Dynamic stabilization of toroidal drops in linear flow

Viscous drops, subject to a linear flow in an immiscible viscous fluid, deform and, under certain conditions, attain stationary shapes. When the outer flow is a combination of rotation and axisymmetric extensional or bi-extensional flow, yielded shapes are flat drops, flat drops with dimples, and toroidal drops. The toroidal drops in compressional flow are highly unstable. Under the combined action of rotation and extensional flow, two toroidal stationary shapes exist at the same conditions.  The torus with higher major radius is stable with respect to the axisymmetric disturbances, while the second one is unstable. Any perturbation of an unstable shape results in either collapse of the torus to a simply connected shape, or to torus extension. 

We applied control methods to stabilize toroidal drops in the linear flow. As a control variable we chose the intensity of axisymmetric (non-rotational) component of the outer flow. The control was designed using a simplified two-state dynamic model and tested on non-simplified dynamic equations for drop deformation. We succeeded to stabilize all the toroidal shapes obtained previously. Moreover, such a dynamic control enabled a construction of new shapes - almost collapsed highly unstable tori - that could not be previously obtained.