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Similarity Solutions for Capillary Pinch-Off in Fluids of Differing Viscosity

Resurgent interest in the classic fluid-mechanical problem of capillary pinch-off of a fluid thread, as recently reviewed in [1], has focused on analysis of the final stage of pinch-off. Theoretical motivation arises from general interest in the self-organization of nonlinear dynamics near finite-time singularities, such as topological transitions, and, in particular, with issues such as universality, scaling, and uniqueness [2,3]. Further motivation arises from diverse technological applications such as emulsification and spray production. Provided molecular dimensions are not reached first, viscous dissipation in the surrounding fluid, however small initially, is asymptotically dominant over inertia. Consequently, pinch-off in surrounding fluid of any viscosity eventually enters a self-similar regime in which the destabilizing capillary pressure is opposed by internal and external viscous dissipation [4‐ 6]. In contrast to the usual assumption that the thread profile and flow field close to pinch-off are determined solely by local dynamics, viscous pinch-off of a thread in surrounding fluid involves an asymptotically large nonlocal contribution to the flow field in the pinching neck. Here we show how to account for this nonlocal contribution and examine the effect of the viscosity ratio on the self-similar pinching structure. Consider a fluid thread of viscosity lh and surface tension g, pinching in surrounding fluid of viscosity h. Under capillary pressure, nonaxisymmetric deformations can be expected to decay more rapidly than the thread pinches; thus only axisymmetric profiles need to be considered. Let the thread have radius hz, t, where z is the axial coordinate, and t is the time to pinch-off. To see how h and z scale with t, note first that in Stokes flow the pressure and velocity fields are both determined by equidimensional Laplacian operators, and thus when only free-surface boundaries are present there is no preferred direction. This suggests that h z as t ! 0. Balancing internal and external viscous stresses against capillary pressure yields the estimate u gh, which suggests that the velocity is constant as t ! 0. Hence, h, z t; the Reynolds