Deformation and breakup of slender drops in linear flows

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.