| A free-surface singularity is a cusp or a corner in the fluid interface profile. Such a singularity could be dynamic, occurring at a particular moment in time, or steady-state, occurring as flow or material parameters approach special limiting values. We here consider two examples of dynamic singularities: film-rupture due to van der Waals interactions and viscous pinch-off. Viscous withdrawal may be an example of a steady-state singularity. As the flow parameter is varied, the steady-state interface converges towards a singular profile. For film-rupture, we find in chapter 2 that a countably infinite number of similarity solutions exist. Only the self-similar solution corresponding to the least oscillatory curvature profile is observed in time-dependent numerical simulations. For surface-tension driven viscous pinch-off (chapter 3), we develop a simplified boundary integral scheme to track dynamics close to pinch-off. We find that self-similar profiles at all viscosity contrasts are asymmetric and conical away from the minimum. The steep cone slope increases monotonically with the viscosity contrast. The shallow cone slope reaches a maximum at moderate viscosity contrasts. Also universal similarity scaling is preserved despite an asymptotically large velocity in the pinching neck driven by nonlocal dynamics. In chapter 4, we consider viscous withdrawal in the limit of small lower-fluid viscosity. A simple long-wavelength model for the steady-state spout profile, based on the assumption that the interface at the spout base is nearly static, is derived and shown to be ill-posed. Chapter 5 presents a simple example of the effect of non-Newtonian rheology on free-surface flow. We show, via numerical simulations and approximate analytic solutions, that rod climbing can occur in nematic liquid crystal in the limit of high flow rates. Unlike rod climbing in a polymeric fluid, different types of surface anchoring at the rotating rod can cause the nematic to climb or dip near a rotating rod. |
