A viscous drop in a planar linear flow -- the role of deformation on streamline topology
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| Format: | Preprint |
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2024
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| author | Narayanan, Sabarish V. Subramanian, Ganesh |
| author_facet | Narayanan, Sabarish V. Subramanian, Ganesh |
| contents | Planar linear flows are a one-parameter family, with the parameter $\hatα\in [-1,1]$ being a measure of the relative magnitudes of extension and vorticity; $\hatα = -1$, $0$ 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 $\hatα\in(0,1]$, the near-field streamlines are closed for $0 \leq \hatα < 1$ and for $λ> λ_c = 2 \hatα / (1 - \hatα)$, $λ$ being the drop-to-medium viscosity ratio; all streamlines are closed for an ambient elliptic linear flow with $\hatα\in[-1,0)$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_02823 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A viscous drop in a planar linear flow -- the role of deformation on streamline topology Narayanan, Sabarish V. Subramanian, Ganesh Fluid Dynamics Planar linear flows are a one-parameter family, with the parameter $\hatα\in [-1,1]$ being a measure of the relative magnitudes of extension and vorticity; $\hatα = -1$, $0$ 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 $\hatα\in(0,1]$, the near-field streamlines are closed for $0 \leq \hatα < 1$ and for $λ> λ_c = 2 \hatα / (1 - \hatα)$, $λ$ being the drop-to-medium viscosity ratio; all streamlines are closed for an ambient elliptic linear flow with $\hatα\in[-1,0)$. 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. |
| title | A viscous drop in a planar linear flow -- the role of deformation on streamline topology |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2406.02823 |