Analytical modeling of capillary flow in tubes of nonuniform cross section

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.

Imbibition of Newtonian Fluids in Paper-like Materials with the Infinitesimal Control Volume Method

Paper-based microfluidic devices are widely used in point-of-care testing applications. Imbibition study of paper porous media is important for fluid controlling, and then significant to the applications of paper-based microfluidic devices. Here we propose an analytical approach based on the infinitesimal control volume method to study the imbibition of Newtonian fluids in commonly used paper-like materials. Three common paper shapes (rectangular paper strips, fan-shaped and circular paper sheets) are investigated with three modeling methods (corresponding to equivalent tiny pores with circle, square and regular triangle cross section respectively). A model is derived for liquid imbibition in rectangular paper strips, and the control equations for liquid imbibition in fan-shaped and circular paper sheets are also derived. The model is verified by imbibition experiments done using the mixed cellulose ester filter paper and pure water. The relation of imbibition distance and time is similar to that of the Lucas−Washburn (L−W) model. In addition, a new porosity measurement method based on the imbibition in circular paper sheets is proposed and verified. Finally, the flow rates are investigated. This study can provide guidance for the design of different shapes of paper, and for better applications of paper-based microfluidic devices.

Synchronization and control of capillary flows in rectangular microchannel with spacers

Liquid control in microchannels is quite important in microfluidic devices used in, for example, lab-on-a-chip and point-of-care applications. Capillary microfluidics, being self-powered, is especially advantageous for use in passive devices, and has attracted significant attention. In this paper, capillary flows in rectangular microchannels with spacers are studied experimentally and theoretically; in particular, capillary flow synchronization (or waiting) behavior is identified and investigated. Based on changes of channel walls, two basic synchronization modes are proposed for flows isolated by spacers in a channel. Experimental results show that the velocities of faster capillary flows are reduced by the liquid pinning effect and that the time delay between two capillary flows is automatically balanced. The synchronization behavior of capillary flows is explained by analyzing the time delay, contact angle variation, and capillary forces. In addition, the quantity of liquid flowing out of the waiting channels is estimated and verified. Then a model for the change in contact angle during synchronization is derived and verified. Finally, we conceive a series of studies of the control of capillary flows for different spacer designs and conduct an experiment to study the dynamic behaviors of a number of capillary flows by adding many spacers in a microchannel. This study expands the applications of capillary microfluidics.

Haines jumps: Pore scale mechanisms

Haines instabilities are sudden jumps of the fluid interface accompanied by fluid redistribution and a transient pressure response. Haines jumps affect global displacement patterns in porous media, promote fingered invasion, increase fluid trapping, and contribute to hysteretic saturation. Detailed analyses and experimental results show that jumps take place when the pressure-volume response is multivalued across the pore throat, during either advancing or receding tests. This situation emerges in "soft systems" such as liquids with entrapped gas bubbles, compliant substrates, and when multiple menisci interact across the porous networks. Elastic deformations and capillarity combine to form the dimensionless elastocapillary number N_{ec} for a given pore geometry and fluid mixture: a system with elastocapillary number N_{ec}<1 is prone to Haines instabilities. Therefore, Haines jumps are more likely to occur not only in soft porous systems, but also when the network topology is characterized by pronounced pore constrictions, and the fluids form small contact angles and generate high interfacial tension.

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.

A Rapid Capillary-Pressure Driven Micro-Channel to Demonstrate Newtonian Fluid Behavior of Zebrafish Blood at High Shear Rates

Blood viscosity provides the rheological basis to elucidate shear stress underlying developmental cardiac mechanics and physiology. Zebrafish is a high throughput model for developmental biology, forward-genetics, and drug discovery. The micro-scale posed an experimental challenge to measure blood viscosity. To address this challenge, a microfluidic viscometer driven by surface tension was developed to reduce the sample volume required (3μL) for rapid (<2 min) and continuous viscosity measurement. By fitting the power-law fluid model to the travel distance of blood through the micro-channel as a function of time and channel configuration, the experimentally acquired blood viscosity was compared with a vacuum-driven capillary viscometer at high shear rates (>500 s⁻¹), at which the power law exponent (n) of zebrafish blood was nearly 1 behaving as a Newtonian fluid. The measured values of whole blood from the micro-channel (4.17cP) and the vacuum method (4.22cP) at 500 s⁻¹ were closely correlated at 27 °C. A calibration curve was established for viscosity as a function of hematocrits to predict a rise and fall in viscosity during embryonic development. Thus, our rapid capillary pressure-driven micro-channel revealed the Newtonian fluid behavior of zebrafish blood at high shear rates and the dynamic viscosity during development.

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.

Intrusion of a Liquid Droplet into a Powder under Gravity

Intrusion of a liquid droplet into a hexagonal close-packed array of spheres under gravity is investigated using analytical methods and volume-of-fluid simulations. Four regimes of ultimate fluid behavior are identified: (A) no liquid imbibition into the bed, (B) trapping of liquid high in the bed, (C) liquid descending to the bottom of the bed, and (D) liquid spreading around the surface of all particles. These regimes are mapped based on the contact angle and Bond number of the system. Many aspects of the dynamics and ultimate liquid behavior are captured using a simplified model of a mass of liquid moving under gravity in a vertical capillary of undulating cross-sectional area. This simplified model is used to form momentum transport equations with four forms of nondimensional time, which are shown to collapse the simulation data with different fluid parameters in different regimes.

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.

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.

Optofluidic characterization of nanoporous membranes

An optofluidic method that accurately identifies the internal geometry of nanochannel arrays is presented. It is based on the dynamics of capillary-driven fluid imbibition, which is followed by laser interferometry. Conical nanochannel arrays in anodized alumina are investigated, which present an asymmetry of the filling times measured from different sides of the membrane. It is demonstrated by theory and experiments that the capillary filling asymmetry only depends on the ratio H of the inlet to outlet pore radii and that the ratio of filling times vary closely as H(7/3). Besides, the capillary filling of conical channels exhibits striking results in comparison to the corresponding cylindrical channels. Apart from these novel results in nanoscale fluid dynamics, the whole method discussed here serves as a characterization technique for nanoporous membranes.

Interfacial jumps and pressure bursts during fluid displacement in interacting irregular capillaries

The macroscopically regular motion of fluid displacement fronts in porous media often results from numerous pore scale interfacial jumps and associated pressure fluctuations. Such rapid pore scale dynamics defy postulated slow viscous energy dissipation and may shape phase entrapment and subsequent macroscopic transport properties. Certain displacement characteristics are predictable from percolation theory; however, insights into rapid interfacial dynamics require mechanistic models for hydraulically interacting pores such as found along fluid displacement fronts. A model for hydraulically coupled sinusoidal capillaries was used to analyze stick-jump interfacial motions with a significant inertial component absent in Darcy-based description of fluid front displacement. High-speed camera provided measurements of rapid interfacial dynamics in sintered glass beads cell during drainage. Interfacial velocities exceeding 50 times mean front velocity were observed in good agreement with model predictions for a pair of sinusoidal capillaries. In addition to characteristic pinning-jumping behavior, interfacial dynamics were sensitive to initial positions within pores at the onset of a jump. Even for a pair of sinusoidal capillaries, minute variations in pore geometry and boundary conditions yield rich behavior of motions, highlighting challenges and potential new insights offered by consideration of pore scale mechanisms in macroscopic description of fluid displacement fronts in porous media.