Fluid Mech. 507, 203212.
Gañán-Calvo, A. M., Barrero, A. & Pantano, C. 1993 The electrohydrodynamics of electrified conical menisci. J. Aero. Sci. 24, S19S20.
Gañán-Calvo, A. M., Lasheras, J. C., Dávila, J. & Barrero, A. 1994 The electrostatic spray emitted from an electrified conical meniscus. J. Aero. Sci. 25, 11211142.
Gañán-Calvo, A. M., Rebollo-Muñoz, N. & Montanero, J. M. 2013 Physical symmetries and scaling laws for the minimum or natural rate of flow and droplet size ejected by Taylor cone-jets. New J. Phys. 15, 033035.
Gilbert, W. 1600 De Magnete. Book 2 (translated by P. F. Mottelay), chap. 2, republished 1958. Dover.
Hartman, R. P. A., Brunner, D. J., Camelot, D. M. A., Marijnissen, J. C. M. & Scarlett, B. 2000 Jet break-up in electrohydrodynamic atomization in the cone-jet mode. J. Aero. Sci. 31, 6595.
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., López-Herrera, J. M., Gañán-Calvo, A. M., Vega, E. J., Montanero, J. M. & Popinet, S. 2012 Numerical simulation of electrospray in the cone-jet mode. Phys. Rev. E 86, 026305.
Herrada, M. A. & Montanero, J. M. 2016 A numerical method to study the dynamics of capillary fluid systems. J. Comput. Phys. 306, 137147.
Higuera, F. J. 2003 Flow rate and electric current emitted by a Taylor cone. J. Fluid Mech. 484, 303327.
Higuera, F. J. 2010 Numerical computation of the domain of operation of an electrospray of a very viscous liquid. J. Fluid Mech. 648, 3552.
Higuera, F. J. 2017 Qualitative analysis of the minimum flow rate of a cone-jet of a very polar liquid. J. Fluid Mech. 816, 428441.
Hoepffner, J. & Paré, G. 2013 Recoil of a liquid filament: escape from pinch-off through creation of a vortex ring. J. Fluid Mech. 734, 183197.
Hohman, M. M., Shin, M., Rutledge, G. & Brenner, M. P. 2001 Electrospinning and electrically forced jets. I. Stability theory. Phys. Fluids 13, 22012220.
Huberman, M. N., Beynon, J. C., Cohen, E., Goldin, D. S., Kidd, P. W. & Zafran, S. 1968 Present status of colloid microthruster technology. J. Spacecr. 5, 3191324.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilites in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Jaworek, A. 2007 Electrospray droplet sources for thin film deposition. J. Mater. Sci. 42, 266297.
Jaworek, A. & Krupa, A. 1999 Classification of the modes of EHD spraying. J. Aero. Sci. 30, 873893.
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81, 206229.
Leib, S. J. & Goldstein, M. E. 1986 Convective and absolute instability of a viscous liquid jet. Phys. Fluids 29, 952954.
López-Herrera, J. M., Gañán-Calvo, A. M. & Herrada, M. A. 2010 Absolute to convective instability transition in charged liquid jets. Phys. Fluids 22, 062002.
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111146.
Mestel, A. J. 1994 Electrohydrodynamic stability of a slightly viscous jet. J. Fluid Mech. 274, 93113.
Mestel, A. J. 1996 Electrohydrodynamic stability of a highly viscous jet. J. Fluid Mech. 312, 311326.
Rahmanpour, M., Ebrahimi, R. & Pourrajabian, A. 2017 Numerical simulation of two-phase electrohydrodynamic of stable Taylor cone-jet using a volume-of-fluid approach. J. Braz. Soc. Mech. Sci. Engng, doi:10.1007/s40430-017-0832-7.
Rayleigh, J. W. S. 1881 On the equilibrium of liquid conducting masses charged with electricity. Proc. R. Soc. Lond. A 5, 110112.
Rayleigh, L. 1878 On the instability of jets. Proc. Lond. Math. Soc. s1‐10, 413.
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 2764.
Scheideler, W. J. & Chena, C.-H. 2014 The minimum flow rate scaling of Taylor cone-jets issued from a nozzle. Appl. Phys. Lett. 104, 024103.
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.
Taylor, G. 1964 Disintegration of water drops in electric field. Proc. R. Soc. Lond. A 280, 383397.
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319352.
Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. A 150, 322337.
Tseng, Y.-H. & Prosperetti, A. 2015 Local interfacial stability near a zero vorticity point. J. Fluid Mech. 776, 536.
Vega, E. J., Montanero, J. M., Herrada, M. A. & Ferrera, C. 2014 Dynamics of an axisymmetric liquid bridge close to the minimum-volume stability limit. Phys. Rev. E 90, 013015.
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.
Xie, J., Jiang, J., Davoodi, P., Srinivasan, M. P. & Wang, C. 2015 Electrohydrodynamic atomization: a two-decade effort to produce and process micro-/nanoparticulate materials. Chem. Engng Sci. 125, 3257.
Yamashita, M. & Fenn, J. B. 1984 Electrospray ion source. Another variation on the free-jet theme. J. Phys. Chem. 88 (20), 44514459.
Yan, F., Farouk, B. & Ko, F. 2003 Numerical modeling of an electrostatically driven liquid meniscus in the cone-jet mode. J. Aero. Sci. 34, 99116.
Yang, W., Duan, H., Li, C. & Deng, W. 2014 Crossover of varicose and whipping instabilities in electrified microjets. Phys. Rev. Lett. 112, 054501.
Yuill, E. M., Ray, N., Saand, S. J., Hieftje, G. M. & Baker, L. A. 2013 Electrospray ionization from nanopipette emitters with tip diameters of less than 100 nm. Anal. Chem. 85, 84988502.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

The steady cone-jet mode of electrospraying close to the minimum volume stability limit

  • A. Ponce-Torres (a1), N. Rebollo-Muñoz (a1), M. A. Herrada (a2), A. M. Gañán-Calvo (a2) and J. M. Montanero (a1)...


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.