ABSTRACT
Time-dependent deformation of a two-dimensional inviscid drop subjected to quasi-steady straining flow, such as that created in Taylor’s four-roller mill (Taylor[3]) at a slowly varying rotation rate of the cylinders, is addressed. Following the spirit of matched asymptotic expansions, the problem is decomposed into the outer Stokes problem with no drop effect, which is solved by the boundary-element method, and the inner free-boundary problem for the drop behaviour in unbounded fluid. Using complex variable techniques, a broad class of explicit solutions is found, which is described by a rational conformal mapping of the unit disc onto the flow domain, with time-dependent coefficients satisfying ordinary differential equations. Some numerical simulations of the resulting equations are implemented, which suggest that, with increasing rotation rate of the cylinders, a sudden cusping of a sufficiently small drop occurs at a critical strain. This cusped configuration survives up to any strain, but when the strain is being gradually removed, the drop becomes suddenly rounded at a lower critical strain. This proves that, under certain circumstances, a two-dimensional inviscid drop placed in quasi-steady creeping flow exhibits hysteresis behaviour.
