Formation of a pointed drop in Taylor's four-roller mill

Abstract

The paper addresses the mathematical modelling of the formation of a pointed drop in a four-roller mill, observed by Taylor (1934) in the Cavendish Laboratory. Since the experiments were carried out with drops of small diameter compared to the mill size, the method of matched asymptotic expansions is applicable. A two-dimensional Stokes flow generated by the rotating rollers in the mill but with no drop effect (outer problem) is computed numerically by a boundary-element method. The local expansion of that flow at the centre of the mill, where the drop is to be positioned, is used as a far field for the flow around the drop in unbounded fluid (inner problem). Employing a plane-flow model and using complex-variable techniques, the explicit solutions previously obtained by the author are adapted to the inner problem. It is proved that, with an increasing rotation rate of the rollers, the drop does develop two apparent cusps on the interface, and its shapes have striking similarities with Taylor's experiments. Response diagrams showing the drop distortion versus the elongational strain demonstrate that these are one-to-one function of each other if the drop diameter is greater than a critical value determined by the size of the mill but cease to be one-to-one otherwise. This behaviour is identified with a sudden transition from a rounded drop to a cusped one at a critical strain.

Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow

“…Here, we set J = G in the non-dimensionalization. The cubic terms provide a more faithful model of the flow generated by the counter-rotating cylinders in G. I. Taylor's seminal four-roller mill experiments, and are derived by local expansion of the flow caused by the rollers about a point at the centre of the mill (Antanovskii 1996). With the internal pressure set to zero, the pressure far from the capsule evolves in time, i.e.…”

“…The analysis is modified to account for the nonlinear terms in the imposed flow, and we present some of the equations without derivation, referring the reader to Antanovskii (1996) and Siegel (2000) for details.…”

“…We also consider an imposed nonlinear strain introduced by Antanovskii (1996) as a model for the far-field flow in Taylor's four-roller mill. Here, steady-state solutions are found to have tip curvatures that increase with capillary number, and, at a critical value Ca c , the capsule exhibits zero-angled cusps.…”

An elliptical-pore model for late-stage planar viscous sintering

“…At the same time, microstructural information on pore shrinkage rates obtained from the solution to each inner problem gives the required macroscopic parameters for the outer flow approximation. Antanovskii (1996) adopted a similar approach based on matched asymptotic expansions in considering the free-surface evolution of a single constant-pressure bubble in the flow field of Taylor's four-roller mill.…”

The evolution of a slender non-axisymmetric drop in an extensional flow

“…Two-dimensional models of an inviscid bubble in a viscous straining flow have received much attention, mainly due to the existence of exact solutions which result from the application of complex variable methods. The first steady solutions were obtained by Richardson (1968) (generalised by Antanovskii 1996) and later extended to include time dependence by Antanovskii (1994) and Tanveer & Vasconcelos (1995).…”