A viscous drop in a planar linear flow: the role of deformation on streamline topology

Planar linear flows are a one-parameter family, with the parameter $\hat {\alpha }\in [-1,1]$ being a measure of the relative magnitudes of extension and vorticity; $\hat {\alpha } = -1$ , $0$ and $1$ correspond to solid-body rotation, simple shear flow and planar extension, respectively. For a neutrally buoyant spherical drop in a hyperbolic planar linear flow with $\hat {\alpha }\in (0,1]$ , the near-field streamlines are closed for $\lambda \gt \lambda _c = 2 \hat {\alpha } / (1 - \hat {\alpha })$ , $\lambda$ being the drop-to-medium viscosity ratio; all streamlines are closed for an ambient elliptic linear flow with $\hat {\alpha }\in [-1,0)$ . We use both analytical and numerical tools to show that drop deformation, as characterized by a non-zero capillary number ( $Ca$ ), destroys the aforementioned closed-streamline topology. While inertia has previously been shown to transform closed Stokesian streamlines into open spiralling ones that run from upstream to downstream infinity, the streamline topology around a deformed drop, for small but finite $Ca$ , is more complicated. Only a subset of the original closed streamlines transforms to open spiralling ones, while the remaining ones densely wind around a configuration of nested invariant tori. Our results contradict previous efforts pointing to the persistence of the closed streamline topology exterior to a deformed drop, and have important implications for transport and mixing.