When two fluid drops come close enough together to touch, surface tension quickly pulls the drops together into one larger drop. This is an example of a singular fluid flow, as the topology of the interface changes at the moment of contact. Similarly, when a pair of bubbles touch, the surface topology changes and a singular flow begins. Since the stress from surface tension depends on the surface curvature, these singularities are often characterised by divergent fluid velocities. Experimental observation or numerical simulation of these flows is therefore difficult due to the high velocities and small lengthscales.

In this thesis, I will find multi-scale theoretical solutions for the singular flows during the initial stages of the coalescence of bubbles and drops, solving for the velocity field in the fluid and the rate of coalescence. Each solution has several lengthscales, and on each lengthscale, we must solve some form of the Navier--Stokes equations. I will employ a variety of analytical and numerical techniques to solve for the flow on each scale. These asymptotic solutions are valid at early times; future numerical simulations of the subsequent flow could be initialised with these solutions, rather than the actual singularity.

In the course of solving for these singular flows, I will also describe the solution for the motion of a stretched fluid edge, the retraction of a narrow fluid wedge, the capillary flow around a parabola, and the effect of a time-dependent force on a fluid half-space. These fundamental flows have applications outside of coalescence, which I will outline throughout the thesis.