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Oblique liquid curtains with a large Froude number

Published online by Cambridge University Press:  19 December 2018

E. S. Benilov Open the ORCID record for E. S. Benilov [Opens in a new window]
E. S. Benilov*
Affiliation:
Department of Mathematics and Statistics, University of Limerick, Ireland V94 T9PX, Ireland
*
†Email address for correspondence: [email protected]
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Abstract

This paper examines two-dimensional liquid curtains ejected at an angle to the horizontal and affected by gravity and surface tension. The flow in the curtain is, generally, sheared. The Froude number based on the injection velocity and the outlet’s width is assumed large; as a result, the streamwise scale of the curtain exceeds its thickness. A set of asymptotic equations for such (slender) curtains is derived and its steady solutions are examined. It is shown that, if the surface tension exceeds a certain threshold, the curtain – quite paradoxically – bends upwards, i.e. against gravity. Once the flow reaches the height where its initial supply of kinetic energy can take it, the curtain presumably breaks up and splashes down.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Wakes/Jets: Jets Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 861 , 25 February 2019 , pp. 328 - 348
Copyright
© 2018 Cambridge University Press 

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References

Benilov, E. S. 1995 Stability of large-amplitude geostrophic flows localized in a thin layer. J. Fluid Mech. 288, 157–174.Google Scholar
Benilov, E. S. 2015 Hydraulic jumps in a shallow flow down a slightly inclined substrate. J. Fluid Mech. 782, 5–24.Google Scholar
Benilov, E. S., Barros, R. & O’Brien, S. B. G. 2016 Stability of thin liquid curtains. Phys. Rev. E 94, 043110.Google Scholar
Benilov, E. S., Lapin, V. N. & O’Brien, S. B. G. 2011 On rimming flows with shocks. J. Engng Maths 75, 49–62.Google Scholar
Brown, D. R. 1961 A study of the behaviour of a thin sheet of moving liquid. J. Fluid Mech. 10, 297–305.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
Decent, S. P., Părău, E. I., Simmons, M. J. H. & Uddin, J. 2018 On mathematical approaches to modelling slender liquid jets with a curved trajectory. J. Fluid Mech. 844, 905–916.Google Scholar
Dyson, R. J., Brander, J., Breward, C. J. W. & Howell, P. D. 2009 Long-wavelength stability of an unsupported multilayer liquid film falling under gravity. J. Engng Maths 64, 237–250.Google Scholar
Entov, V. M. & Yarin, A. L. 1984 The dynamics of thin liquid jets in air. J. Fluid Mech. 140, 91.Google Scholar
Finnicum, D. S., Weinstein, S. J. & Ruschak, K. J. 1993 The effect of applied pressure on the shape of a two-dimensional liquid curtain falling under the influence of gravity. J. Fluid Mech. 255, 647–665.Google Scholar
Friedlander, S. & Vishik, M. M. 1991 Instability criteria for the flow of an inviscid incompressible fluid. Phys. Rev. Lett. 66, 2204–2206.Google Scholar
Goren, S. L. 1966 Development of the boundary layer at a free surface from a uniform shear flow. J. Fluid Mech. 25, 87–95.Google Scholar
Khayat, R. E. 2014 Free-surface jet flow of a shear-thinning power-law fluid near the channel exit. J. Fluid Mech. 748, 580–617.Google Scholar
Kochin, N. E., Kibel, I. A. & Roze, N. V. 1964 Theoretical Hydromechanics. Wiley.Google Scholar
Leblanc, S. 1997 Stability of stagnation points in rotating flows. Phys. Fluids 9, 3566–3569.Google Scholar
Lhuissier, H., Brunet, P. & Dorbolo, S. 2016 Blowing a liquid curtain. J. Fluid Mech. 795, 784–807.Google Scholar
Li, X. 1993 Spatial instability of plane liquid sheets. Chem. Engng Sci. 48, 2973–2981.Google Scholar
Li, X. 1994 On the instability of plane liquid sheets in two gas streams of unequal velocities. Acta Mech. 106, 137–156.Google Scholar
Lifschitz, A. 1991 Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity. Phys. Lett. A 157, 481–487.Google Scholar
Odulo, A. B. 1979 Long non-linear waves in the rotating ocean of variable depth. Dokl. Akad. Nauk SSSR 248, 1439–1442.Google Scholar
Ramos, J. I. 1996 Planar liquid sheets at low Reynolds numbers. Intl J. Numer. Meth. Fluids 22, 961–978.Google Scholar
Roche, J. S., Grand, N. L., Brunet, P., Lebon, L. & Limat, L. 2006 Perturbations on a liquid curtain near break-up: wakes and free edges. Phys. Fluids 18, 082101.Google Scholar
Sevilla, A. 2011 The effect of viscous relaxation on the spatiotemporal stability of capillary jets. J. Fluid Mech. 684, 204–226.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
Tillett, J. P. K. 1968 On the laminar flow in a free jet of liquid at high Reynolds numbers. J. Fluid Mech. 32 (02), 273.Google Scholar
Wallwork, I. M.2001 The trajectory and stability of a spiralling liquid jet. PhD thesis, University of Birmingham.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
Zakharov, V. E. 1981 On the Benney equations. Physica D 3, 193–202.Google Scholar