# Bifurcation dynamics of a particle-encapsulating droplet in shear flow

## TL;DR

This paper investigates the complex bifurcation behavior of a particle encapsulated within a deforming droplet under shear flow, revealing new equilibrium states and the underlying hydrodynamic interactions through a dynamical systems approach.

## Contribution

It introduces a detailed numerical analysis of bifurcations in particle-encapsulating droplets, highlighting new equilibrium solutions and the role of hydrodynamic forces in their emergence.

## Key findings

- Identification of multiple equilibrium solutions including eccentric and time-periodic states.
- Discovery of bifurcation points such as supercritical pitchfork and Hopf bifurcations.
- Elucidation of hydrodynamic interactions driving the bifurcations.

## Abstract

To understand the behavior of composite fluid particles such as nucleated cells and double-emulsions in flow, we study a finite-size particle encapsulated in a deforming droplet under shear flow as a model system. In addition to its concentric particle-droplet configuration, we numerically explore other eccentric and time-periodic equilibrium solutions, which emerge spontaneously via supercritical pitchfork and Hopf bifurcations. We present the loci of these solutions around the codimenstion-two point. We adopt a dynamical system approach to model and characterize the coupled behavior of the two bifurcations. By exploring the flow fields and hydrodynamic forces in detail, we identify the role of hydrodynamic particle-droplet interaction which gives rise to these bifurcations.