Deformation and breakup of slender drops in linear flows

We study the deformation and breakup of a low-viscosity slender drop in a linear flow, v∞ = L.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 D: mm, where D =1/2 (L+ L ^T), andm is the orientation of the drop, and rotates as if it were a material element in a hypothetical flow M= GO + SI, where Ω =1/2(L^T-L), and G is a known function of the drop length. The approximations involved in the model are quite good whenM has only one eigenvalue with a positive real part, and somewhat less precise whenM 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 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.