Dynamics of an elastic, nearly inextensible membrane driven by active inclusions

Vesicles encapsulating colloids or microswimmers provide a model platform for studying membrane–particle interactions relevant to protocell mobility and precision delivery. We develop an analytical framework for the dynamics of deformable, closed interfaces driven by internally generated fluid motion. Focusing on a nearly spherical enclosure bounded by an elastic, weakly extensible membrane, we derive coupled solutions for the internal Stokes flow and interfacial deformation induced by active or force-driven particles. The theory yields explicit expressions for the enclosure's locomotion, flow field, and shape response under generic internal forcing. Beyond the linear regime, we identify the onset of nonaxisymmetric steady configurations and delineate their stability boundaries. Near the instability threshold, the weakly nonlinear dynamics reveal shape–flow coupling that governs self-propulsion and morphological transitions. These results provide a unified description of active vesicle and droplet mechanics, linking internal forcing to emergent deformation and motion in soft fluid–structure systems.