Lift force on a spherical droplet in a viscous linear shear flow

We study numerically the flow around a spherical droplet set fixed in a linear shear flow with moderate shear rates ( $Sr\leq 0.5$ , $Sr$ being the ratio between the velocity difference across the drop and the relative velocity) over a wide range of external Reynolds numbers ( $0.1<{{Re}}\leq 250$ , ${{Re}}$ based on the slip velocity and the viscosity of the external fluid) and drop-to-fluid viscosity ratios ( $0.01\leq \mu ^\ast \leq 100$ ). The flow structure, the vorticity field and their intrinsic connection with the lift force are analysed. Specifically, the results on lift force are compared with the low- ${{Re}}$ solution derived for droplets of arbitrary $\mu ^\ast$ , as well as prior data at finite ${{Re}}$ available in both the clean-bubble limit ( $\mu ^\ast \to 0$ ) and the solid-sphere limit ( $\mu ^\ast \to \infty$ ). Notably, at ${{Re}}=O(100)$ , the lift force exhibits a non-monotonic transition from $\mu ^\ast \to 0$ to $\mu ^\ast \to \infty$ , peaking at $\mu ^\ast \approx 1$ . This behaviour is related to an internal three-dimensional flow bifurcation also occurring under uniform-flow conditions, which makes the flow to evolve from axisymmetric to biplanar symmetric. This flow bifurcation occurs at low-but-finite $\mu ^\ast$ when the internal Reynolds number ( ${{Re}}^i$ , based on the viscosity of the internal fluid) exceeds approximately 300. In the presence of shear, the corresponding imperfect bifurcation enhances the extensional rate of the flow in the wake. Consequently, the streamwise vortices generated behind the droplet can be more intense compared with those behind a clean bubble. Given the close relation between the lift and these vortices, a droplet with ${{Re}}=O(100)$ and $\mu ^\ast \approx 1$ typically experiences a greater lift force than that in the inviscid limit.