The shape of a two-dimensional viscous drop deforming in several time-dependent
flow fields, including that due to a potential vortex, has been studied. Vortex flow
was approximated by linearizing the induced velocity field at the drop centre, giving
rise to an extensional flow with rotating axes of stretching. A generalization of the
potential vortex, a flow we have called rotating extensional flow, occurs when the
frequency of revolution of the flow is varied independently of the shear rate. Drops
subjected to this forcing flow exhibit an interesting resonance phenomenon. Finally
we have studied drop deformation in an oscillatory extensional flow.
Calculations were performed at small but non-zero Reynolds numbers using an
ADI front-tracking/finite difference method. We investigate the effects of interfacial
tension, periodicity, viscosity ratio, and Reynolds number on the drop dynamics.
The simulation reveals interesting behaviour for steady stretching flows, as well as
time-dependent flows. For a steady extensional flow, the drop deformation is found to
be non-monotonic with time in its approach to an equilibrium value. At sufficiently
high Reynolds numbers, the drop experiences multiple growth–collapse cycles, with
possible axes reversal, before reaching a final shape. For a vortex flow, the long-time deformation reaches a steady value, and the drop attains a revolving steady
elliptic shape. For rotating extensional flows as well as oscillatory extensional flows,
the maximum value of deformation displays resonance with variation in parameters,
first increasing and then decreasing with increasing interfacial tension or forcing
frequency. A simple ODE model with proper forcing is offered to explain the observed
phenomena.