Hysteretic response of supported drops during forced oscillations

Viscous liquid drops undergoing forced oscillations have been shown to exhi bit hysteretic deformation under certain conditions both in experiments and by solution of simplified model equations that can only provide a qualitat ive description of their true response. The first hysteretic deformation re sults for oscillating pendant drops obtained by solving the full transient, nonlinear Navier-Stokes system are presented herein using a sweep procedur e in which either the forcing amplitude A or frequency Omega is first incre ased and then decreased over a given range. The results show the emergence of turning-point bifurcations in the parameter space of drop deformation ve rsus the swept parameter. For example, when a sweep is carried out by varyi ng Omega while holding A fixed, the first turning point occurs at Omega = O mega(u) as Omega is being increased and the second one occurs at Omega = Om ega(l) < Omega(u) as Omega is being decreased. The two turning points shift further from each other and toward lower values of the swept parameter as Reynolds number Re is increased. These turning points mark the ends of a hy steresis range within which the drop may attain either of two stable steady oscillatory states limit cycles-as identified by two distinct solution bra nches. In the hysteresis range, one solution branch, referred to as the upp er solution branch, is characterized by drops having larger maximum deforma tions compared to those on the other branch, referred to as the lower solut ion branch. Over the range Omega(l) less than or equal to Omega less than o r equal to Omega(u), the sweep procedure enables detection of the upper sol ution branch which cannot be found if initially static drops are set into o scillation as in previous studies of forced oscillations of supported and c aptive drops, or liquid bridges. The locations of the turning points and th e associated jumps in drop response amplitudes observed at them are studied over the parameter ranges 0.05 less than or equal to A less than or equal to 0.125, 20 less than or equal to Re less than or equal to 40, and gravita tional Bond number 0 less than or equal to G less than or equal to 1. Criti cal forcing amplitudes for onset of hysteresis are also determined for thes e Re values. The new findings have important ramifications in several pract ical applications. First, that Omega(u) - Omega(l) increases as Re increase s overcomes the limitation which is inherent to the current practice of inf erring the surface tension and/or viscosity of a bridge/drop liquid from me asurement of its resonance frequencies (Chen & Tsamopoulos 1993; Mollot et nl. 1993). Moreover, that the value of A for onset of hysteresis can be as low as 5% of the drop radius, or lower, has important implications for othe r free-surface flows such as coating flows.