Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T06:23:38.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Equilibrium shapes and stability of captive annular menisci

Published online by Cambridge University Press:  21 April 2006

J. A. Tsamopoulos ,
A. J. Poslinski  and
M. E. Ryan
J. A. Tsamopoulos
Affiliation:
Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
A. J. Poslinski
Affiliation:
Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
M. E. Ryan
Affiliation:
Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Equilibrium shapes and stability of annular fluid menisci held together by surface tension are analysed by applying asymptotic and computer-aided techniques from bifurcation theory. The shapes and locations of the menisci are governed by the Young–Laplace equation. These shapes are grouped together into families of like symmetry that branch from the basic family of annular shapes at specific values of the aspect ratio, α. Multiple equilibrium shapes exist over certain values of α. The inner, outer or both the inner and outer interfaces may possess either a cylindrical or sinusoidal equilibrium shape. Changes in applied pressure, fluid volume, or gravitational Bond number break families of the same symmetry which now develop limit points. Numerical calculations rely on a finite-element representation of the interfaces and the results compare very well with asymptotic analysis which is valid for small deformations. The results are important for the blow moulding process and are invaluable in understanding its dynamics. These dynamics are expected to be considerably different from the dynamics of a liquid jet first analysed by Rayleigh.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 197 , December 1988 , pp. 523 - 549
Copyright
© 1988 Cambridge University Press

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

Abbott, J. P.1978 An efficient algorithm for the determination of certain bifurcation points. J. Comput. Appl. Math. 4, 19–26.Google Scholar
Boussinesq, J.1869 C. R. Acad. Sci. Paris 69, 128.
Brown, R. A. & Scriven, L. E.1980 The shapes and stability of captive rotating drops. Phil. Trans. R. Soc. Lond. A 297, 51–79.Google Scholar
Chandrasekhar, S.1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.
Esser, P. D., Paul, D. D. & Abdel-Khalik, S. I. 1981 Stability of the lithium “waterfall” first wall protection concept for inertial confinement fusion reactors. Nucl. Technol./Fusion 1, 285–294.Google Scholar
Friedman, B.1956 Principles and Techniques of Applied Mathematics. Wiley Press.
Hoffman, M. A., Takahashi, R. K. & Monson, R. D.1980 Annular liquid jet experiments. Trans. ASME I: J. Fluids Engng 102, 344–349.Google Scholar
Ioos, G. & Joseph, D. D.1980 Elementary Stability and Bifurcation Theory. Springer Press.
Keller, H. B.1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz), pp. 359–384. Academic.
Laplace, P. S.1805 Theory of Capillary Attraction. Supplement to the tenth book of Celestial Mechanics (translated and annotated by N. Bowditch, 1839), 1966 reprint by Chelsea. New York.
Lee, C. P. & Wang, T. G.1986 A theoretical model for the annular jet instability. Phys. Fluids 29, 2076–2085.Google Scholar
MACSYMA 1977 Reference Manual. Laboratory of Computer Science, Massachusetts Institute of Technology.
Matkowsky, B. J. & Reiss, E. L.1977 Singular perturbations of bifurcations. SIAM J. Appl. Maths 33, 230–255.Google Scholar
Nayfeh, A. H.1970 Nonlinear stability of a liquid jet. Phys. Fluids 13, 841–847.Google Scholar
Plateau, J. A. F.1873 Statique Expérimentale et Théorique des Liquides Soumix aux Seules Forces Moleculaires. Gauthier-Villars Press.
Rayleigh, Lord1879 On the stability of jets. Proc. Lond. Math. Soc. 10, 4–13.Google Scholar
Rayleigh, Lord1892 On the stability of cylindrical fluid surfaces. Phil. Mag. 34, 177–180.Google Scholar
Reiss, E. L.1977 Imperfect bifurcation. In Applications of Bifurcation theory (ed. P. H. Rabinowitz), pp. 37–71. Academic.
Strang, G. & Fix, G. J.1973 An Analysis of the Finite Element Method. Prentice Hall Press.
Thomas, P. D. & Brown, R. A.1987 LU decomposition of matrices with augmented dense constraints. Int. J. Num. Meth. Engng 24, 1451–1459.Google Scholar
Tsamopoulos, J. A., Akylas, T. R. & Brown, R. A.1985 Dynamics of charged drop break-up. Proc. R. Soc. Lond. A 401, 67–88.Google Scholar
Ungar, L. H. & Brown, R. A.1982 The dependence of the shape and stability of captive rotating drops on multiple parameters. Phil. Trans. R. Soc. Lond. A 306, 347–370.Google Scholar
Young, T.1805 Essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. A 95, 65–87.Google Scholar