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The withdrawal of a liquid or the translation of a liquid slug in a capillary tube leads to the deposition of a thin film on the inner wall. When particles or contaminants are present in the liquid, they deposit and contaminate the tube if the liquid film is sufficiently thick. In this article, we experimentally investigate the condition under whic...
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... schematic of a particle fully wetted by a liquid of thickness h * around the stagnation point is shown in Fig. 8. Below, we detail the amplitude and orientation of the different forces to estimate their influence on particle entrainment. We consider the situation in the frame of reference moving with the front. Besides, close to the entrainment threshold, the velocity of the particle nearly vanished at the stagnation point before being able to be ...
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... drag force acts along the x-axis and acts as a driving force to entrain the particle (see Fig. ...
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... illustrated in Fig. 8 the capillary force can be decomposed in its y component that pushes the particle toward the surface of the capillary tube 13) and an axial force acting to keep the particle in the liquid reservoir (in the direction of the ...
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... schematic of a particle fully wetted by a liquid of thickness h * around the stagnation point is shown in Fig. 8. Below, we detail the amplitude and orientation of the different forces to estimate their influence on particle entrainment. We consider the situation in the frame of reference moving with the front. Besides, close to the entrainment threshold, the velocity of the particle nearly vanished at the stagnation point before being able to be ...
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... drag force acts along the x-axis and acts as a driving force to entrain the particle (see Fig. ...
Citations
... These forces may also be considered for determining the physics underpinning the coating of fibres using bridge patterns via the capillary number Ca, which has long been known to control the coating of many thin structures and flat plates according to the Landau-Levich-Derjaguin law (Levich & Landau 1942;Deriagin & Levi 1964). A better understanding of the film deposited by flowing bridges may also allow them to be functionalized to deposit suspended particulate via capillary deposition, a concept already explored in the laboratory (Jeong et al. 2020) and used to deliver drugs in a circular geometry (Kim et al. 2017), but unexplored for fibrous systems. Furthermore, the coating of dry fibres with flowing bridges remains unexplored but is expected to yield valuable physical insight akin to that found for beads coating vertical fibres and fibre bundles (Leonard et al. 2023) Finally, it is worth emphasizing the exceptional degree of control and the myriad of patterns that extend well beyond the scope of what has been highlighted in this paper. ...
Liquid bridges are formed when a flowing liquid interacts with multiple parallel fibres, as relevant to heat and mass transfer applications that utilize flow down fibre arrays. We perform a comprehensive experimental study of flowing liquid bridges between two vertical fibres whose spacing is controlled dynamically in our experimental apparatus. The bridge patterns exhibit a regular periodic spacing typical of absolute instability for low flow rates, but become spatially inhomogeneous above a critical flow rate where the base flow is convectively unstable. The shapes of individual bridges and their associated dynamics are measured, as they depend upon the liquid properties, and fibre geometry/spacing. The bridge length scales similarly to static bridges between parallel fibres. The bridge dynamics exhibits a dependence on viscosity and scale with the impedance. A simple energy balance is used to derive a scaling relationship for the bridge velocity that captures the general trend of our experimental data. Finally, we demonstrate that these scalings similarly apply when the fibres are dynamically separated or brought together.
... This point also corresponds to the location that separates the flow into two regions: a region that continues into the coating film and a region where the fluid recirculates into the liquid bath. The particles that make it past the stagnation point and that are of a size comparable to h * see a sharp increase in frictional forces, which allows for the capillary forces to be overcome [35]. At the limit of small capillary numbers, the thickness at the stagnation point and the film thickness are related through h * ≈ 3h [17]. ...
... This point also corresponds to the location that separates the flow into two regions: a region that continues into the coating film and a region where the fluid recirculates into the liquid bath. The particles that make it past the stagnation point and that are of size comparable to h * see a sharp increase in frictional forces, which allows for the overcoming of capillary forces [35]. In the limit of small capillary numbers, the thickness at the stagnation point and the film thickness are related through h * ≈ 3h [17]. ...
Sorting elongated anisotropic particles, such as fibers, dispersed in suspensions poses significant challenges as they present two characteristic dimensions: length and diameter. Fibers in suspension usually align with the flow, leading to diameter-based filtration when passing through a sieve. Modifying the flow conditions by introducing more mixing so that fibers are arbitrarily oriented can lead to sorting by diameter and length simultaneously, resulting in a lower filtration quality. In this paper, we demonstrate that capillary filtration by dip coating can be utilized to selectively sort fibers by length or by diameter in a controlled manner. Using the withdrawal of a flat substrate from a fiber suspension, we demonstrate that fibers are primarily sorted by their diameters. When considering cylindrical substrates, fibers can be sorted by length under appropriate conditions due to the orientation adopted by the fibers during their entrainment. We report guidelines for designing this filtration process and obtaining good sorting efficiency.
... The deposition of a continuous thin liquid film through the displacement of the liquid phase by air in a confined geometry is a phenomenon that has been a crucial objective in many engineering and biologicalrelevant settings, such as in enhanced oil recovery, 1,2 drug delivery, 3,4 gas-assisted injection molding, 5 coating processes, 6,7 and biomedical engineering. [8][9][10][11] Flow configurations employed to investigate the aforementioned applications typically involve the formation of a thin liquid film upon a long bubble advancing into a tube prefilled with fluids. ...
Thin‐film deposition of fluids is ubiquitous in a wide range of engineering and biological applications, such as surface coating, polymer processing, and biomedical device fabrication. While the thin viscous film deposition in Newtonian fluids has been extensively investigated, the deposition dynamics in frequently encountered non‐Newtonian complex fluids remain elusive, with respect to predictive scaling laws for the film thickness. Here, we investigate the deposition of thin films of shear‐thinning viscoelastic fluids by the motion of a long bubble translating in a circular capillary tube. Considering the weakly elastic regime with a shear‐thinning viscosity, we provide a quantitative measurement of the film thickness with systematic experiments. We further harness the recently developed hydrodynamic lubrication theory to quantitatively rationalize our experimental observations considering the effective capillary number Cae and the effective Weissenberg number Wie, which describe the shear‐thinning and the viscoelastic effects on the film formation, respectively. The obtained scaling law agrees reasonably well with the experimentally measured film thickness for all test fluids. Our work may potentially advance the fundamental understanding of the thin‐film deposition in a confined geometry and provide valuable engineering guidance for processes that incorporate thin‐film flows and non‐Newtonian fluids.
... The empirical parameters were determined by the experimental data of various liquids. [24,25] When the gravitational and inertial forces are negligible for a liquid in a capillary tube, the liquid film thickness h I 1 is deposited on the inner wall of a tube (tube radius: b) by competition between viscous and capillary forces. It depends on the capillary number. ...
... The h I 1 value in the tube has been predicted by the Bretherton law for Ca � 1, h I 1 =b ¼ 1:34Ca 2=3 . Recently, it has been known that h I 1 in the tube has been calculated by the following empirical equation through experiments [4,24,25] : ...
... The capillary forces are given by the surface tension γ and the receding and advancing contact angles θ r and θ a , respectively. [4,24,25] The gravitational force is given by the liquid density and the gravitational constant. The value L is estimated by Eq. (4): ...
... The viscous and frictional forces drag the particle into the coating film, while the capillary force pushes it back into the bulk of liquid. 56 When increasing the withdrawal velocity U , the thickness h of the film and at the stagnation point h * = 3 h increase. When the film thickness becomes larger than a fraction of the diameter of a spherical particle, isolated particles start to be entrained in the coating film. ...
The dip coating of suspensions made of monodisperse non-Brownian spherical particles dispersed in a Newtonian fluid leads to different coating regimes depending on the ratio of the particle diameter to the thickness of the film entrained on the substrate. In particular, dilute particles dispersed in the liquid are entrained only above a threshold value of film thickness. In the case of anisotropic particles, in particular fibers, the smallest characteristic dimension will control the entrainment of the particle. Furthermore, it is possible to control the orientation of the anisotropic particles depending on the substrate geometry. To test the hypotheses, we performed dip-coating experiments with dilute suspensions of non-Brownian fibers with different length-to-diameter aspect ratios. We characterize the number of fibers entrained on the surface of the substrate as a function of the withdrawal velocity, allowing us to estimate a threshold capillary number below which all the particles remain in the liquid bath. Besides, we measure the angular distribution of the entrained fibers for two different substrate geometries: flat plates and cylindrical rods. We then measure the film thickness for more concentrated fiber suspensions. The entrainment of the fibers on a flat plate and a cylindrical rod is primarily controlled by the smaller characteristic length of the fibers: their diameter. At first order, the entrainment threshold scales similarly to that of spherical particles. The length of the fibers only appears to have a minor influence on the entrainment threshold. No preferential alignment is observed for non-Brownian fibers on a flat plate, except for very thin films, whereas the fibers tend to align themselves along the axis of a cylindrical rod for a large enough ratio of the fiber length to the radius of the cylindrical rod.
... The transport of long gas bubbles or liquid drops in confined geometries plays an important role in many engineering and biological settings, such as enhanced oil recovery (Tran et al. 2016;Grassia 2019;Majeed et al. 2021), coating processes (Yu, Khodaparast & Stone 2017;Jeong et al. 2020), drug delivery (Hernot & Klibanov 2008;Gao et al. 2016), biomechanics and biomedical devices (Clanet, Héraud & Searby 2004;Chao, Jin & Fan 2020;Ma et al. 2020;Li et al. 2021). When such a long bubble of length L R translates at a constant speed U in a circular capillary of radius R, the bubble forms a symmetrical bullet shape, commonly called a Taylor bubble, and a thin film of liquid is generated between the bubble and capillary. ...
... Pioneering investigations on this topic were conducted by Bretherton (1961) and Taylor (1961). For a long bubble translating in confined geometries with small dimensions where gravity plays a negligible role, the dynamics is characterized by the interplay between the viscosity and surface tension, as captured by the definition of the capillary number, Ca = μU/σ , where μ is the fluid viscosity and σ is the surface tension (Aussillous & Quéré 2000;Jeong et al. 2020). For Ca 1, Bretherton (1961) found that the thickness of the thin liquid film h scales as h/R ∼ Ca 2/3 in regimes where inertia effects are negligible compared to surface tension and viscous effects. ...
The motion of a long gas bubble in a confined capillary tube is ubiquitous in a wide range of engineering and biological applications. While the understanding of the deposited thin viscous film near the tube wall in Newtonian fluids is well developed, the deposition dynamics in commonly encountered non-Newtonian fluids remains much less studied. Here, we investigate the dynamics of a confined bubble moving in shear-thinning fluids with systematic experiments, varying the zero-shear-rate capillary number in the range of O(10^{-3}\unicode{x2013}10^2) considering the zero-shear-rate viscosity. The thickness of the deposited liquid film, the bubble speed and the bubble front/rear menisci are measured, which are further rationalized with the recent theoretical studies based on appropriate rheological models. Compared with Newtonian fluids, the film thickness decreases for both the carboxymethyl cellulose and Carbopol solutions when the shear-thinning effect dominates. We show that the film thickness follows the scaling law from Aussillous & Quéré ( Phys. Fluids , vol. 12, no. 10, 2000, pp. 2367–2371) with an effective capillary number , considering the characteristic shear rate in the film as proposed by Picchi et al. ( J. Fluid Mech. , vol. 918, no. A7, 2021, pp. 1–30). is calculated by the Carreau number and the power-law index from the Carreau–Yasuda rheological model. The shear-thinning effect also influences the bubble speed and delays the transition to the parabolic region in the bubble front and rear menisci. In particular, a high degree of undulations on the bubble surface results in an intricate rear viscosity distribution for the rear meniscus and the deviation between the experiments and theory may require a further investigation to resolve the axial velocity field. Our study may advance the fundamental understandings and engineering guidelines for coating processes involving thin-film flows and non-Newtonian fluids.
... The transport of long gas bubbles or liquid drops in confined geometries plays an important role in many engineering and biological settings, such as enhanced oil recovery (Tran et al. 2016;Grassia 2019;Majeed et al. 2021), coating processes (Yu et al. 2017;Jeong et al. 2020), drug delivery (Hernot & Klibanov 2008;Gao et al. 2016), biomechanics and biomedical devices (Clanet et al. 2004;Chao et al. 2020;Ma et al. 2020;Li et al. 2021). When such a long bubble of length L >> R translates at a constant speed U in a circular capillary of radius R, the bubble forms a symmetrical bullet shape, commonly called a Taylor bubble, and a thin film of liquid is generated between the bubble and capillary. ...
... Pioneering investigations on this topic were conducted by Bretherton (1961) and Taylor (1961). For a long bubble translating in confined geometries with small dimensions where gravity plays a negligible role, the dynamics are characterized by the interplay between the viscosity and surface tension, as captured by the definition of the capillary number, Ca = µU/σ, where µ is the fluid viscosity and σ is the surface tension (Aussillous & Quéré 2000;Jeong et al. 2020). For Ca ≪ 1, Bretherton (1961) found that the thickness of the thin liquid film h scales as h/R ∼ Ca 2/3 , in regimes where inertia effects are negligible compared to surface tension and viscous effects. ...
The motion of a long gas bubble in a confined capillary tube is ubiquitous in a wide range of engineering and biological applications. While the understanding of the deposited thin viscous film near the tube wall in Newtonian fluids is well developed, the deposition dynamics in commonly encountered non-Newtonian fluids remains much less studied. Here, we investigate the dynamics of a confined bubble moving in shear-thinning fluids with systematic experiments, varying the zero-shear-rate capillary number in the range of considering the zero-shear-rate viscosity. The thickness of the deposited liquid film, the bubble speed and the bubble front/rear menisci are measured, which are further rationalized with the recent theoretical studies based on appropriate rheological models. Compared with Newtonian fluids, the film thickness decreases for both the carboxymethyl cellulose and Carbopol solutions when the shear-thinning effect dominates. We show that the film thickness follows the scaling law from \citet{aussillous2000quick} with an effective capillary number , considering the characteristic shear rate in the film as proposed by \citet{picchi2021motion}. is calculated by the Carreau number and the power-law index from the Carreau-Yasuda rheological model. The shear-thinning effect also influences the bubble speed and delays the transition to the parabolic region in the bubble front and rear menisci. In particular, a high degree of undulations on the bubble surface results in intricate rear viscosity distribution for the rear meniscus and the deviation between the experiments and theory may require a further investigation to resolve the axial velocity field. Our study may advance the fundamental understandings and engineering guidelines for coating processes involving thin-film flows and non-Newtonian fluids.
... However, at present, most of the previous studies on migration of microplastics have been carried out through column experiments, which offer macroscopic transport properties but not give mechanistic insight at the microscale. Only a few studies on the migration and deposition of microplastics have been performed through visualized experiments in rectangular channels and Hele-Shaw cells (e.g., Jeong et al., 2020;Wu et al., 2021). ...
Microplastics are ubiquitous in the natural environment and have the potential to endanger the natural environment, ecology and even human health. A series of microfluidic experiments by using soft lithography technology were carried out to investigate the effect of flow rate, particle volume fraction, particle size and pore/throat ratio on microplastics migration and deposition at the pore scale. We discovered a range of deposition patterns of the spherical microplastics from no particle deposition, to discontinuous particle layer, and to continuous particle layers in the retained liquid in the pores, depending on the particle size and volume fraction. Several metrics, including air saturation, probability of particle detainment, expansion ratio and thickness of residual liquid, were quantified to examine the role of various parameters on particle migration and retention of microplastics. At low flow rates (Q = 0.05 μL/min), microplastics migration and deposition were sensitive to changes in particle volume fraction, particle size and pore/throat ratio. In contrast, at high flow rates (Q > 5 μL/min), the migration and retention of particles were mainly controlled by strongly channelized air invasion pattern, while the particle volume fraction, particle size and pore/throat size have only secondary influence. At intermediate range of flow rates, microplastics migration and deposition were dramatically impacted by flow rate, particle volume fraction, particle size and pore/throat ratio. This work improves the understanding of the mechanisms of particle migration and retention in porous media and can provide a reference for more accurate assessment of the exposure levels and times of microplastics in soil and groundwater systems.
... Interestingly, the fluid mechanics and flow topology underlying the dip coating process share many common features with the flow of a plug of liquid in a tube, including the presence of a stagnation point governing the coating film and the evolution of the film thickness h with the capillary number Ca. 38 As a result, the entrainment threshold of isolated particles in the coating film during the dip coating of a plate is consistent with the entrainment threshold in a capillary tube. 36,39 However, the relevance of the different coating regimes and the properties of the coating film remain more elusive for a plug of particulate suspension at a moderate volume fraction pushed by an immiscible fluid in cylindrical capillary tubes. Yet, such an approach could allow coating the inner wall of tubes, giving them some surface properties. ...
... The only constraint is that the diameter of the particles should be small compared to the diameter of the tube. 39 Similarly, the viscosity of the interstitial fluid can be rescaled through the capillary number and will not modify the dynamics, 26 therefore we kept the same interstitial fluid throughout the study. ...
... 34,38 Previous studies in the dip coating configuration and in the present capillary tubes configuration but with isolated spherical particles have reported that the threshold for single particle entrainment are similar. 36,39 . In the case of non-dilute suspensions, the coating of the capillary tube can be predicted at first order using the average volume fraction, and the corresponding effective viscosity. ...
The displacement of a suspension of particles by an immiscible fluid in a capillary tube or in a porous media is a canonical configuration that finds application in a large number of natural and industrial applications, including water purification, dispersion of colloids and microplastics, coating and functionalization of tubings. The influence of particles dispersed in the fluid on the interfacial dynamics and on the properties of the liquid film left behind remain poorly understood. Here, we study the deposition of a coating film on the walls of a capillary tube induced by the translation of a suspension plug pushed by air. We identify the different deposition regimes as a function of the translation speed of the plug, the particle size, and the volume fraction of the suspension. The thickness of the coating film is characterized, and we show that similarly to dip coating, three coating regimes, liquid only, heterogeneous, and thick films, are observed. We also show that, at first order, the thickness of films thicker than the particle diameter can be predicted using the effective viscosity of the suspension. Nevertheless, we also report that for large particles and concentrated suspensions, a shear-induced migration mechanism leads to local variations in volume fraction and modifies the deposited film thickness and composition.
