Wake instability of a fixed spherical droplet with a high drop-to-fluid viscosity ratio

Direct numerical simulations of a uniform flow past a fixed spherical droplet are performed to investigate the parameter range within which the axisymmetric flow becomes unstable due to an external flow bifurcation. The hydrodynamics is governed by three dimensionless numbers: the viscosity ratio, $\mu^\ast$, and the external and internal Reynolds numbers, $\Rey^e$ and $\Rey^i$, respectively. The drop-to-fluid density ratio is related to these parameters as $\rho^\ast=\mu^\ast \Rey^i/\Rey^e$. This study focuses on highly viscous droplets with $\mu^\ast \geq 5$, where wake instability is driven by the vorticity flux transferred from the droplet surface into the surrounding fluid. By analysing the wake structure, we confirm that the onset of the external bifurcation is linked to the tilting of the azimuthal vorticity, $\omega_\phi$, in the wake and that the bifurcation occurs once the isocontours of $\omega_\phi$ align nearly perpendicular to the symmetry axis. We propose an empirical criterion for predicting the onset of the external bifurcation, formulated in terms of the maximum vorticity on the external side of the droplet surface. This criterion is applicable for sufficiently high $\Rey^i$ and holds over a wide range of $\mu^\ast$ and $\Rey^e$. Additionally, we examine the bifurcation sequence for two specific external Reynolds numbers, $\Rey^e=300$ and $\Rey^e=500$, and show that, beyond a critical viscosity ratio, the axisymmetric wake first transitions to a steady planar-symmetric state before undergoing a secondary Hopf bifurcation. Finally, we highlight the influence of $\Rey^i$ on external bifurcation and show that, at moderate $\Rey^i$, wake instability may set in at a lower vorticity threshold than predicted by our criterion. These findings provide new insights into the external flow bifurcation of viscous droplets.