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Bifurcation of swirl in liquid cones

Published online by Cambridge University Press:  26 April 2006

Vladimir Shtern  and
Antonio Barrero
Vladimir Shtern
Affiliation:
Department of Energy Engineering and Fluid Mechanics, University of Seville, E-41012 Seville, Spain
Antonio Barrero
Affiliation:
Department of Energy Engineering and Fluid Mechanics, University of Seville, E-41012 Seville, Spain
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Abstract

We show that rotation appears owing to bifurcation in primarily pure meridional steady motion of viscous incompressible fluid. This manifestation of the laminar axisymmetric 'swirl dynamo’ occurs in flows inside liquid conical menisci with the cone half-angle θc < 90°. The liquid flows towards the cone apex near the surface and moves away along the axis driven by (i) surface shear stresses (typical for electrosprays) or (ii) by body Lorentz forces (e.g. in the process of cathode eruption). When the motion intensity increases and passes a critical value, new swirling regimes appear resulting from the supercritical pitchfork bifurcation. This agrees with recent observations of swirl in Taylor cones. We find that when the swirl Reynolds number Γc reaches a threshold value, flow separation occurs and the meridional motion becomes two-cellular with inflows near both the surface and the axis, and an outflow near the cone θ = θs, 0 < θs < θc. In the limit of high Γc, the angular thickness of the near-surface cell tends to zero. In case (i) the swirl is concentrated near the surface while the motion inside the inner cell becomes purely meridional with the radial velocity being uniform. We also study the two-phase flow of a liquid inside and a gas outside the meniscus. Flow separation occurs in both media and then swirl is concentrated near the interface. In case (ii) we reveal another interesting effect: a cascade of flow separations near the axis. As the driving forces increase, meridional motion becomes multi-cellular although very slow in comparison with swirl. To cover all ranges of parameters we combine numerical calculations and asymptotic analyses.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 300 , 10 October 1995 , pp. 169 - 205
Copyright
© 1995 Cambridge University Press

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References

Bailey, A. G. 1988 Electrostatic Spraying in Liquids John Wiley.
Barrero, A., Fernandez-feria, R., Gañan-Calvo, A. M. & Fernandez de la mora, J. 1995a The role of liquid viscosity and electrical conductivity on the motions inside Taylor cones. Manuscript in preparation.
Barrero, A., Gañan-Calvo, A. M. & Davila, J. 1995b Measurements of droplet size and charge in the electrospraying of liquids. I. Scaling laws for polar liquids. Aerosol Sci. Technol. (submitted).Google Scholar
Bojarevics, V., Freibergs, J. A., Shilova, E. I. & Shcherbinin, E. V. 1989 Electrically Induced Vortical Flows Kluwer.
Bojarevics, V., Sharamkin, V. L. & Shcherbinin, E. V. 1977 Effect of longitudinal magnetic field on the medium motion in electrical arc welding. Magnetohydrodynamics 13, 115–120.Google Scholar
Burrgraf, O. R. & Foster, M. R. 1977 Continuation or breakdown in tornado-like vortices. J. Fluid Mech. 80, 685–704.Google Scholar
Fenn, J. B., Mann, M., Meng, C. K., Wong, C. F. & Whitehouse, C. 1989 Electrospray ionization for mass spectrometry of large biomolecules. Science 246, 64–71.Google Scholar
Fernandez de la mora, J. 1992 The effect of charge emission from electrified liquid cones. J. Fluid Mech. 243, 561–574.Google Scholar
Fernandez de la mora, J., Fernandez-Feria, R. & Barrero, A. 1991 Theory of incompressible conical vortices at high Reynolds numbers, Bull. Am. Phys. Soc. 36, 2619.Google Scholar
Fernandez de la mora, J. & Loscertales, I. G. 1994 The current emitted by highly conducting Taylor cones. J. Fluid Mech. 260, 155–184.Google Scholar
Fernandez-Feria, R., Fernandez de la mora, J. & Barrero, A. 1995 Conically similar swirling flows at high Reynolds numbers flows. Part 1. One-cell solutions, high Reynolds number. SIAM J. Appl. Maths (submitted).Google Scholar
Funakoshi, M. & Inoue, S. 1988 Surface waves due to resonant horizontal oscillation. J. Fluid Mech. 192, 219–247.Google Scholar
Gañan-Calvo, A. M. 1995 The electrohydrodynamics of the cone-jet electrospraying. J. Fluid Mech. (submitted).Google Scholar
Goldshtik, M. A. 1960 A paradoxical solution of the Navier-Stokes equations. Appl. Mat. Mech. (Sov.) 24, 610–621.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1988 Conical flows of fluid with variable viscosity. Proc. R. Soc. Lond. A 419, 91–106.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1989 Self-similar hydromagnetic dynamo. Sov. Phys., J. Exp. Theor. Phys. 96, 1728–1743.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1990 Collapse in conical viscous flows. J. Fluid Mech. 218, 483–508.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1991 Bifurcation in near-star convection. Eur. J. Mech. B/Fluids 10 (n 2, Suppl., 49–54.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1993 Axisymmetric hydromagnetic dynamo. Prog. Astronaut. Aeronaut. 149, 87–102.Google Scholar
Goldshtik, M. A., Shtern, V. N. & Zhdanova, E. M. 1984 Onset of self-rotation in a submerged jet. Sov. Phys. Dokl. 277, 815–818.Google Scholar
Hayati, I., Bailey, A. I. & Tadros, Th. F. 1986a Mechanism of stable jet formation in electrohydrodynamic atomisation. Nature 319, 41–43.Google Scholar
Hayati, I., Bailey, A. & Tadros, Th. F. 1986b Investigations into the mechanism of electrohydrodynamic spraying of liquids. II Mechanism of stable jet formation and electrical forces acting on a liquid cone. J. Colloid Interface Sci. 117, 222–230.Google Scholar
Kawakubo, M., Tsuchia, Y., Sugaya, M. & Matsumura, K. 1978 Formation of a vortex around a sink: A kind of phase transition in a nonequilibrium open system. Phys. Lett. A 68, 65–66.Google Scholar
Long, R. R. 1961 A vortex in an infinite viscous fluid. J. Fluid Mech. 11, 611–624.Google Scholar
Lundquist, S. 1969 On the hydromagnetic viscous flow generated by a diverging electric current. Ark. Fys. 40, 85–95.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Narain, J. P. & Uberoi, M. S. 1971 Magnetohydrodynamics of conical flows. Phys. Fluids 25, 2687–2692.Google Scholar
Ogawa, A. 1993 Vortex flows, pp. 215–221. CRC Press.
Pantano, C., Gañan, A. M. & Barrero, A. 1994 Zerouth order, electrostatic solution for electrospraying in cone-jet mode. J. Aerosol Sci. (to appear).Google Scholar
Paull, R. & Pillow, A. F. 1985 Conically similar viscous flows. Part 3. Characterisation of axial causes in swirling flows and one-parameter flow generated by a uniform half-line source of kinematic swirl angular momentum. J. Fluid Mech. 155, 359–380.Google Scholar
Petrunin, A. A. & Shtern, V. N. 1993 Bifurcations in MHD flow generated by electric current discharge. Prog. Astronaut. Aeronaut. 149, 116–130.Google Scholar
Shcherbinin, E. V. 1969 On a kind of exact solution in the magnetohydrodynamics. Magnetohydrodynamics 5 (4), 46–58.Google Scholar
Shercliff, J. A. 1970 Fluid motion due to an electric current source. J. Fluid Mech. 40, 241–250.Google Scholar
Shtern, V. 1995 Cosmic jets as a pump for magnetic field. Phys. Lett. A (accepted).Google Scholar
Shtern, V. & Barrero, A. 1994 Striking features of fluid flows in Taylor cones related to electrosprays. J. Aerosol Sci. 25, 1049–1063.Google Scholar
Shtern, V. & Barrero, A. 1995 Instability nature of swirl appearance in liquid cones. Phys. Rev. E 52, 627–635.Google Scholar
Shtern, V., Goldshtik, M. & Hussain, F. 1994 Generation of swirl due to symmetry breaking. Phys. Rev. E 49, 2881–2886.Google Scholar
Shtern, V. & Hussain, F. 1993 Hysteresis in a swirling jet as a tornado model. Phys. Fluids A 5, 2183–2195.Google Scholar
Shtern, V. & Hussain, F. 1995 Hysteresis in swirling jets. J. Fluid Mech. (submitted).Google Scholar
Sozou, C. 1971 On fluid motion induced by electric current source. J. Fluid Mech. 49, 25–32.Google Scholar
Sozou, C. 1992 On solutions relating to conical vortices over a plane wall. J. Fluid Mech. 244, 633–644.Google Scholar
Sozou, C. & Pickering, W. M. 1976 Magnetohydrodynamic flow due to the discharge of an electric current in a hemispherical container. J. Fluid Mech. 73, 641–650.Google Scholar
Squire, B. 1952 Some viscous fluid flow problems. 1. Jet emerging from a hole in a plane wall. Phil. Mag. 43, 942–945.Google Scholar
Taylor, G. I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383–397.Google Scholar
Torrance, K. E. 1979 Natural convection in thermally stratified enclosures with localised heating from below. J. Fluid Mech. 95, 477–495.Google Scholar
Yatseev, V. N. 1950 On a class of exact solutions of the viscous fluid motion equations. Sov. Phys., J. Exp. Theor. Phys. 20, 2147–2157.Google Scholar
Zeleny, J. 1917 Instability of electrified liquid surface. Phys. Rev. 10, 1–6.Google Scholar
Zhigulev, V. N. 1960 On ejection effect due to an electric current discharge. Sov. Phys. Dokl. 130, 280–283.Google Scholar