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Universal size and shape of viscous capillary jets: application to gas-focused microjets

  • A. M. GAÑÁN-CALVO (a1), C. FERRERA (a1) (a2) and J. M. MONTANERO (a3)


The size and shape of capillary microjets are analysed theoretically and experimentally. We focus on the particular case of gas-focused viscous microjets, which are shaped by both the pressure drop in the axial direction occurring in front of the discharge orifice, and the tangential viscous stress caused by the difference between the velocities of the co-flowing gas stream and liquid jet behind the orifice. The momentum equation obtained from the slender approximation reveals that the momentum injected into the jet in these two regions is proportional to the ratio of the pressure drop to the orifice diameter. Thus, the liquid-driving forces can be reduced to a single term in the momentum equation. Besides, the size and shape of gas-focused microjets were experimentally measured. The experiments indicated that the Weber number has a minor influence on the jet diameter for steady, stable jets, while both the axial coordinate and the Reynolds number affect its size significantly. When the experimental results are expressed in terms of conveniently scaled variables, one obtains a remarkable collapse of all measured jet diameters into a single curve. The curve matches a universal self-similar solution of the momentum equation for a constant driving force, first calculated by Clarke (Mathematika, vol. 12, 1966, p. 51) and not yet exploited in the field of steady tip-streaming flows, such as flow focusing and electrospray. This result shows that the driving force or motor mentioned above attains a rather homogeneous value at the region where the gas-focused microjet develops. The approach used in this work can also be applied to study other varied microjet generation means (e.g. co-flowing, electrospray and electrospinning).


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Barenblatt, G. I. 2003 Scaling. Cambridge University Press.
Basaran, O. A. 2002 Small-scale free surface flows with breakup: drop formation and emerging applications. AlChE J. 48, 18421848.
Cabezas, M. G., Bateni, A., Montanero, J. M. & Neumann, A. W. 2006 Determination of surface tension and contact angle from the shapes of axisymmetric fluid interfaces without use of apex coordinates. Langmuir 22, 1005310060.
Clarke, N. S. 1966 A differential equation in fluid mechanics. Mathematika 12, 5153.
Clarke, N. S. 1968 Two-dimensional flow under gravity in a jet of viscous liquid. J. Fluid Mech. 31, 481500.
Dzenis, Y. 2004 Spinning continuous fibers for nanotechnology. Science 304, 19171919.
Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865929.
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 179.
Ferrera, C., Cabezas, M. G. & Montanero, J. M. 2006 An experimental analysis of the linear vibration of axisymmetric liquid bridges. Phys. Fluids 18, 082105.
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. 1999 The surface charge in electrospraying: its nature and its universal scaling laws. J. Aero. Sci. 30, 863872.
Gañán-Calvo, A. M. 2008 Unconditional jetting. Phys. Rev. E 78, 026304.
Gañán-Calvo, A. M. & Montanero, J. M. 2009 Revision of capillary cone-jet physics: electrospray and flow focusing. Phys. Rev. E 79, 066305.
Gañán-Calvo, A. M., Pérez-Saborid, M., López-Herrera, J. M. & Gordillo, J. M. 2004 Steady high viscosity liquid micro-jet production and fiber spinning using co-flowing gas conformation. Eur. Phys. J. B 39, 131137.
Glauert, M. B. & Lighthill, M. J. 1955 The axisymmetric boundary layer on a long thin cylinder. Proc. R. Soc. Lond. A 230, 188203.
Gordillo, J. M., Pérez-Saborid, M. & Gañán-Calvo, A. M. 2001 Linear stability of co-flowing liquid-gas jets. J. Fluid Mech. 448, 2351.
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.
Higuera, F. J. 2003 Flow rate and electric current emitted by a Taylor cone. J. Fluid Mech. 484, 303327.
Lasheras, J. C. & Hopfinger, E. J. 2000 Liquid jet instability and atomization in a coaxial gas stream. Annu. Rev. Fluid Mech. 32, 275308.
Martín-Banderas, L., Rodríguez-Gil, A., Cebolla, A., Chávez, S., Berdún-Álvarez, T., Fernandez-Garcia, J. M., Flores-Mosquera, M. & Gañán-Calvo, A. M. 2006 Towards high-throughput production of uniformly encoded microparticles. Adv. Mater. 18, 559564.
de la Mora, J. F. 2007 The fluid dynamics of Taylor cones. Annu. Rev. Fluid Mech. 39, 217243.
Morse, P. M. & Feshback, H. 1953 Methods of Classical Physics. McGraw-Hill.
Rosell-Llompart, J. & Gañán-Calvo, A. M. 2008 Turbulence in pneumatic flow focusing and flow blurring regimes. Phys. Rev. E 77, 036321.
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.
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26, 65102.
Zeleny, J. 1917 Instability of electrified liquid surfaces. Phys. Rev. 10, 16.
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Universal size and shape of viscous capillary jets: application to gas-focused microjets

  • A. M. GAÑÁN-CALVO (a1), C. FERRERA (a1) (a2) and J. M. MONTANERO (a3)


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