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

TL;DR

This paper investigates how deformation of a neutrally buoyant spherical drop in planar linear flows affects the topology of streamlines, revealing that deformation destroys closed streamlines and creates complex flow patterns with implications for transport and mixing.

Contribution

It provides a combined analytical and numerical analysis showing that drop deformation alters streamline topology, contradicting previous assumptions of topology persistence.

Findings

Deformation destroys closed streamline topology around the drop.

Only some closed streamlines become open spirals, others form nested tori.

Results have implications for transport and mixing in fluid flows.

Abstract

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 $0 \leq \hat{\alpha} < 1$ and for $\lambda > \lambda_c = 2 \hat{\alpha} / (1 - \hat{\alpha})$, $\lambda$ 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 spiraling 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 spiraling 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.