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On the breakup of spiralling liquid jets

Published online by Cambridge University Press:  15 January 2019

Yuan Li ,
Grigori M. Sisoev  and
Yulii D. Shikhmurzaev
Yuan Li
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
Grigori M. Sisoev
Affiliation:
Institute of Mechanics, Lomonosov Moscow State University, Moscow 119192, Russia
Yulii D. Shikhmurzaev*
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
*
†Email address for correspondence: [email protected]
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Abstract

The generation of drops from a jet spiralling out of a spinning device, under the action of centrifugal force, is considered for the case of small perturbations introduced at the inlet. Close to the inlet, where the disturbances can be regarded as small, their propagation is found to be qualitatively similar to that of a wave propagating down a straight jet stretched by an external body force (e.g. gravity). The dispersion equation has the same parametric dependence on the base flow, but the base flow is, of course, different. Further down the jet, where the amplitude of the disturbances becomes finite and eventually resulting in drop formation, the flow appears to be quite complex. As shown, for the regular/periodic process of drop generation, the wavelength corresponding to the frequency at the inlet, increasing as the wave propagates down the stretching jet, determines, in general, not the volume of the resulting drop but the sum of volumes of the main drop and the satellite droplet that follows the main one. The proportion of the total volume forming the main drop depends on how far down the jet the drops are produced, i.e. on the magnitude of the inlet disturbance. The volume of the main drop is found to be a linear function of the radius of the unperturbed jet evaluated at the point where the drop breaks away from the jet. This radius, and the corresponding velocity of the base flow, have to be found simultaneously with the jet’s trajectory by using a jet-specific non-orthogonal coordinate system described in detail in Shikhmurzaev & Sisoev (J. Fluid Mech., vol. 819, 2017, pp. 352–400). Some characteristic features of the nonlinear dynamics of the drop formation are discussed.

JFM classification

Instability: Absolute/convective instability Drops and Bubbles: Breakup/coalescence Drops and Bubbles: Drops
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 364 - 384
Copyright
© 2019 Cambridge University Press 

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References

Ahmed, M. & Youssef, M. S. 2012 Characteristics of mean droplet size produced by spinning disk atomizers. Trans. ASME J. Fluids Engng 134, 071103.Google Scholar
Ahmed, M. & Youssef, M. S. 2014 Influence of spinning cup and disk atomizer configurations on droplet size and velocity characteristics. Chem. Engng Sci. 107, 149–157.Google Scholar
Ambravaneswaran, B., Phillips, S. D. & Basaran, O. A. 2000 Theoretical analysis of a dripping faucet. Phys. Rev. Lett. 85, 5332–5335.Google Scholar
Aziz, A. K. 1972 Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Academic Press.Google Scholar
Bechtel, S. E., Carlson, C. D. & Forest, M. G. 1995 Recovery of the Rayleigh capillary instability from slender 1-D inviscid and viscous models. Phys. Fluids 7 (12), 2956–2971.Google Scholar
Borthakur, M. P., Biswas, G. & Bandyopadhyay, D. 2017 Formation of liquid drops at an orifice and dynamics of pinch-off in liquid jets. Phys. Rev. E 96, 013115.Google Scholar
Castrejón-Pita, J. R., Castrejón-Pita, A. A., Thete, S. S., Sambath, K., Hitchings, I. M., Hinch, J., Lister, J. R. & Basaran, O. A. 2015 Plethora of transitions during breakup of liquid filaments. Proc. Natl Acad. Sci. USA 112, 4582–4587.Google Scholar
Chakraborty, I., Rubio-Rubio, M., Sevilla, A. & Gordillo, J. M. 2016 Numerical simulation of axisymmetric drop formation using a coupled level set and volume of fluid method. Intl J. Multiphase Flow 84, 54–65.Google Scholar
Cheong, B. S. & Howes, T. 2004 Capillary jet instability under the influence of gravity. Chem. Engng Sci. 59, 2145–2157.Google Scholar
Chesnokov, Y. G. 2000 Nonlinear development of capillary waves in a viscous liquid jet. J. Tech. Phys. 45 (8), 787–794.Google Scholar
Decent, S. P., King, A. C., Simmons, M. J. H., Părău, E. I., Wallwork, I. M., Gurney, C. J. & Uddin, J. 2009 The trajectory and stability of a spiralling liquid jet: viscous theory. Appl. Maths Model. 33, 4283–4302.Google Scholar
Decent, S. P., King, A. C. & Wallwork, I. M. 2002 Free jets spun from a prilling tower. J. Engng Maths 42, 265–282.Google Scholar
Entov, V. M. & Yarin, A. L. 1984 The dynamics of thin liquid jets in air. J. Fluid Mech. 140, 91–111.Google 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.Google Scholar
Frankel, I. & Weihs, D. 1987 Influence of viscosity on the capillary instability of a stretching jet. J. Fluid Mech. 185, 361–383.Google Scholar
Frost, A. R. 1981 Rotary atomization in the ligament formation mode. J. Agric. Engng Res. 26, 63–78.Google Scholar
van Hoeve, W., Gekle, S., Snoeijer, J., Versluis, M., Brenner, M. P. & Lohse, D. 2010 Breakup of diminutive Rayleigh jets. Phys. Fluids 22, 122003.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Kistler, S. F. & Scriven, L. E. 1984 Coating flow theory by finite element and asymptotic analysis of the Navier–Stokes system. Intl J. Numer. Meth. Fluids 4, 207–229.Google Scholar
Le Dizès, S. & Villermaux, E. 2017 Capillary jet breakup by noise amplification. J. Fluid Mech. 810, 281–306.Google Scholar
Leib, S. J. & Goldstein, M. E. 1986a Convective and absolute instability of a viscous liquid jet. Phys. Fluids 29 (4), 952–954.Google Scholar
Leib, S. J. & Goldstein, M. E. 1986b The generation of capillary instabilities on a liquid jet. J. Fluid Mech. 168, 479–500.Google Scholar
Lenard, P. 1887 Über die Schwingungen fallander Tropfen. Ann. Phys. 30, 209–243.Google Scholar
Li, Y., Sisoev, G. M. & Shikhmurzaev, Y. D. 2018 Spinning disk atomization: theory of the ligament regime. Phys. Fluids 30, 092101.Google Scholar
Li, Y. & Sprittles, J. E. 2016 Capillary breakup of a liquid bridge: identifying regimes and transitions. J. Fluid Mech. 797, 29–59.Google Scholar
Mellado, P., McIlwee, H. A. & Badrossamay, M. A. 2011 A simple model for nanofiber formation by rotary jet-spinning. Appl. Phys. Lett. 99, 203107.Google Scholar
Mikami, T., Cox, R. G. & Mason, S. G. 1975 Breakup of extending liquid threads. Intl J. Multiphase Flow 2, 113–138.Google Scholar
Pearson, J. R. A. 1985 Mechanics of Polymer Processing. Applied Science Publishers.Google Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. R. Soc. Lond. A 10, 4–12.Google Scholar
Rutland, D. F. & Jameson, G. J. 1970 Theoretical predictions of the sizes of drops formed in the breakup of capillary jets. Chem. Engng Sci. 25, 1689–1698.Google Scholar
Saleh, S. N., Ahmed, S. M., Al-Mosuli, D. & Barghi, S. 2015 Basic design methodology for a prilling tower. Can. J. Chem. Engng 93, 1403–1409.Google Scholar
Sauter, U. S. & Buggisch, H. W. 2005 Stability of initially slow viscous jets driven by gravity. J. Fluid Mech. 533, 237–257.Google Scholar
Senchenko, S. & Bohr, T. 2005 Shape and stability of a viscous thread. Phys. Rev. E 71, 056301.Google Scholar
Senuma, S., Lowe, C., Zweifel, Y., Hilborn, J. G. & Marison, I. 2000 Alginate hydrogelmicrospheres and microcapsules prepared by spinning disk atomization. Biotechnol. Bioengng 67, 616–622.Google Scholar
Senuma, Y. & Hilborn, J. G. 2002 High-speed imaging of drop formation from low viscosity liquids and polymer melts in spinning disk atomization. Polym. Engng Sci. 42, 969–982.Google Scholar
Shikhmurzaev, Y. D. 2005 Capillary breakup of liquid threads: a singularity-free solution. IMA J. Appl. Maths 70, 880–907.Google Scholar
Shikhmurzaev, Y. D. 2007 Capillary Flows with Forming Interfaces. Chapman & Hall.Google Scholar
Shikhmurzaev, Y. D. & Sisoev, G. M. 2017 Spiralling liquid jets: verifiable mathematical framework, trajectories and peristaltic waves. J. Fluid Mech. 819, 352–400.Google Scholar
Wallwork, I. M., Decent, S. P., King, A. C. & Schulkes, R. M. S. M. 2002 The trajectory and stability of a spiralling liquid jet: Part 1. Inviscid theory. J. Fluid Mech. 459, 43–65.Google Scholar
Weber, C. 1931 Zum Zerfall eines Flüssigkeitsstrables. Z. Angew. Math. Mech. 11, 136–154.Google Scholar