E. John Hinch

Significance: Fluid flows, governed by nonlinear equations, permit formation of singularities. Often, singularities are artifacts of neglecting physical effects. However, free-surface flows exhibit observable singularities including filament pinch-off. As filaments thin, slightly (highly) viscous filaments are expected from theory to transition from an inertial (viscous) regime where viscosity (density) is negligible to an inertial–viscous regime where viscous and inertial effects are important. Previous works show this transition either does not occur or occurs for filament radii well below theoretical predictions. We demonstrate that thinning filaments unexpectedly pass through a number of intermediate transient regimes, thereby delaying onset of the final regime. The findings raise the question if similar dynamical transitions arise in problems that are not necessarily hydrodynamic in nature.

We study the solitary wave solutions in a thin film of a power-law fluid coating a vertical fibre. Different behaviours are observed for shear-thickening and shear-thinning fluids. For shear-thickening fluids, the solitary waves are larger and faster when the reduced Bond number is smaller. For shear-thinning fluids, two branches of solutions exist for a certain range of the Bond number, where the solitary waves are larger and faster on one and smaller and slower on the other as the Bond number decreases. We carry out an asymptotic analysis for the large and fast-travelling solitary waves to explain how their speeds and amplitudes change with the Bond number. The analysis is then extended to examine the stability of the two branches of solutions for the shear-thinning fluids.

Kalliadasis & Chang (J. Fluid Mech., vol. 261, 1994, pp. 135–168) showed within lubrication theory that if a liquid film is thicker than a critical value then drops will accelerate and grow, whereas with thinner films the drops fall at a constant velocity. As the thickness of the film increases to the critical value, the drops move faster and are larger. We revisit their asymptotic analysis of these large drops. While we recover their results for the leading-order and first correction, we do not agree on further corrections. In particular we find it necessary to evaluate the third correction, which they do not consider, before we obtain a first approximation to the dependence of the speed on the non-dimensional control parameter. We proceed to two further corrections in order to improve this first approximation to the speed.

The final stages of pinchoff and breakup of dripping droplets of near-inviscid Newtonian fluids are studied experimentally for pure water and ethanol. High-speed imaging and image analysis are used to determine the angle and the minimum neck size of the cone-shaped extrema of the ligaments attached to dripping droplets in the final microseconds before pinchoff. The angle is shown to steadily approach the value of 18.0 \ifmmode\pm\else\textpm\fi{}0.4$\circ${}, independently of the initial flow conditions or the type of breakup. The filament thins and necks following a $\tau${}2/3 law in terms of the time remaining until pinchoff, regardless of the initial conditions. The observed behavior confirms theoretical predictions.

Granular pastes are dense dispersions of non-colloidal grains in a simple or a complex fluid. Typical examples are the coating, gluing or sealing mortars used in building applications. We study the rupture of a thick layer of mortar paste in a simple pulling test where the paste is confined between two flat surfaces. It is shown that, depending on the rheological properties of the paste and the plate separation velocity, two main failure modes are obtained. The first mode is the inwards shear flow of the paste with viscous fingering instabilities, similarly to what has been observed with Newtonian fluids and with non-Newtonian colloidal suspensions or polymer solutions. The second failure mode is stemming from the expansion of bubbles, similarly to what has been observed in soft adhesive polymer layers and, more recently, in highly viscous fluids. It is shown that the crossover between the two failure modes is determined by the conditions required to generate a pressure drop able to trigger the growth of pre-existing micro-bubbles smaller than the inter-granular distance.