Surface-tension-driven coalescence

• Main Author: Thompson, Alice B.
• Published: University of Nottingham 2012
• Subjects: 530.427; QA299 Analysis
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.559602

When fluid droplets coalesce, the flow is initially controlled by a balance between surface tension and viscosity. For low viscosity fluids such as water, the viscous lengthscale is quickly reached, yielding a new balance between surface tension and inertia. Numerical and asymptotic calculations have shown that there is no simply connected solution for the coalescence of inviscid fluid drops surrounded by a void, as large amplitude capillary waves cause the free surface to pinch off. We analyse in detail a linearised version of this free boundary problem. For zero density surrounding fluid, we find asymptotic solutions to the leading order linear problem for small and large contact point displacement. In both cases, this requires the solution of a mixed type boundary value problem via complex variable methods. For the large displacement solution, we match this to a WKB analysis for capillary waves away from the contact point. The composite solution shows that the interface position becomes self intersecting for sufficiently large contact point displacement. We identify a distinguished density ratio for which flows in the coalescing drops and surrounding fluid are equally important in determining the interface shape. We find a large displacement solution to the leading order two-fluid problem with a multiple-scales analysis, using a spectral method to solve the leading order periodic oscillator problem for capillary waves. This is matched to a single-parameter inner problem, which we solve numerically to obtain the correct boundary conditions for the secularity equations. We find that the composite solution for the two-fluid problem is simply connected for arbitrarily large contact-point displacement, and so zero density surrounding fluid is a singular limit.