Soft Lubrication Flow from a Rigid Particle Approaching a Vesicle Membrane

As a versatile candidate for drug delivery, giant unilamellar vesicles encapsulating magnetic particles (magGUVs) under an inhomogeneous magnetic field showed great potential for targeted and controlled cargo release in microfluidics (Malik et al., 2025, Nanoscale). Motivated by these results, we investigate the elasto-hydrodynamics of a deformable vesicle encapsulating a rigid particle subjected to a constant driving force to move toward the vesicle membrane. Employing analytical calculations, boundary integral simulations, and asymptotic analysis of the soft lubrication flow in the narrowing gap between the particle and the elastic interface, we show that the particle persistently outpaces the vesicle, thereby driving drainage of the interstitial fluid between the particle and the membrane. This drainage proceeds in a symmetric, self-similar fashion, regardless of initial asymmetries. As the gap thins, membrane tension increases monotonically and may eventually exceed the lysis threshold, leading to rupture over long timescales. Our results provide a quantitative estimate, in terms of vesicle deformability (reduced area) and relative particle size, for the maximum delivery distance achievable in the magGUV system before membrane rupture compromises encapsulation.