Displacement of a three-dimensional immiscible droplet in a duct

The displacement of a three-dimensional immiscible droplet subject to gravitational forces in a duct is studied with the lattice Boltzmann method. The effects of the contact angle and capillary number (the ratio of viscous to surface forces) on droplet dynamics are investigated. It is found that there exists a critical capillary number for a droplet with a given contact angle. When the actual capillary number is smaller than the critical value, the droplet moves along the wall and reaches a steady state. When the capillary number is greater than the critical value, one or more small droplets pinch off from the wall or from the rest of the droplet, depending on the contact angle and the specific value of the capillary number. As the downstream part of the droplet is pinching off, a bottleneck forms and its area continues decreasing until reaching zero. The general trend found in a previous two-dimensional study that the critical capillary number decreases as the contact angle increases is confirmed. It is shown that at a fixed capillary number above the critical value, increasing the contact angle results in a larger first-detached portion. At a fixed contact angle, increasing the capillary number results in an increase of the size of the first detached droplet for $\theta\,{=}\,78^\circ$ and $\theta\,{=}\,90^\circ$, but a decrease for $\theta\,{=}\,118^\circ$. It is also found that the droplet is stretched longer as the capillary number becomes larger. For a detaching droplet, the maximal velocity value occurs near the bottleneck between the up-and downstream parts of the droplet and the shear stress there reaches a local maximum. The three-dimensional effects are most clearly seen for $\theta\,{=}\, 90^\circ$, where the wetted length and wetted area vary in the opposite direction and the shape of the interface between the wall and the droplet is distorted severely from the original round shape.