Particle encapsulation due to thread breakup in Stokes flow

The capillary instability of a liquid thread containing a regular array of spherical particles along the centreline is considered with reference to microencapsulation. The thread interface may be clean or occupied by an insoluble surfactant. The main goal of the analysis is to illustrate the effect of the particle spacing on the growth rate of axisymmetric perturbations and identify the structure of the most unstable modes. A normal-mode linear stability analysis based on Fourier expansions for Stokes flow reveals that, at small particle separations, the interfacial profiles are nearly pure sinusoidal waves whose growth rate is nearly equal to that of a pure thread devoid of particles. Higher harmonics suddenly enter the normal modes for moderate and large particle separations, elevating the growth rates and yielding a stability diagram that consists of a sequence of superposed pure-thread lobes. A complementary numerical stability analysis based on the boundary integral formulation for Stokes flow reveals the strong stabilizing effect of particles whose radius is comparable to the thread radius. Numerical simulations of the finite-amplitude motion based on the boundary integral method demonstrate that thread breakup leads to particles coated with annular layers of different thicknesses.