A generalized analytical model for estimating the rate of imbibition/drainage of wetting/nonwetting fluids in capillaries

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

It has been determined experimentally and numerically that a nonwetting slug in a tapered capillary tube, under the sole action of capillary force, self-propels itself toward the wider end of the tube until an equilibrium state is reached. The aim of this work is to highlight the state of the slug at equilibrium in terms of configuration and location. Furthermore, it turns out that gravity adds richness to this phenomenon, and more fates become possible. A modified Bond number is developed that determines the relative importance of gravity and capillarity for this system. According to the magnitude of the Bond number, three more fates are possible. Therefore, in a tapered capillary tube held vertically upward with its wider end at the top, in the absence of gravity or under microgravity conditions, the nonwetting slug moves upward toward the wider end of the tube until it reaches equilibrium with the two menisci part of a single sphere. The location of the slug at equilibrium in this case represents the farthest fate among the other fates. When gravity exists yet capillarity dominates, the slug still moves upward toward the wider end. However, in this case, the two menisci become parts of two different spheres of different curvatures. For this scenario, the slug climbs upward but reaches a lower level compared to the previous scenario. On the other hand, when gravity dominates, the slug experiences a net downward pull toward the narrower end of the tube and starts to move in the direction of gravity until capillary force establishes a balance, then it stops. When gravity sufficiently dominates, it pulls the slug downward until it completely drains off the tube. A computational fluid dynamics (CFD) analysis is conducted in order to build a framework for verification exercises. Excellent agreements between the results of the developed model and the CFD analysis are obtained. A fate map and a scheme are developed to identify these four fates based on two Bond numbers; namely, the initial Bond number and that associated with the slug when it is at the exit.

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

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.

Capillary imbibition and flow of wetting liquid in irregular capillaries: A 100-year review

Capillary imbibition, such as plant roots taking up water, reservoir rocks absorbing brine and a tissue paper wiping stains, is pervasive occurred in nature, engineering and industrial fields, as well as in our daily life. This phenomenon is earliest modeled through the process that wetting liquid is spontaneously propelled by capillary pressure into regular geometry models. Recent studies have attracted more attention on capillary-driven flow models within more complex geometries of the channel, since a detailed understanding of capillary imbibition dynamics within irregular geometry models necessitates the fundamentals to fluid transport mechanisms in porous media with complex pore topologies. Herein, the fundamentals and concepts of different capillary imbibition models in terms of geometries over the past 100 years are reviewed critically, such as circular and non-circular capillaries, open and closed capillaries with triangular/rectangular cross-sections, and heterogeneous geometries with axial variations. The applications of these models with appropriate conditions are discussed in depth accordingly, with a particular emphasize on the capillary flow pattern as a consequence of capillary geometry. In addition, a universal model is proposed based on the dynamic wetting condition and equivalent cylindrical geometry to describe the capillary imbibition process in terms of various solid topologies. Finally, future research is suggested to focus on analyzing the dynamics during corner flow, the snap-off of wetting fluid, the capillary rise of non-Newtonian fluids and applying accurate physical simulation methods on capillary-driven flow processes. Generally, this review provides a comprehensive understanding of the capillary-driven flow models inside various capillary geometries and affords an overview of potential advanced developments to enhance the current understanding of fluid transport mechanisms in porous media.

Capillary-Driven Ejection of a Droplet from a Micropore into a Channel: A Theoretical Model and a Computational Fluid Dynamics Verification

In this work, the problem of re-ejection of a permeating droplet through a membrane pore back to the feed channel when the transmembrane pressure (TMP) becomes zero is investigated. This problem is important in the context of oily water filtration using membranes. In particular, in the novel periodic feed pressure technique (PFPT), which has been proposed to combat membrane fouling, the TMP alternates between the operating value and zero in a periodic manner. During the period in which TMP is high, filtration occurs, and when it is zero, cleaning commences. We are particularly interested in what happens to a droplet, initially undergoing permeation, when the TMP becomes zero. It is evident that when the TMP is zero the meniscus inside the pore reverses its motion toward the feed channel rather than toward the permeate side by the action of interfacial tension force. A theoretical model is built to determine the rate at which the meniscus inside the pore advances when the TMP is zero. The conservation of momentum equation is used to establish a one-dimensional model that updates the location of the meniscus with time. The derived model considers both quasi-static and dynamic scenarios. In addition, the model accounts for both the viscosity contrast between the two fluids, as well as the gravity. A computational fluid dynamics (CFD) simulation has been built to provide a framework for model verification and validation. The model, based on quasi-static conditions, provides an overall similar trend to that obtained via CFD analysis. Nevertheless, the quasi-static model predicts a more rapid meniscus advancement inside the pore than the CFD simulation. When the dynamic contact angle is incorporated, very good matching is observed.

Fog Harvesting with Highly Wetting and Nonwetting Vertical Strips

Highly wetting and nonwetting substrates have been widely used in fogwater collection systems for enhanced water harvesting. In this work, fog harvesting substrates comprising PVC strips of different wetting properties and widths ranging from 1-5 mm were vertically aligned and spaced apart at regular intervals to give the same solid area fraction of 0.8. Evaluation of the water collection efficiencies of the tested configurations revealed that 1 mm wide superhydrophilic strips was the most efficient, achieving double the amount of water harvested compared with 2.8 mm wide strips. This finding was attributed to the low Stokes numbers of the aerosol particle distribution of the fog which tended to result in them being brought by the flow streamlines toward the air gaps between the strips. Stagnant flow regions at the edges of each strip, revealed through potential flow calculations, then caused higher liquid imbibition and impaction there for water harvesting. It was also found that the Cassie nonwetting substrates that originally exhibited contact angles of 161° transformed to Wenzel wetting with zero contact angle within 60 min of fog interception. Optical profilometry revealed no obvious difference in surface roughness between the central region and edges of the strips, indicating that surface morphology was unlikely to be a contributing factor for enhanced water collection at the edges. The findings here indicated that highly wetting vertical strip architectures with narrow widths (1 mm) were favorable over wider strips for water harvesting provided that clogging and re-entrainment were not significant factors.