  Although the behavior of fluid-filled vesicles in steady flows has been
extensively studied, far less is understood regarding the shape dynamics of
vesicles in time-dependent oscillatory flows. Here, we investigate the
nonlinear dynamics of vesicles in large amplitude oscillatory extensional
(LAOE) flows using both experiments and boundary integral (BI) simulations. Our
results characterize the transient membrane deformations, dynamical regimes,
and stress response of vesicles in LAOE in terms of reduced volume (vesicle
asphericity), capillary number ($\Ca$, dimensionless flow strength), and
Deborah number ($\De$, dimensionless flow frequency). Results from single
vesicle experiments are found to be in good agreement with BI simulations
across a wide range of parameters. Our results reveal three distinct dynamical
regimes based on vesicle deformation: pulsating, reorienting, and symmetrical
regimes. We construct phase diagrams characterizing the transition of vesicle
shapes between pulsating, reorienting, and symmetrical regimes within the
two-dimensional Pipkin space defined by $\De$ and $\Ca$. Contrary to
observations on clean Newtonian droplets, vesicles do not reach a maximum
length twice per strain rate cycle in the reorienting and pulsating regimes.
The distinct dynamics observed in each regime result from a competition between
the flow frequency, flow time scale, and membrane deformation timescale. By
calculating the particle stresslet, we quantify the nonlinear relationship
between average vesicle stress and strain rate. Additionally, we present
results on tubular vesicles that undergo shape transformation over several
strain cycles. Broadly, our work provides new information regarding the
transient dynamics of vesicles in time-dependent flows that directly informs
bulk suspension rheology.
