  Lipid vesicles play a key role in fundamental biological processes. Despite
recent progress, we lack a complete understanding of the non-equilibrium
dynamics of vesicles due to challenges associated with long-time observation of
shape fluctuations in strong flows. In this work, we present a flow-phase
diagram for vesicle shape and conformational transitions in planar extensional
flow using a Stokes trap, which enables control over the center-of-mass
position of single or multiple vesicles in precisely defined flows [Shenoy,
Rao, Schroeder, \textit{PNAS}, 113(15):3976-3981, 2016]. In this way, we
directly observe the non-equilibrium conformations of lipid vesicles as a
function of reduced volume $\nu$, capillary number $Ca$, and viscosity contrast
$\lambda$. Our results show that vesicle dynamics in extensional flow are
characterized by the emergence of three distinct shape transitions, including a
tubular to symmetric dumbbell transition, a spheroid to asymmetric dumbbell
transition, and quasi-spherical to ellipsoid transition. The experimental phase
diagram is in good agreement with recent predictions from simulations
[Narsimhan, Spann, Shaqfeh, \textit{J. Fluid Mech.}, 2014, \textbf{750}, 144].
We further show that the phase boundary of vesicle shape transitions is
independent of the viscosity contrast. Taken together, our results demonstrate
the utility of the Stokes trap for the precise quantification of vesicle
stretching dynamics in precisely defined flows.
