Simulation of hydrodynamically interacting particles confined by a spherical cavity

We present a theoretical framework to model the behavior of a concentrated colloidal dispersion confined inside a spherical cavity. Prior attempts to model such behavior were limited to a single enclosed particle and attempts to enlarge such models to two or more particles have seen limited success owing to the challenges of accurately modeling many-body and singular hydrodynamic interactions. To overcome these difficulties, we have developed a set of hydrodynamic mobility functions that couple particle motion with hydrodynamic traction moments that, when inverted and combined with near-field resistance functions, form a complete coupling tensor that accurately captures both the far-field and near-field physics and is valid for an arbitrary number of spherical particles enclosed by a spherical cavity of arbitrary relative size a/R, where a and R are the particle and cavity size, respectively. This framework is then utilized to study the effect of spherical confinement on the self-and entrained motion of the colloids, for a range of particle-to-cavity size ratios. The self-motion of a finite-size enclosed particle is studied first, recovering prior results published in the literature: The hydrodynamic mobility of the particle is greatest at the center of the cavity and decays as (a/R)/(1-y2), where y is the particle distance to the cavity center. Near the cavity wall, the no-slip surfaces couple strongly and mobility along the cavity radius vanishes as xi (=) R - (a + y), where y is center-to-center distance from particle to cavity. Corresponding motion transverse to the cavity radius vanishes as [ln(1/xi)](-1). The effect of confinement on entrainment of a particle in the flow created by the motion of others is also studied, where we find that confinement exerts a qualitative effect on the strength and anisotropy of entrainment of a passive particle dragged by the flow of a forced particle. As expected, entrainment strength decays with increased distance from the forced particle. Surprisingly, however, there is a separation beyond which entrainment changes sign. For some configurations, the passive particle is dragged along with the forced particle, and at others, it is driven in the opposite direction, consistent with observations of recirculating flowand reverse particle migration in eukaryotic cells. The mobility functions presented here can be utilized to model the motion of any number of enclosed particles, making them ideal for use in dynamic simulation.