Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-17T18:20:31.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Transience to instability in a liquid sheet

Published online by Cambridge University Press:  22 October 2010

N. S. BARLOW ,
B. T. HELENBROOK  and
S. P. LIN
N. S. BARLOW*
Affiliation:
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA
B. T. HELENBROOK
Affiliation:
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA
S. P. LIN
Affiliation:
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Series solutions are found which describe the evolution to absolute and convective instability in an inviscid liquid sheet flowing in a quiescent ambient gas and subject to a localized perturbation. These solutions are used to validate asymptotic stability predictions for sinuous and varicose disturbances. We show how recent disagreements in growth predictions stem from assumptions made when arriving at the Fourier integral response. Certain initial conditions eliminate or reduce the order of singularities in the Fourier integral. If a Gaussian perturbation is applied to both the position and velocity of a sheet when the Weber number is less than one, we observe absolutely unstable sinuous waves which grow like t1/3. If only the position is perturbed, we find that the sheet is stable and decays like t−2/3 at the origin. Furthermore, if both the position and velocity of a sheet are perturbed in the absence of ambient gas, we observe a new phenomenon in which sinuous waves neither grow nor decay and varicose waves grow like t1/2 with a convective instability.

JFM classification

Instability: Absolute/convective instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 666 , 10 January 2011 , pp. 358 - 390
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barlow, N. S., Helenbrook, B. T., Lin, S. P. & Weinstein, S. J. 2010 An interpretation of absolutely and convectively unstable waves using series solutions. Wave Motion doi:10.1016/j.wavemoti.2010.04.003.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. McGraw-Hill.Google Scholar
Bers, A. 1983 Space–time evolution of plasma instabilities-absolute and convective. In Handbook of Plasma Physics, vol. 1: Basic Plasma Physics 1 (ed. Galeev, A. A. & Sagdeev, R. Z.), chap. 3.2, pp. 451–517. North-Holland.Google Scholar
Bers, A. & Briggs, R. J. 1964 Criteria for determining absolute instabilities and distinguishing between amplifying and evanescent waves. Bull. Am. Phys. Soc. 9, 304.Google Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Brown, D. R. 1961 A study of the behaviour of a thin sheet of moving liquid. J. Fluid Mech. 10, 297–305.CrossRefGoogle Scholar
Copson, E. T. 1962 An Introduction to the Theory of Functions of a Complex Variable. Oxford University Press.Google Scholar
Crapper, G. D., Dombrowski, N., Jepson, W. P. & Pyott, G. A. D. 1973 A note on the growth of Kelvin–Helmholtz waves on thin liquid sheets. J. Fluid Mech. 57, 671–672.CrossRefGoogle Scholar
De Luca, L. & Costa, M. 1997 Instability of a spatially developing liquid sheet. J. Fluid Mech. 331, 127–144.CrossRefGoogle Scholar
Gaster, M. 1968 Growth of disturbances in both space and time. J. Fluid Mech. 11 (4), 723–727.Google Scholar
Hagerty, W. W. & Shea, J. F. 1955 A study of the stability of plane fluid sheets. J. Appl. Mech. 22, 509–514.CrossRefGoogle Scholar
Huerre, P. 1987 Spatio-temporal instabilities in closed and open flows. In Instabilities and Nonequilibrium Structures (ed. Tirapegui, E. & Villarroel, D.), vol. 1, pp. 141–177. D. Reidel.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Jazayeri, S. A. & Li, X. 2000 Nonlinear instability of plane liquid sheets. J. Fluid Mech. 406, 281–308.CrossRefGoogle Scholar
Le Grand-Piteira, N., Brunet, P., Lebon, L. & Limat, L. 2006 Propagating wave pattern on a falling liquid curtain. Phys. Rev. Lett. E 74 (2), 026305.CrossRefGoogle ScholarPubMed
Lin, S. P. 2003 Breakup of Liquid Sheets and Jets. Cambridge University Press.CrossRefGoogle Scholar
Lin, S. P. & Jiang, W. Y. 2003 Absolute and convective instability of a radially expanding liquid sheet. Phys. Fluids 15 (6), 1745–1754.CrossRefGoogle Scholar
Lin, S. P., Lian, Z. W. & Creighton, B. J. 1990 Absolute and convective instability of a liquid sheet. J. Fluid Mech. 220, 673–689.CrossRefGoogle Scholar
Lin, S. P. & Wang, Z. L. 2008 Three types of linear theories for atomizing liquids. Atomization Sprays 18 (3), 273–286.Google Scholar
Luchini, P. 2004 Is a plane liquid curtain algebraically absolutely unstable? Phys. Fluids 16 (7), 2154–2157.CrossRefGoogle Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill Book Company.Google Scholar
Nicholson, D. R. 1975 Parametric instabilities in inhomogeneous plasma. PhD thesis, Department of Physics, University of California, Berkely, CA.CrossRefGoogle Scholar
Rayleigh, Lord 1896 Theory of Sound, 2nd edn., vol. 2. Macmillan and Co.Google Scholar
Roche, J. S., Grand, N. Le, Brunet, P., Lebon, L. & Limat, L. 2006 Perturbations on a liquid curtain near break-up: wakes and free edges. Phys. Fluids 18 (8), doi:10.1063/1.2238867.CrossRefGoogle Scholar
Rosenbluth, M. N., White, R. B. & Liu, C. S. 1973 Temporal evolution of a three-wave parametric instability. Phys. Rev. Lett. 31, 1190–1193.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Seeley, R. T. 1966 An Introduction to Fourier Series and Integrals. W.A. Benjamin.Google Scholar
Squire, H. B. 1953 Investigation of the instability of a moving liquid film. Brit. J. Appl. Phys. 4, 167–169.CrossRefGoogle Scholar
Sturrock, P. A. 1958 Kinematics of growing waves. Phys. Rev. 112 (5), 1488–1503.CrossRefGoogle Scholar
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. Part II. Waves on fluid sheets. Proc. R. Soc. Lond. A 253, 296–312.Google Scholar