Investigation of the Different Regimes Associated with the Growth of an Interface at the Exit of a Capillary Tube into a Reservoir: Analytical Solutions and CFD Validation

Abstract

The emergence of a droplet from a capillary tube opening into a reservoir is an important phenomenon in several applications. In this work, we are particularly interested in this phenomenon in an attempt to highlight the physics behind droplet appearance. The emergence of a droplet from a tube opening into a reservoir under quasi-static conditions passes through three stages. The first stage starts when the meniscus in the tube reaches the exit. At this moment, the meniscus intersects the wall of the tube at the equilibrium contact angle. The interface then develops until its radius of curvature becomes equal to the tube radius. During this stage, the capillary pressure increases. In the second stage, the interface continues to evolve with its radius of curvature increasing until the static contact angle with respect to the surface of the reservoir is achieved. This marks the end of the second stage and the start of the third in which the contact line (CL) starts to depart the tube opening along the reservoir surface and the contact angle remains constant. Analytical models for the three stages have been derived based on the law of conservation of linear momentum. The models account for pressure, gravitational, capillary, and viscous forces, while inertia force is ignored. The model can predict the profiles of the mean velocity in the tube, the capillary pressure, and the evolution of the contact angle. In addition, a computational fluid dynamics (CFD) simulation has been conducted to provide a framework for validation and verification of the developed model. The CFD simulation shows qualitative behavior in terms of snapshots of the emerging droplet with time similar to that speculated by the analytical model. In addition, quantitative comparisons with respect to velocity, pressure, and volume profiles of the droplet show very good agreement, which builds confidence in the modeling approach.

Fates of a Nonwetting Slug in Tapered Microcapillaries under Gravity and Zero Gravity Conditions: Dynamics, Asymptotic Equilibrium Analysis, and Computational Fluid Dynamics Verifications

“…These are (1) when Bo ≪ 1.0, there is an equilibrium state, and, in this case, the slug moves toward the wider end of the tube (i.e., upward) until reaching an equilibrium position, at which case, both menisci assume the same curvature, i.e., R 1 = R 2 = R eq , (2) when Bo < 1.0, there is an equilibrium state, and the slug moves toward the wider end (i.e., upward) and stops before reaching the same extent as that of the previous case, and in this case R 1 < R 2 , (3) when Bo > 1.0, gravity dominates and it pulls the slug downward toward the narrower end of the tube until an equilibrium state is reached, and (4) when Bo ≫ 1.0, there is no equilibrium state, and the slug drains downward in the direction of the gravity. It is to be noted that, in this scenario, the slug drains out of the tube in two possible ways; namely, (1) as a continuous phase or (2) as an emerging droplet. − The first case occurs when the slug drains into a reservoir filled with the same fluid as the slug, and the second occurs when the slug drains into a reservoir filled with a wetting fluid. However, these two possibilities will not affect the fate of the slug.…”

“…In order to provide verifications of the highlighted fate criteria, CFD analyses are conducted on four cases representing the four fates. Details of the CFD setup, governing equations, mesh resolution and sensitivity, boundary conditions, validation, and time stepping can be found in several previous studies, e.g., refs , , and – and will not be repeated here for the sake of brevity. It is important to note that the capillary length, defined as √(γ/ρ g ), manifests an important length scale that can be used to check the sphericity of the meniscus.…”