Numerical Simulations of Capillary-Driven Flows in Nonuniform Cross-Sectional Capillaries

The Fastest Capillary Flow in Root-like Networks under Gravity

Capillary flow has garnered significant attention due to its unique dynamic characteristics that require no external force. Creating a quantitative analytical model to evaluate capillary flow behaviors in root-like networks is essential for enhancing fluid control properties in functional textiles. In this study, we explore the capillary dynamics within root-like networks under the influence of gravity and derive the most rapid capillary flow via structural optimization. The flow time in a capillary is dominated by the capillary pressure, viscous pressure loss, and gravity, each of which exhibits diverse sensitivities to the structures of root-like networks. We scrutinize various structural parameters to understand their impact on capillary flow in root-like networks. Subsequently, optimal structural parameters (namely, the mother tube diameter and diameter ratio) are identified to minimize capillary flow time. Moreover, we discovered that the correlation between flow time and distance for capillary flow in root-like networks does not obey the classical Lucas-Washburn equation. These results affirm that root-like networks can enhance capillary flow, providing critical insights for numerous capillary-flow-dependent engineering applications.

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.

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.

Effect of Fluid Properties on Contact Angles in the Eagle Ford Shale Measured with Spontaneous Imbibition

Models of fluid flow are used to improve the efficiency of oil and gas extraction and to estimate the storage and leakage of carbon dioxide in geologic reservoirs. Therefore, a quantitative understanding of key parameters of rock-fluid interactions, such as contact angles, wetting, and the rate of spontaneous imbibition, is necessary if these models are to predict reservoir behavior accurately. In this study, aqueous fluid imbibition rates were measured in fractures in samples of the Eagle Ford Shale using neutron imaging. Several liquids, including pure water and aqueous solutions containing sodium bicarbonate and sodium chloride, were used to determine the impact of solution chemistry on uptake rates. Uptake rate analysis provided dynamic contact angles for the Eagle Ford Shale that ranged from 51 to 90° using the Schwiebert-Leong equation, suggesting moderately hydrophilic mineralogy. When corrected for hydrostatic pressure, the average contact angle was calculated as 76 ± 7°, with higher values at the fracture inlet. Differences in imbibition arising from differing fracture widths, physical liquid properties, and wetting front height were investigated. For example, bicarbonate-contacted samples had average contact angles that varied between 62 ± 10° and ∼84 ± 6° as the fluid rose in the column, likely reflecting a convergence-divergence structure within the fracture. Secondary imbibitions into the same samples showed a much more rapid uptake for water and sodium chloride solutions that suggested alteration of the clay in contact with the solution producing a water-wet environment. The same effect was not observed for sodium bicarbonate, which suggested that the bicarbonate ion prevented shale hydration. This study demonstrates how the imbibition rate measured by neutron imaging can be used to determine contact angles for solutions in contact with shale or other materials and that wetting properties can vary on a relatively fine scale during imbibition, requiring detailed descriptions of wetting for accurate reservoir modeling.

Open-Channel Capillary Trees and Capillary Pumping

Velocity of capillary flow in closed or open channels decreases as the flow proceeds down the length of the channel, varying as the inverse of the square root of time or as the inverse of travel distance. In order to increase the flow rate-and extend the duration of the flow-capillary pumps have been designed by mimicking the pumping principle of paper or cotton fibers. These designs provide a larger volume available for the wicking of the liquids. In microsystems for biotechnology, different designs have been developed based on experimental observation. In the present paper, the mechanisms at the basis of capillary pumping are investigated using a theoretical model for the flow in an open-channel "capillary tree" (i.e., an ensemble of channels with bifurcations mimicking the shape of a tree). The model is checked against experiments. Rules for obtaining better designs of capillary pumps are proposed; specifically, we find (1) when using a capillary tree with identical channel cross-sectional areas throughout, it is possible to maintain nearly constant flow rates throughout the channel network, (2) flow rate can be increased at each branch point of a capillary tree by slightly decreasing the areas of the channel cross section and decreasing the channel lengths at each level of ramification within the tree, and (3) higher order branching (trifurcations vs bifurcations) amplify the flow rate effect. This work lays the foundation for increasing the flow rate in open microfluidic channels driven by capillary flow; we expect this to have broad impact across open microfluidics for biological and chemical applications such as cell culture, sample preparation, separations, and on-chip reactions.

Equilibrium structures of water molecules confined within a multiply connected carbon nanotube: a molecular dynamics study

Water confinement inside a carbon nanotube (CNT) has been one of the most exciting subjects of both experimental and theoretical interest. Most of the previous studies, however, considered CNT structures with simple cylindrical shapes. In this paper, we report a classical molecular dynamics study of the equilibrium structural arrangement of water molecules confined in a multiply connected carbon nanotube (MCCNT) containing two Y-junctions. We investigate the structural arrangement of the water molecules in the MCCNT in terms of the density of water molecules and the average number of hydrogen bonds per water molecule. Our results show that the structural rearrangement of the H2O molecules takes place several angstroms ahead of the Y-junction, rather than only at the CNT junction itself. This phenomenon arises because it is difficult to match the boundary condition for hydrogen bonding in the region where two different hydrogen-bonded structures are interconnected with each other.

Synergistic actions of mixed small and large pores for capillary absorption through biporous polymeric materials

Water absorption in porous media is an important process involved in numerous materials for various applications, such as in the building industry, food processing and bioengineering. Designing new materials with appropriate absorption properties requires an understanding of how absorption behavior depends on both the material's morphology and the properties of the solid matrix, i.e. hydrophilic/hydrophobic nature and swelling/deformation properties. Although the basic principles of imbibition are well-known for simple porous systems, much less is known about absorption in complex porous systems, in particular those containing several coexisting porous phases, such as wood for example. Here, water absorption is studied for model porous organic materials exhibiting several degrees of hydrophobicity and two pore size levels, either as monoporous materials (large or small pores) or as biporous materials (mixed large and small pores). The interconnected biporous structure is designed via a double porogen templating approach using cubic sodium chloride particles as templates for the generation of the larger pore size (250-400 μm) and i-PrOH as a porogenic solvent for the smaller pore size (2-5 μm). While absorption for the small pore material is well described by the classical Washburn theory, the large pore material shows a drastic reduction in the imbibition rate. This behavior is attributed to the slow breakthrough mechanism for the water interface at sharp edge connections between pores. Remarkably, this slow regime is suppressed for the biporous material and the imbibition rate is even higher than the sum of rates obtained for its monoporous counterparts, which highlights the synergistic action of mixed small and large pores.

Nanojunction Effects on Water Flow in Carbon Nanotubes

We report on the results of extensive molecular dynamics simulation of water imbibition in carbon nanotubes (CNTs), connected together by converging or diverging nanojunctions in various configurations. The goal of the study is to understand the effect of the nanojunctions on the interface motion, as well as the differences between what we study and water imbibition in microchannels. While the dynamics of water uptake in the entrance CNT is the same as that of imbibition in straight CNTs, with the main source of energy dissipation being the friction at the entrance, water uptake in the exit CNT is more complex due to significant energy loss in the nanojunctions. We derive an approximate but accurate expression for the pressure drop in the nanojunction. A remarkable difference between dynamic wetting of nano- and microjunctions is that, whereas water absorption time in the latter depends only on the ratios of the radii and of the lengths of the channels, the same is not true about the former, which is shown to be strongly dependent upon the size of each segment of the nanojunction. Interface pinning-depinning also occurs at the convex edges.

Modeling the abnormally slow infiltration rate in mesoporous films

The rate of infiltration of fluids into bimodal porous systems was modeled to decrease as the ratio of the pore size to pore neck size was increased.

Capillary Displacement of Viscous Liquids in Geometries with Axial Variations

Axial variations in geometry and presence of viscous displaced fluid are known to alter the diffusive-dynamics of capillary imbibition of a wetting liquid. We here show that the coupled effect of axially varying capillary geometry and finite viscosity of the displaced fluid can lead to significant variations in both short and long time dynamics of imbibition. Based on a theoretical model and lattice Boltzmann simulations, we analyze capillary displacement of a viscous liquid in straight and diverging capillaries. At short times, the imbibition length scales proportionally with time as opposed to the diffusive-dynamics of imbibition of a single wetting liquid. Whereas, at long times, geometry-dependent power-law behavior occurs which qualitatively resembles single liquid imbibition. The distance at which the crossover between these two regimes occurs depends strongly on the viscosities of the imbibing and the displaced liquid. Additionally, our simulations show that the early time imbibition dynamics are also affected by the dynamic contact angle of the meniscus.

Capillary Flow Resistors: Local and Global Resistors

The use of capillary systems in space and biotechnology applications requires the regulation of the capillary flow velocity. It has been observed that constricted sections act as flow resistors. In this work, we also show that enlarged sections temporarily reduce the velocity of the flow. In this work, the theory of the dynamics of capillary flows passing through a constricted or an enlarged channel section is presented. It is demonstrated that the physics of a capillary flow in a channel with a constriction or an enlargement is different and that a constriction acts as a global flow resistor and an enlargement as a local flow resistor. The theoretical results are checked against experimental approaches.

Filling of charged cylindrical capillaries

We provide an analytical model to describe the filling dynamics of horizontal cylindrical capillaries having charged walls. The presence of surface charge leads to two distinct effects: It leads to a retarding electrical force on the liquid column and also causes a reduced viscous drag force because of decreased velocity gradients at the wall. Both these effects essentially stem from the spontaneous formation of an electric double layer (EDL) and the resulting streaming potential caused by the net capillary-flow-driven advection of ionic species within the EDL. Our results demonstrate that filling of charged capillaries also exhibits the well-known linear and Washburn regimes witnessed for uncharged capillaries, although the filling rate is always lower than that of the uncharged capillary. We attribute this to a competitive success of the lowering of the driving forces (because of electroviscous effects), in comparison to the effect of weaker drag forces. We further reveal that the time at which the transition between the linear and the Washburn regime occurs may become significantly altered with the introduction of surface charges, thereby altering the resultant capillary dynamics in a rather intricate manner.

Dynamics of a capillary invasion in a closed-end capillary

The position of fluid invasion in an open capillary increases as the square root of time and ceases when the capillary and hydrostatic forces are balanced, when viscous and inertia terms are negligible. Although this fluid invasion into open-end capillaries has been well described, detailed studies of fluid invasion in closed-end capillaries have not been explored thoroughly. Thus, we demonstrated, both theoretically and experimentally, a fluid invasion in closed-end capillaries, where the movement of the meniscus and the invasion velocity are accompanied by adiabatic gas compression inside the capillary. Theoretically, we found the fluid oscillations during invasion at short time scales by solving the one-dimensional momentum balance. This oscillatory motion is evaluated to determine which physical forces dominate the different conditions, and is further described by a damped driven harmonic oscillator model. However, this oscillating motion is not observed in the experiments. This inconsistency is due to the following: first, a continuous decrease in the radius of the curvature caused by decreasing the invasion velocity and increasing pressure inside the closed-end capillary, and second, the shear stress increase in the short time scale by the plug like velocity profile within the entrance length. The viscous term of modified momentum equation can be written as K(8μl/rc(2))(dl/dt) by using the multiplying factor K, which represents the increase of shear stress. The K is 7.3, 5.1, and 4.8 while capillary aspect ratio χc is 740, 1008, and 1244, respectively.

Geometry-induced asymmetric capillary flow

When capillary flow occurs in a uniform porous medium, the depth of penetration is known to increase as the square root of time. However, we demonstrate in this study that the depth of penetration in multi-section porous layers with variation in width and height against the flow time is modified from this diffusive-like response, and liquids can pass through porous systems more readily in one direction than the other. We show here in a model and an experiment that the flow time for a negative gradient of cross-sectional widths is smaller than that for a positive gradient at the given total height of porous layers. The effect of width and height of local layers on capillary flow is quantitatively analyzed, and optimal parameters are obtained to facilitate the fastest flow.

Treelike networks accelerating capillary flow

Transport in treelike networks has received wide attention in natural systems, oil recovery, microelectronic cooling systems, and textiles. Existing studies are focused on transport behaviors under a constant potential difference (including pressure, temperature, and voltage) in a steady state [B. Yu and B. Li, Phys. Rev. E 73, 066302 (2006); J. Chen, B. Yu, P. Xu, and Y. Li, Phys. Rev. E 75, 056301 (2007)]. However, dynamic (time-dependent) transport in such systems has rarely been concerned. In this work, we theoretically investigate the dynamics of capillary flow in treelike networks and design the distribution of radius and length of local branches for the fastest capillary flow. It is demonstrated that capillary flow in the optimized tree networks is faster than in traditional parallel tube nets under fixed constraints. As well, the flow time of the liquid is found to increase approximately linearly with penetration distance, which differs from Washburn's classic description that flow time increases as the square of penetration distance in a uniform tube.

Inverse problem of capillary filling

The inverse problem of capillary filling, as defined in this work, consists in determining the capillary radius profile from experimental data of the meniscus position l as a function of time t. This problem is central in diverse applications, such as the characterization of nanopore arrays or the design of passive transport in microfluidics; it is mathematically ill posed and has multiple solutions; i.e., capillaries with different geometries may produce the same imbibition kinematics. Here a suitable approach is proposed to solve this problem, which is based on measuring the imbibition kinematics in both tube directions. Capillary filling experiments to validate the calculation were made in a wide range of length scales: glass capillaries with a radius of around 150  μm and anodized alumina membranes with a pores radius of around 30  nm were used. The proposed method was successful in identifying the radius profile in both systems. Fundamental aspects also emerge in this study, notably the fact that the l(t)∝t1/2 kinematics (Lucas-Washburn relation) is not exclusive of uniform cross-sectional capillaries.

Optimal design of porous structures for the fastest liquid absorption

Porous materials engineered for rapid liquid absorption are useful in many applications, including oil recovery, spacecraft life-support systems, moisture management fabrics, medical wound dressings, and microfluidic devices. Dynamic absorption in capillary tubes and porous media is driven by the capillary pressure, which is inversely proportional to the pore size. On the other hand, the permeability of porous materials scales with the square of the pore size. The dynamic competition between these two superimposed mechanisms for liquid absorption through a heterogeneous porous structure may lead to an overall minimum absorption time. In this work, we explore liquid absorption in two different heterogeneous porous structures [three-dimensional (3D) circular tubes and porous layers], which are composed of two sections with variations in radius/porosity and height. The absorption time to fill the voids of porous constructs is expressed as a function of radius/porosity and height of local sections, and the absorption process does not follow the classic Washburn's law. Under given height and void volume, these two-section structures with a negative gradient of radius/porosity against the absorption direction are shown to have faster absorption rates than control samples with uniform radius/porosity. In particular, optimal structural parameters, including radius/porosity and height, are found that account for the minimum absorption time. The liquid absorption in the optimized porous structure is up to 38% faster than in a control sample. The results obtained can be used a priori for the design of porous structures with excellent liquid management property in various fields.

Capillary rise dynamics of aqueous glycerol solutions in glass capillaries: a critical examination of the Washburn equation

The classic description of capillary rise given by the Washburn equation was recently questioned in the light of experimental evidence for a velocity dependent dynamic contact angle at a moving contact line. We present a systematic investigation of the capillary rise dynamics of glycerol and aqueous glycerol solutions in vertical glass capillaries of various radii. For pure glycerol, the results of our experiments are in almost perfect agreement with the predictions of the Washburn equation using independently measured values for the liquid and capillary parameters. For aqueous glycerol solutions we observe discrepancies between the theoretical expectations and the experimental results, which are increasing with the water content of the solution. A thorough analysis, combined with scaling arguments, allows us to conclude that dynamic contact angle effects alone cannot provide a consistent explanation for these discrepancies. Rather, they can be perfectly accounted for if the mixture flowing in the capillary would have an effective, increased viscosity (in respect to the nominal value). We suggest and briefly discuss various mechanisms that could contribute to this observed behavior.

Different regimes in vertical capillary filling

In this paper, we identify that the different regimes encountered in a vertical capillary filling or a capillary-rise problem are determined entirely by two dimensionless parameters: Ohnesorge number (Oh) and Bond number (Bo). The initial universal inertial regime, which has been analyzed in our recent paper [Das et al., Phys. Rev. E 86, 067301 (2012)], is followed by any one of three possible regimes, dictated by the ratio Oh/Bo. For Oh/Bo>>1, the viscous effects dominate the gravitational effects, and one encounters the classical Washburn regime. For the other limit, i.e., Oh/Bo<<1, the viscous effects are insignificant and there is no Washburn regime. On the contrary, the inertial regime transits to the oscillatory regime with the filling length ℓ oscillating about the Jurin height (~1/Bo), which is the maximum height attained by a liquid column in vertical capillary filling, with the viscous effects (~Oh) dictating the nature of the oscillations. For Oh/Bo~1, we get a behavior intermediate of these two extreme regimes. Finally, we identify the correct force picture that drives the oscillatory regime and in the process achieve quantitative match with the experimental results, that was precluded in the previous studies.

On the influence of pore structure on the free-imbibition of sessile drops into nanoporous substrates

We report here a model experimental study on the influence of pore structure on the free-imbibition of sessile drops into nanoporous substrates. The work takes advantage of the existence of distinct pore structures on the two sides of a nanoporous alumina membrane: straight parallel channels versus a denser and tortous network. We show first that the spreading which coexists with the free-imbibition predominates in the early stage well follows on both sides the power-law scaling with time predicted by the universal Tanner's law. More interestingly, we found also that the imbibition rate scales in a similar way with the time on both sides of the membrane, showing that the pore structure does not affect qualitatively the free-imbibition kinetics. On the other hand, our results clearly show that the pore structure has a quantitative impact on the imbibition rate, which increases markedly from the A side (dense network of short and tortuous pores) to the side B (straight vertical channels). This latter result shows that, as regards the free-imbibition, the topology of the pores has a preeminent impact on their volume, which is here comparable for both sides of the membrane. More unexpectedly, this quantitative impact of the pore structure on the imbibition rate seems to display a certain sensitivity to the viscosity of the liquid.