Bifurcation Dynamics of a Particle-Encapsulating Droplet in Shear Flow

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 codimension-two point. We adopt a dynamic 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.

Figure 1

(a) Sketch: a spherical particle moving inside a droplet under shear. A snapshot of the composite system cut by the $y=0$ (b) and $x=0$ (c) plane.

Figure 2

Evolution of the in-plane and spanwise displacements $d_{xz}$ (green) and $d_y$ (red) between the centers of particle (blue) and droplet (pink), for size ratio $\alpha =0.4$ and $\mathrm{Ca}=0.2$ (first row), 0.275 (second row), and 0.29 (third row). The second (third) column shows the profile of the composite system at equilibrium, cut by $x=0$ ($y=0$) plane. The particle position is shown at different instants within a period for $\mathrm{Ca}=0.275$, the insets of (d) and (f) display the limit cycle solution.

Figure 3

Equilibrium displacements ${\mathrm{\Delta }}_y$, ${\mathrm{\Delta }}_{xz}^{\min /\max }$ versus Ca for $\alpha =0.3$ (a), 0.4 (b), and 0.5 (c); ${}^{\sim }$ denotes their counterparts of the decoupled simulations. (d) Linear fitting of ${\tilde{\mathrm{\Delta }}}_y$ and ${\tilde{\mathcal{A}}}_{xz}={\tilde{\mathrm{\Delta }}}_{xz}^{\max }-{\tilde{\mathrm{\Delta }}}_{xz}^{\min }$ versus ${[{\mathrm{Ca}}_c(\alpha )-\mathrm{Ca}]}^{1/2}$.

Figure 4

(a) Parametric portrait displaying the loci of four solution types in the $(\mathrm{Ca},\alpha )$ space. (b) Close-up of (a) in the vicinity of the codimension-two point $({\mathrm{Ca}}_{*},{\alpha }_{*})$ from where six bifurcation curves originate. The inset shows the bifurcation topology in the $({\mu }_1,{\mu }_2)$ space.

Figure 5

Droplet-induced disturbance flow ${\widehat{\mathbf{u}}}_d$ of concentric systems in the extensional flow, shown on the $y=0$ (a) and $x=z$ (b) plane, for $\alpha =0.5$, $\mathrm{Ca}=0.1$ and 0.2. (c) Hydrodynamic force $F_Y(\mathrm{Ca})$ exerted on the particle with a fixed offset $(0,d_y^{\mathrm{fix}}>0,0)$ inside the droplet under shear; $F_Y^p$ and $F_Y^{\eta }$ denote the pressure and viscous parts of $F_Y$, respectively, and $F_Y^{\eta }{|}_s$ the shear force of $F_Y^{\eta }$. The composite profile is shown on the $z=0$ plane for $\mathrm{Ca}=0.15$ (d) and 0.325 (e).