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

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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.

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