Wetting considerations in capillary rise and imbibition in closed square tubes and open rectangular cross-section channels

The spontaneous capillary-driven filling of microchannels is important for a wide range of applications. These channels are often rectangular in cross-section, can be closed or open, and horizontal or vertically orientated. In this work, we develop the theory for capillary imbibition and rise in channels of rectangular cross-section, taking into account rigidified and non-rigidified boundary conditions for the liquid–air interfaces and the effects of surface topography assuming Wenzel or Cassie-Baxter states. We provide simple interpolation formulae for the viscous friction associated with flow through rectangular cross-section channels as a function of aspect ratio. We derive a dimensionless cross-over time, T _c, below which the exact numerical solution can be approximated by the Bousanquet solution and above which by the visco-gravitational solution. For capillary rise heights significantly below the equilibrium height, this cross-over time is T _c ≈ (3X _e/2)^2/3 and has an associated dimensionless cross-over rise height X _c ≈ (3X _e/2)^1/3, where X _e = 1/G is the dimensionless equilibrium rise height and G is a dimensionless form of the acceleration due to gravity. We also show from wetting considerations that for rectangular channels, fingers of a wetting liquid can be expected to imbibe in advance of the main meniscus along the corners of the channel walls. We test the theory via capillary rise experiments using polydimethylsiloxane oils of viscosity 96.0, 48.0, 19.2 and 4.8 mPa s within a range of closed square tubes and open rectangular cross-section channels with SU-8 walls. We show that the capillary rise heights can be fitted using the exact numerical solution and that these are similar to fits using the analytical visco-gravitational solution. The viscous friction contribution was found to be slightly higher than predicted by theory assuming a non-rigidified liquid–air boundary, but far below that for a rigidified boundary, which was recently reported for imbibition into horizontally mounted open microchannels. In these experiments we also observed fingers of liquid spreading along the internal edges of the channels in advance of the main body of liquid consistent with wetting expectations. We briefly discuss the implications of these observations for the design of microfluidic systems.     

Demarcating wetting states in textured microchannels under flow conditions by Poiseuille number

In this work, flow friction in microchannels decorated with micropillars was investigated experimentally, with an interest to understand the wetting transition through two simple means: Poiseuille number and scaling laws. Different wetting states were demarcated by qualitatively assessing the behaviour of Poiseuille number (Po = f·Re, where f is friction factor and Re is Reynolds number), which are further corroborated by confocal microscopy-based measurements and numerical simulations. The wetting transition ensued smoothly with an increase in Re, independent of the gas fraction (a ratio of area covered by the liquid–gas interface to the total projected area), for moderate gas fractions, whereas an early breakdown of the Cassie–Baxter state occurred irrespective of Re at high gas fractions. Additionally, the scaling laws were found to correlate well with the underlying state of the flow. Our observations revealed that the liquid–gas interface exhibits a partial slip, contrary to the common notion that it is shear free. It is inferred that an increase in effective flow area leads to a reduction in flow friction in textured microchannels. The present work underlines three important outcomes. The first is the identification of wetting states in flow conditions shown by tracking the Poiseuille number. The second is that the liquid–gas interface is deduced to behave like a partial slip boundary. The third is that a textured microchannel can be worse than an enlarged dimension microchannel.     

Liquid and gas flows in microchannels of varying cross section: a comparative analysis of the flow dynamics and design perspectives

This paper presents a comparative study of the flow of liquid and gases in microchannels of converging and diverging cross sections. Towards this, the static pressure across the microchannels is measured for different flow rates of the two fluids. The study includes both experimental and numerical investigations, thus providing several useful insights into the local information of flow parameters as well. Three different microchannels of varying angles of convergence/divergence (4°, 8° and 12°) are studied to understand the effect of the angle on flow properties such as pressure drop, Poiseuille number and diodicity. A comparison of the forces involved in liquid and gas flows shows their relative significance and effect on the flow structure. A diodic effect corresponds to a difference in the flow resistance in a microchannel of varying cross section, when the flow is subjected alternatively to converging and diverging orientations. In the present experiments, the diodic effect is observed for both liquid and gas as working fluids. The effect of governing parameters—Reynolds number and Knudsen number, on the diodicity is analysed. Based on these results, a comparison of design perspectives that may be useful in the design of converging/diverging microchannels for liquid and gas flows is provided.     

Liquid film thickness measurement underneath a gas slug with miniaturized sensor matrix in a microchannel

Measurement of liquid film thickness is essential for understanding the dynamics of two-phase flow in microchannels. In this work, a miniaturized sensor matrix with impedance measurement and MEMS technology to measure the thin liquid film underneath a bubble in the air–water flow in a horizontal microchannel has been developed. This miniaturized sensor matrix consists of 5 × 5 sensors where each sensor is comprised of a transmitter and a receiver electrode concentrically. The dimension and performance of the sensor electrodes were optimized with simulation results. The maximum diameter of the sensor ring is 310 µm, allowing a measurable range of liquid film thickness up to 83 µm. These sensors were distributed on the surface of a wafer with photolithography technology, covering a total length of 8 mm and a width of 2 mm. A spatial resolution of 0.5 × 2.0 mm² and a temporal resolution of 5 kHz were achieved for this sensor matrix with a measurement accuracy of 0.5 µm. A series of microchannels with different heights were used in the calibration in order to achieve the signal-to-thickness characteristics of each sensor. This delicate sensor matrix can provide detailed information on the variation of film thickness underneath gas–water slug directly, accurately and dynamically.     

Pressure drop of three-phase liquid–liquid–gas slug flow in round microchannels

In this paper we present a model for the calculation of pressure drop of three-phase liquid–liquid–gas slug flow in microcapillaries of a circular cross section. Introduced models consist of terms attributing for frictional and interfacial pressure drop, incorporating the presence of a stagnant thin film at the wall of the channel. Different formulations of the interfacial pressure drop equation were employed, using expressions developed by Bretherton (J Fluid Mech 10:166–188, 1961), Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989). Models were validated experimentally using oleic acid–water–nitrogen and heptane–water–nitrogen three-phase flows in round Teflon or Radel R microchannels of 254- and 508-µm nominal inner diameter, for capillary numbers Ca _b between 10^?4 and 4.9 × 10^?1 and Reynolds numbers Re between 0.095 and 300. Best agreement between measured and calculated values of pressure drop, with relative error between ?22 and 19 % or ?20 and 16 %, is reached for Warnier’s or Ratulowski and Chang’s interfacial pressure drop equation, respectively. The results prove that three-phase slug flow pressure drop can be successfully predicted by extending existing two-phase slug flow correlations. Good agreement of Bretherton’s equation was reached only at lower Ca numbers, indicating that an extension of the interfacial pressure drop equation as performed by Warnier et al. (Microfluid Nanofluid 8:33–45, 2010) or Ratulowski and Chang (Phys Fluids A 1:1642–1655, 1989) for higher capillary numbers is necessary. Additionally it was demonstrated that pressure drop increases substantially if dry slug flow occurs or if microchannels with significant surface roughness are employed. Those influences were not accounted for in the models presented.     

Effect of wall roughness on performance of microchannel applied in microfluidic device

The influence of wall roughness on flow and heat transfer performance in microchannels at low Reynolds number is investigated in this paper. Two sizes of PMMA microchannels are fabricated by microinjection molding and the width is 20 μm and 800 μm respectively. The surface profile of bottom wall is described by the two-dimensional fractal geometry method and it is found there is error within 5% between surface roughness obtained by the fractal geometry method and actual roughness. Then, the effects of dimensionless relative roughness (5–7.5%), fractal dimension (1.5–1.8), aspect ratio (0.025–4) on the flow resistance and heat transfer performance are analyzed by numerical and experimental method respectively. Reynolds number considered here are 10–60. The results show that the better flow performance and heat transfer performance can be obtained with high aspect ratio of rectangular microchannel. However, increasing surface roughness not only increases the heat transfer performance, but also introduces a large flow resistance, which makes the friction coefficient rise sharply. As a result, surface roughness has great influence on the flow and heat transfer performance, and the most suitable surface morphology should be obtained according to the specific application.

     

Hydrodynamics of gas–liquid Taylor flow in rectangular microchannels

The effect of fluid properties and operating conditions on the generation of gas–liquid Taylor flow in microchannels has been investigated experimentally and numerically. Visualisation experiments and 2D numerical simulations have been performed to study bubble and slug lengths, liquid film hold-up and bubble velocities. The results show that the bubble and slug lengths increase as a function of the gas and liquid flow rate ratios. The bubble and slug lengths follow the model developed by Garstecki et?al. (Lab chip 6:437–446, 2006) and van Steijn et?al. (Chem Eng Sci 62:7505–7514, 2007), however, the model coefficients appear to be dependent on the liquid properties and flow conditions in some cases. The ratio of the bubble velocity to superficial two-phase velocity is close to unity, which confirms a thin liquid film under the assumption of a stagnant liquid film. Numerical simulations confirm the hypothesis of a stagnant liquid film and provide information on the thickness of the liquid film.     

Holdup characteristics of two-phase parallel microflows

Two-phase parallel microflows, i.e., stratified flow and core-annular flow, have many applications in lab-on-chip devices. These include transport and reaction processes, such as liquid–liquid extraction and phase transfer catalysis. The phase holdup (fraction of the microchannel volume occupied by a specified phase) is a key parameter of these flow systems. In this work, mathematical models based on fundamental principles are used to predict the phase holdup in stratified flow and core-annular flow. For stratified flow, a two-dimensional model of flow in a rectangular channel of arbitrary aspect ratio is considered. A simpler one-dimensional model of stratified flow between infinite parallel plates is also analyzed. In the case of core-annular flow, axisymmetry is assumed in the model. The results of the models agree well with published experimental results. The dependence of phase holdup on the flow-rate fraction (the primary operating variable which can be controlled experimentally) is studied in detail. The nature of this relationship varies with the ratio of fluid viscosities and the channel’s aspect ratio (in the case of stratified flow). In the literature, the holdup is sometimes erroneously assumed to be identical to the flow-rate fraction. It is shown that this is not possible in the case of core-annular flow, while in stratified flow it is true only for a unique critical flow-rate. This critical flow-rate is viscosity dependent. The aspect ratio of the channel is found to have a considerable influence on the holdup in stratified flow when the fluids have different viscosities. However, even in such cases, there exists a point of geometric invariance at which the holdup is independent of the aspect ratio. At this point, the simple one-dimensional model of stratified flow can predict the holdup with complete accuracy.     

Novel microstructured evaporation device

Microfluidic devices have become more and more important in the field of thermal or chemical process engineering within the last years (by Schubert et al. Microscale Thermophys Eng 5:17–39, 2001). Cooling is one point of research where microchannels and other microstructured geometries provide significant advantages compared to conventional devices as they offer much higher possible surface to volume ratios and short characteristic distances. Therefore, an intense amount of heat can be transferred by these devices, which can be significantly increased by phase transition. Thus, evaporation of a fluid flow in microchannels allows very compact, fast, and powerful cooling devices. In this research study a novel microstructured evaporator geometry consisting of curved microchannels was evaluated on its evaporation properties compared to previous studies dealing with evaporation of R134a (Tetrafluoroethane) in straight microchannels (by Wibel et al. Chem Eng J 167:705–712, 2011). This novel evaporator design takes advantage of the strong centrifugal forces acting on the (evaporating) two phase flow in the curved microchannels. Due to the feasibility of very small radii of curved microchannels, strong centrifugal forces can be obtained for the fluid flow inside the microstructures. Additionally, those forces are boosted as flow velocities within the channels become higher due to the volume increase induced by evaporation. Therefore, a phase separation could take place inside the microstructure with a higher liquid fraction of evaporating coolant near one side of the curved channels during the transition to vapor. A high liquid fraction inside the evaporator is aimed by an intended removal of the evaporated gas phase from the microstructure. Experimental results of the evaporation of water and R134a as coolants demonstrate the potential of this curved geometry in comparison to evaporation in straight channels. Optical investigations of the new micro evaporator concept by high-speed videography (by Maikowske et al. Appl Therm Eng 30:1872–1876, 2010) are carried out for further improvements of the design. Various bubble formations and movements of the evaporating fluid flow were studied for various vapor fractions by using different fluids. These investigations show how the separation and extraction of the vapor fraction of the novel microstructure concept could be improved.     

A micro PDMS flow sensor based on time-of-flight measurement for conductive liquid

Transdermal extraction of interstitial fluid (ISF) offers an attractive method for non-invasive blood glucose monitoring. In order to calculate blood glucose concentration accurately, precise volume measurement of transdermally extracted ISF is required due to human skin’s varying permeability. In this paper, we presented a novel flow sensor fabricated from polydimethylsiloxane (PDMS), designed to measure the volume of conductive liquid. The flow sensor consists of two pairs of metal electrodes, which are fabricated in the PDMS channel. The volume of liquid is measured utilizing the time-of-flight of the two electrode pairs’ resistance while the liquid is flowing through the flow sensor. 1–14 μL normal saline solution was measured, the flow sensor measured volumes correlate very well (R ² = 0.9996 and R ² = 0.9975 for vacuum pump and syringe pump situations respectively) with the actual volumes. And the coefficient of variation for 10 times 10 μL normal saline solution measurement is 0.0077 (vacuum pump) and 0.0381 (syringe pump), respectively. The demonstrated flow sensor provides excellent functionality for conductive liquid.     

Flow-induced deformation of compliant microchannels and its effect on pressure–flow characteristics

We report theoretical and experimental investigations of flow through compliant microchannels in which one of the walls is a thin PDMS membrane. A theoretical model is derived that provides an insight into the physics of the coupled fluid–structure interaction. For a fixed channel size, flow rate and fluid viscosity, a compliance parameter \(f_{\text{p}}\) is identified, which controls the pressure–flow characteristics. The pressure and deflection profiles and pressure–flow characteristics of the compliant microchannels are predicted using the model and compared with experimental data, which show good agreement. The pressure–flow characteristics of the compliant microchannel are compared with that obtained for an identical conventional (rigid) microchannel. For a fixed channel size and flow rate, the effect of fluid viscosity and compliance parameter \(f_{\text{p}}\) on the pressure drop is predicted using the theoretical model, which successfully confront experimental data. The pressure–flow characteristics of a non-Newtonian fluid (0.1 % polyethylene oxide solution) through the compliant and conventional (rigid) microchannels are experimentally measured and compared. The results reveal that for a given change in the flow rate, the corresponding modification in the viscosity due to the shear thinning effect determines the change in the pressure drop in such microchannels.     

Liquid flow through microchannels with grooved walls under wetting and superhydrophobic conditions

Recent developments in superhydrophobic surfaces have enabled significant reduction in the frictional drag for liquid flow through microchannels. There is an apparent risk when using such surfaces, however, that under some conditions the liquid meniscus may destabilize and, consequently, the liquid will wet the entire patterned surface. This paper presents analytical and experimental results that compare the laminar flow dynamics through microchannels with superhydrophobic walls featuring ribs and cavities oriented both parallel and transverse to the direction of flow under both wetting and non-wetting conditions. The results show the reduction in the total frictional resistance is much greater in channels when the liquid phase does not enter the cavity regions. Further, it is demonstrated that the wetting and non-wetting cavity results represent limiting cases between which the experimental data lie. Generalized expressions enabling prediction of the classical friction factor-Reynolds number product as a function of the relevant governing dimensionless parameters are also presented for both the superhydrophobic and wetting states. Experimental results are presented for a range of parameters in the laminar flow regime.     

Debottlenecking level control for tanks in series

Smooth operation of an industrial liquid–liquid extraction process with three tanks in series posed an interesting control problem: for control of each tank's liquid level through PI manipulation of each corresponding outlet flow rate, the conflicting objectives of keeping both level and flow rate fluctuations within bounds became impossible to satisfy when the main disturbance (feed flow rate) exceeded a certain magnitude. A solution had to be found, preferably using the existing PI controllers and no additional equipment such as flow sensors and advanced control hardware, to keep cost down. A solution was found based on implementation of a simple modified control structure that employs the same PI controllers. This structure essentially aggregates the three tanks into one larger tank, whose virtual level is controlled by a PI controller, supported by the remaining two PI controllers that keep the differences between tank levels to a minimum, so that the three tanks can behave as virtually one. In addition to theoretical analysis of the proposed structure, plant validation is presented. The questions of how this structure could be reached through numerical optimization are addressed, whereas the broader question of automating the intuitive process of solution generation is raised.     

Design optimization of micromixer with square-wave microchannel on compact disk microfluidic platform

This paper presents a passive micromixer on a compact disk (CD) microfluidic platform that performs plasma mixing function. The driving force of CD microfluidic platform including, the centrifugal force due to the system rotation, the Coriolis force as a function of the rotation angular frequency and velocity of liquid. Numerical simulations are performed to investigate the flow characteristics and mixing performance of three CD microfluidic mixers with square-wave, curved and zig-zag microchannels, respectively. Of the three microchannels, the square-wave microchannel is found to yield the best mixing performance, and is therefore selected for design optimization. Four CD microfluidic micromixers incorporating square-wave PDMS microchannels with different widths in the x- and y-directions are fabricated using conventional photolithography techniques. The mixing performance of the four microchannels is investigated both numerically and experimentally. The results show that given an appropriate specification of the microchannel geometry and a CD rotation speed of 2,000 rpm, a mixing efficiency of more than 93 % can be obtained within 5 s.     

Optimal fuzzy sliding-mode control for bio-microfluidic manipulation

In biometric and biomedical applications, a special transporting mechanism must be designed for the micro total analysis system (μTAS) to move samples and reagents through the microchannels that connect the unit procedure components in the system. An important issue for this miniaturization and integration is the microfluid management technique, i.e., microfluid transportation, metering, and mixing. In view of this, an optimal fuzzy sliding-mode control (OFSMC) based on the 8051 microprocessor is designed and a complete microfluidic manipulated biochip system is implemented in this study, with a pneumatic pumping actuator, two feedback-signal photodiodes and flowmeters for better microfluidic management. This new technique successfully improved the efficiency of biochemical reaction by increasing the effective collision into the probe molecules as the target molecules flow back and forth. The new technique was used in DNA extraction. When the number of Escherichia coli cells was 2×10²–10⁴ in 25 μl of whole blood, the extraction efficiency of immobilized beads with solution flowing back and forth was 600-fold larger than that of free beads.     

Second-order gaseous slip flow models in long circular and noncircular microchannels and nanochannels

This paper significantly extends previous studies to the transition regime by employing the second-order slip boundary conditions. A simple analytical model with second-order slip boundary conditions for a normalized Poiseuille number is proposed. The model can be applied to either rarefied gas flows or apparent liquid slip flows. The developed simple models can be used to predict the Poiseuille number, mass flow rate, tangential momentum accommodation coefficient, pressure distribution of gaseous flow in noncircular microchannels and nanochannels by the research community for the practical engineering design of microchannels and nanochannels. The developed second-order models are preferable since the difficulty and “investment” is negligible compared with the cost of alternative methods such as molecular simulations or solutions of Boltzmann equation. Navier–Stokes equations with second-order slip models can be used to predict quantities of engineering interest such as the Poiseuille number, tangential momentum accommodation coefficient, mass flow rate, pressure distribution, and pressure drop beyond its typically acknowledged limit of application. The appropriate or effective second-order slip coefficients include the contribution of the Knudsen layers in order to capture the complete solution of the Boltzmann equation for the Poiseuille number, mass flow rate, and pressure distribution. It could be reasonable that various researchers proposed different second-order slip coefficients because the values are naturally different in different Knudsen number regimes. It is analytically shown that the Knudsen’s minimum can be predicted with the second-order model and the Knudsen value of the occurrence of Knudsen’s minimum depends on inlet and outlet pressure ratio. The compressibility and rarefaction effects on mass flow rate and the curvature of the pressure distribution by employing first-order and second-order slip flow models are analyzed and compared. The condition of linear pressure distribution is given.     

Liquid–liquid micro-dispersion in a double-pore T-shaped microfluidic device

For further understanding the dispersion process in the T-shaped microfluidic device, a double-pore T-shaped microchannel was designed and tested with octane/water system to form monodispersed plugs and droplets in this work. The liquid–liquid two-phase flow patterns were investigated and it was found that only short plugs, relative length L/w < 1.4, were produced. Additionally, the droplets flow was realized at phase ratios (F _C /F _D) just higher than 0.5, which is much smaller than that in the single-pore T-shaped microchannels. A repulsed effect between the initial droplets was observed in the droplet formation process and the periodic fluctuation flow of the dispersed phase was discussed by analyzing the resistances. Besides, the effect of the two-phase flow rates on the plug length and the droplet diameter was investigated. Considering the mutual effect of the initial droplets and the equilibrium between the shearing force with the interfacial tension, phase ratio and Ca number were introduced into the semi-empirical models to present the plug and droplet sizes at different operating conditions.     

Gaseous slip flow in long microchannels

An analytic and experimental investigation into gaseous flow with slight rarefaction through long microchannels is undertaken. A two-dimensional (2-D) analysis of the Navier-Stokes equations with a first-order slip-velocity boundary condition demonstrates that both compressibility and rarefied effects are present in long microchannels. By undertaking a perturbation expansion in ϵ, the height-to-length ratio of the channel, and using the ideal gas equation of state, it is shown that the zeroth-order analytic solution for the streamwise mass flow corresponds well with the experimental results. Also, the effect of slip upon the pressure distribution is derived, and it is obtained that this slip velocity leads directly to a wall-normal migration of mass. The fabrication of wafer-bonded microchannels that possess well-controlled surface structure is described, and a means for accurately measuring the mass how through the channels is presented. Experimental results obtained with this mass-flow measurement technique for streamwise helium mass flow through microchannels 52.25-μm wide, 1.33-μm deep, and 7500-μm long for a pressure range of 1.6-4.2 atmospheres (outlet pressures at atmospheric) are presented and shown to compare favorably with the analysis     

Analysis of combined pressure-driven electroosmotic flow through square microchannels

In this paper three-dimensional single-phase liquid flow through microchannels with a square-shaped cross-section driven by simultaneous application of pressure gradient and electroosmotic pumping mechanism is studied. The governing system of equations consists of the electric potential field and flow field equations. The solution procedure involves three steps. First, the net charge distribution on the cross-section of the microchannel is computed by solving two-dimensional Poisson–Boltzmann equation using the finite element method. Then, using the computed fluid’s charge distribution, the magnitude of the resulting body force due to interaction of an external electric field with the charged fluid particles is calculated along the microchannel. In the third step, the flow equations are solved by considering three-dimensional Navier–Stokes equations with an electrokinetic body force. The computations reveal that the flow pattern in the microchannel is significantly different from the parabolic velocity profile of the laminar pressure-driven flow. The effect of the liquid bulk ionic concentration and the external electric field strength on flow patterns through the square-shaped microchannels is also investigated.