Khakhar, D. V. ; Ottino, J. M.
(1986)
*Deformation and breakup of slender drops in linear flows*
Journal of Fluid Mechanics, 166
.
pp. 265-285.
ISSN 0022-1120

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Official URL: http://journals.cambridge.org/action/displayAbstra...

Related URL: http://dx.doi.org/10.1017/S0022112086000149

## Abstract

We study the deformation and breakup of a low-viscosity slender drop in a linear flow, $\overline{\boldmath v}^{\infty} \overline{\boldmath L}\cdot\overline{\boldmath x} $, assuming that the drop remains axisymmetric. We find that the drop stretches as if it were immersed in an axisymmetric extensional flow with a strength $\overline{\boldmath D}:\overline{\boldmath m}\overline{\boldmath m} $, where $\overline{\boldmath D}=\frac{1}{2}(\overline{\boldmath L}+\overline{\boldmath L}^T)$, and $\overline{\boldmath m} $ is the orientation of the drop, and rotates as if it were a material element in a hypothetical flow $\overline{\boldmath M}=G\overline{\boldmath D}+\overline{\Omega} $, where $\overline{\Omega} = \frac{1}{2}(\overline{\boldmath L}^T - \overline{\boldmath L})$, and G is a known function of the drop length. The approximations involved in the model are quite good when $\overline{\boldmath M}$ has only one eigenvalue with a positive real part, and somewhat less precise when $\overline{\boldmath M}$ has two eigenvalues with positive real parts. In the suitable limits the model reduces to Buckmaster's (1973) model for axisymmetric extensional flow and to the linear-axis version of the more general model proposed by Hinch & Acrivos (1980) for simple shear flow. In establishing a criterion for breakup for all linear flows, we find that the relevant quantity that specifies the flow is the largest positive real part of the eigenvalues of $\overline{\boldmath M}$, which depends on the drop length and the imposed flow. Our predictions are in reasonable agreement with the recent experimental data of Bentley (1985) for general two-dimensional linear flows and those of Grace (1971) for simple shear and hyperbolic extensional flow. We also present calculations for a class of three-dimensional flows as an illustration of the behaviour of three-dimensional flows in general.

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ID Code: | 17415 |

Deposited On: | 16 Nov 2010 08:35 |

Last Modified: | 04 Jun 2011 05:56 |

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