A complete quasistatic model for the permeation of a droplet between two reservoirs

The permeation of a droplet between two reservoirs through a capillary tube is investigated. A one-dimensional model is developed and a computational fluid dynamics (CFD) analysis is conducted to highlight the different regimes associated with this process. Three configurations of the droplet during permeation/emergence can be identified. For a permeating droplet, they are, namely (1) a movement of the contact line along the surface of the reservoir with a constant contact angle with the radius of curvature of the interface decreasing, (2) once the contact line reaches the capillary tube, it pauses (i.e., pins) with the radius of curvature continues to decrease and the contact angle changes, and (3) a brief process of increasing the radius of curvature until the droplet totally penetrates into the tube. These steps are repeated but in an opposite order as the droplet emerges from the tube into the bottom reservoir. A force balance over a control volume that spans the capillary tube indicates that the pressure at the top and at the bottom of the tube changes with time in connection with the continuous change of the curvature of the droplet at the two reservoirs. Furthermore, during the movement of the advancing interface in the capillary tube, the interfacial tension force opposes the driving pressure force, whereas, when the receding interface clears off the tube the opposite occurs. Comparisons between the developed model and the CFD analysis match very well throughout the whole permeation process, which builds confidence in the modeling approach.