Flow in a containerless liquid system: Ring-sheared drop with finite surface shear viscosity

The ring-sheared drop is a flow configuration for microgravity, where surface tension provides containment and shear in the bulk is driven primarily by the action of surface shear viscosity. A drop is constrained by two thin contact rings, i.e., one stationary at a southern latitude and the other at the same latitude but in the north and rotating. Since we consider a microgravity setting, the drop is not restricted to being small. Furthermore, we allow for arbitrarily small surface shear viscosity, so that in general the interfacial and bulk flows are viscously coupled. Our numerical simulations show that even small surface shear viscosity (quantified nondimensionally by a Boussinesq number) can produce a significant meridional bulk flow at moderate ring rotation rates (quantified by a Reynolds number Re). At very low Re, the bulk flow is viscously dominated and surface viscosity makes very little difference. At high Re, the secondary flow is very weak if the surface viscosity is negligible and the flow tends toward solid-body rotation.

Figure 1

(a) Schematic of the ring-sheared drop and (b) a three-dimensional view of the flow in the limit of a large surface shear viscosity. Colors in (b) represent the azimuthal velocity ranging from yellow (minimum) to red (maximum). The vectors in (b) show the secondary flow in the $(r,z)$-meridional plane.

Figure 2

Numerical profiles of surface azimuthal velocity $v^s$ at $\mathrm{Re}=10$ and $\mathrm{Re}=1000$, both with $\mathrm{Bo}=100$ compared to the analytic profile from (8).

Figure 3

Profiles of the surface azimuthal velocity $v^s$ for $\mathrm{Re}$ and Bo as indicated.

Figure 4

Contours of vortex lines $\gamma$ (left) and azimuthal vorticity $\eta$ (right), for $\mathrm{Re}$ and Bo, as indicated. There are 15 levels in $\gamma \in [0,0.6]$ and 50 levels in $\eta \in [-7,7]$.

Figure 5

Variation of the viscous dissipation rate $\mathrm{\Phi }$ with Bo and $\mathrm{Re}$ as indicated.