Swinging, tumbling, and phase-lagging of multicomponent membranes in a shear flow*

Biological membranes are host to proteins and molecules which can form domain-like structures, resulting in spatially-varying material properties such as bending rigidity and spontaneous curvature. Such membranes can exhibit intricate shapes at equilibrium, suggesting the possibility of rich dynamics when such a body is placed into a flow. Under the assumption of small deformations we develop a reduced order model to describe the full fluid-structure interaction between a viscous background shear flow and a vesicle with spatially varying bending stiffness and curvature, which is solved exactly to predict the membrane's time-dependent deformation and inclination angles. A critical ratio which links the flow rate, internal viscosity, and the gradient in material properties is derived, which identifies a rapid transition in the membrane dynamics: from a swinging motion (which includes tangential tank-treading) to a rigid body tumbling behavior, passing through a transition regime which features tumbling and periodic phase-lagging of the membrane material relative to the body's long axis. Full numerical simulations are used to probe the theoretical predictions, which appear valid even when studying substantially deformed membranes.