Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T23:34:44.829Z Has data issue: false hasContentIssue false
  • English
  • Français

A numerical study of the deformation and burst of a viscous drop in an extensional flow

Published online by Cambridge University Press:  19 April 2006

J. M. Rallison  and
A. Acrivos
J. M. Rallison
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
A. Acrivos
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, California 94305
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the deformation and conditions for breakup of a liquid drop of viscosity λμ freely suspended in another liquid of viscosity μ with which it is immiscible and which is being sheared. The problem at zero Reynolds number is formulated exactly as an integral equation for the unknown surface velocity, which is shown to reduce to a particularly simple form when Δ = 1. This equation is then solved numerically, for the case in which the impressed shear is a radially symmetric extensional flow, by an improved version of the technique used, for Δ = 0, by Youngren & Acrivos (1976) so that we model the time-dependent distortion of an initially spherical drop. It is shown that, for a given Δ, a steady shape is attained only if the dimensionless group Ω ≡4πGμa/γ lies below a critical value Ωc(Δ), where G refers to the strength of the shear field, a is the radius of the initial spherical drop and γ is the interfacial tension. On the other hand, when Ω > Ωc the drop extends indefinitely along its long axis. The numerical results for Δ = 0·3, 0·5, 1, 2, 10 and 100 are in good agreement with the predictions of the small deformation analysis by Taylor (1932) and Barthès-Biesel & Acrivos (1973) and, at the smaller Δ, with those of slender-body theory (Taylor 1964; Acrivos & Lo 1978).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 89 , Issue 1 , 14 November 1978 , pp. 191 - 200
Copyright
© 1978 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

Acrivos, A. & Lo, T. A. 1978 Deformation and breakup of a single slender drop in an extensional flow. J. Fluid Mech. 86, 641.Google Scholar
Barthes-Biesel, D. & Acrivos, A. 1973 Deformation and burst of a liquid drop freely suspended in a linear shear field. J. Fluid Mech. 61, 1.Google Scholar
Buckmaster, J. D. 1972 Pointed bubbles in slow viscous flow. J. Fluid Mech. 55, 385.Google Scholar
Buckmaster, J. D. 1973 The bursting of pointed drops in slow viscous flow. J. Appl. Mech. E 40, 18.Google Scholar
Buckmaster, J. D. & Flaherty, J. E. 1973 The bursting of two-dimensional drops in slow viscous flow. J. Fluid Mech. 60, 625.Google Scholar
Cox, R. G. 1969 The deformation of a drop in a general time dependent fluid flow. J. Fluid Mech. 37, 601.Google Scholar
Frankel, N. A. & Acrivos, A. 1970 The constitutive equation for a dilute emulsion. J. Fluid Mech. 44, 65.Google Scholar
Grace, H. P. 1971 Dispersion phenomena in high viscoslty immiscible fluid systems and applications of static mixers as dispersion devices in such systems. Engng Found. 3rd Res. Conf. Mixing, Andover, New Hampshire.
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon & Breach.
Mikami, T., Cox, R. G. & Mason, S. G. 1975 Breakup of extending liquid threads. Int. J. Multiphase Flow 2, 113.Google Scholar
Richardson, S. 1968 Two-dimensional bubbles in slow viscous flows. J. Fluid Mech. 33, 475.Google Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. Roy. Soc. A 138, 41.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. Roy. Soc. A 146, 501.Google Scholar
Taylor, G. I. 1964 Conical free surfaces and fluid interfaces. Proc. 11th Int. Cong. Appl. Mech., Munich.
Tomotika, S. 1936 Breaking up of a drop of viscous liquid immersed in another viscous fluid which is extending at a uniform rate. Proc. Roy. Soc. A 153, 302.Google Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69, 377 (corrigendum 69, 813).Google Scholar
Youngren, G. K. & Acrivos, A. 1976 On the shape of a gas bubble in a viscous extensional flow. J. Fluid Mech. 76, 433.Google Scholar