Influence of surface tension on the instabilities and bifurcations of a particle in a drop under shear

While the deformation regimes under flow of anuclear cells, like red blood cells, have been widely analyzed, the dynamics of nuclear cells are less explored. The objective of this work is to investigate the interplay between the stiff nucleus, modeled here as a rigid spherical particle and the surrounding deformable cell membrane, modeled for simplicity as an immiscible droplet, subjected to an external unbounded plane shear flow. A three-dimensional boundary integral implementation is developed to describe the interface-structure interaction characterized by two dimensionless numbers: the capillary number $Ca$, defined as the ratio of shear to capillary forces and and the particle-droplet size ratio. For large $Ca$, i.e. very deformable droplets, the particle has a stable equilibrium position at the center of the droplet. However, for smaller $Ca$, both the plane symmetry and the time invariance are broken and the particle migrates to a closed orbit located off the symmetry plane, reaching a limit cycle. For even smaller capillary numbers, the time invariance is restored and the particle reaches a steady equilibrium position off the symmetry plane. This series of bifurcations is analyzed and possible physical mechanisms from which they originate are discussed.