Deformable droplet under Poiseuille flow: Role of flow direction, channel inclination and off-centre dynamics

The dynamics of a confined, deformable droplet with an imposed external flow (upward, quiescent and downward) is numerically investigated using an in-house solver based on the Level Set method. This is the first comprehensive investigation unravelling (i) the oscillatory dynamics of an off-centered droplet under an external flow, (ii) the hydrodynamics of a droplet subjected to a downward flow, and (iii) the effect of channel orientation. Both co-current and counter-current droplet motions are explored. While a freely falling initially off-centered droplet demonstrates an oscillatory trajectory with significant shape deformation, we show that an imposed downward flow dampens the oscillations and minimizes deformation resulting in a greater terminal speed of the droplet. Conversely, an upward, counter-current flow causes greater droplet deformation with increased oscillations. Moreover, reducing the channel inclination leads to asymmetric droplet shapes with uneven film thickness on either side of the channel and a higher residence time. The other key findings include (i) transition from convex to concave droplet tail in both upward and downward flows, dictated by the strength of external flow and $\mathrm{Bo}$. (ii) Formation of a streamlined droplet accompanied by an increase in the film thickness on increasing viscosity ratio in both co-current and counter-current droplet flows and (iii) the generation of novel flow maps in the density ratio - pressure drop parameter space delineating three distinct regions: downward sinking, upward rising, and stationary droplet, depending on the relative strength of buoyancy and external flow.