Critical Capillary Number Research Articles

Droplet deformation and breakup in shear-thinning viscoelastic fluid under simple shear flow

A three-dimensional lattice Boltzmann method, which couples the color-gradient model for two-phase fluid dynamics with a lattice diffusion-advection scheme for the elastic stress tensor, is developed to study the deformation and breakup of a Newtonian droplet in the Giesekus fluid matrix under simple shear flow. This method is first validated by the simulation of the single-phase Giesekus fluid in a steady shear flow and the droplet deformation in two different viscoelastic fluid systems. It is then used to investigate the effect of Deborah number De, mobility parameter α, and solvent viscosity ratio β on steady-state droplet deformation. We find for 0.025<α<0.5 that as De increases, the steady-state droplet deformation decreases until eventually approaching the one in the pure Newtonian case with the viscosity ratio of 1/β, which is attributed to the strong shear-thinning effect at high De. While for lower α, the droplet deformation exhibits a complex nonmonotonic variation with De. Under constant De, the droplet deformation decreases monotonically with α but increases with β. Force analysis shows that De modifies the droplet deformation by altering the normal viscous and elastic stresses at both poles and equators of the droplet, while α mainly alters the normal stresses at the poles. Finally, we explore the roles of De and α on the critical capillary number Cacr of the droplet breakup. By establishing both Ca–De and Ca–α phase diagrams, we find that the critical capillary number increases with De or α except that a plateau critical capillary number is observed in Ca–De phase diagram.

Effect of surfactants on droplet generation in a microfluidic T-junction: A lattice Boltzmann study

Droplet generation in a T-junction with surfactants is simulated using our recently developed lattice Boltzmann method. The method is first used to explore the effect of surfactant concentration ψb on droplet generation. As ψb increases, droplet generation tends to shift from squeezing to dripping regime and then to jetting regime. In the clean system, the upstream pressure varies almost periodically with time. However, in the surfactant-laden system, the upstream pressure no longer varies periodically but overall increases with time for droplet generation in squeezing and dripping regimes. This is because the addition of surfactants results in an additional pressure drop between the front and rear of the generated droplet. Then, droplet generation in both clean and surfactant-laden systems is compared to explore the surfactant role under different values of the capillary number Ca. In either clean or surfactant-laden system, the pressure upstream of the junction rapidly decreases as Ca increases. In the presence of surfactants, the upstream pressure overall increases with time for droplet generation in squeezing and dripping regimes, but the increased amplitude decreases with Ca. Finally, we establish the phase diagrams describing how the droplet generation regime varies with flow rate ratio and Ca in both clean and surfactant-laden systems. It is found that the addition of surfactants reduces the critical capillary number distinguishing squeezing from dripping and the critical capillary number distinguishing dripping from jetting.

Lattice Boltzmann simulation of binary three-dimensional droplet coalescence in a confined shear flow

Small-scale microscopic phenomena determine the behavior of large-scale droplets, which brings great challenges to accurately simulate the droplet coalescence process. In this paper, the mesoscopic lattice Boltzmann method based on the phase field theory is used to simulate the collision and coalescence of binary three-dimensional droplets in a confined shear flow. The numerical prediction of droplet coalescence behavior was first compared with the experimental result, and good agreement was reported. Then, we investigated the influences of a comprehensive range of capillary numbers (0.01≤Ca≤0.5) and Reynolds numbers (0.01≤Re≤10) on the shearing dynamics of binary droplets and also provided a quantitative description of droplet behavior in terms of the droplet deformation parameter and relative trajectory. A shearing regime diagram is further constructed based on the coupling effect of Ca and Re, which reveals three distinct types of droplet behaviors, including coalescence, breakup after the coalescence, and non-coalescence. Concretely, three different patterns of droplets can be completely captured with the variation of Ca at low Re; only two types of coalescence and non-coalescence can be observed for a medium Re, and two droplets just slide over each other without the occurrence of the coalescence when Re is sufficiently large. Also, we identified two critical capillary numbers in the lower Re region and one critical capillary number in the middle Re region, respectively, characterizing flow type transitions from the coalescence to breakup, from the breakup to the non-coalescence, and from the coalescence to the non-coalescence. It is found that all the capillary numbers decrease with Re.

Significance of viscous coalescence in migmatites of the Assam-Meghalaya Gneissic Complex, eastern India

ABSTRACT The magnetite ocelli preserved in the Chandrapur area of the Assam-Meghalaya Gneissic Complex, eastern India, display viscous coalescence. The viscous coalescence phenomenon generally occurs below a critical capillary number, which is governed by the size of the coalescing droplets. The smaller the size of the coalescing droplets, the greater the possibility that they will exhibit viscous coalescing. From our results we infer that intrusion of younger pegmatitic magma into the much older polyphase deformed quartzofeldspathic gneiss of Chandrapur initiated localized partial melting in the gneissic rocks surrounding the intrusions. This localized partial melting produced small magma pools or leucocratic neosome, which was followed by intermingling between the in situ melt (leucocratic neosome) and external melt (pegmatite), leading to chaotic mixing between the two magmatic phases. Chaotic mixing produced thin veins or filaments of the pegmatitic magma as a result of stretching and folding dynamics. Gradually, the thin filaments underwent capillary instability to produce discrete viscous swirls or ocelli. The ocelli consist of leucocratic minerals like K-feldspar, plagioclase, and quartz, with crystals of magnetite at the center representing magnetite ocelli. The mineralogical assemblage of the ocelli matches that of the pegmatitic rocks. After their formation, some of the smaller magnetite ocelli underwent very gentle collisions due to the effect of capillary and viscous forces. Such collisions produced pairs, clusters, or linear structures that are now preserved in the migmatites of the study area.

Stability and tip streaming of a surfactant-loaded drop in an extensional flow. Influence of surface viscosity

We study numerically the nonlinear stationary states of a droplet covered with an insoluble surfactant in a uniaxial extensional flow. We calculate both the eigenfunctions to reveal the instability mechanism and the time-dependent states resulting from it, which provides a coherent picture of the phenomenon. The transition is of the saddle-node type, both with and without surfactant. The flow becomes unstable under stationary linear perturbations. Surfactant considerably reduces the interval of stable capillary numbers. Inertia increases the droplet deformation and decreases the critical capillary number. In the presence of the surfactant monolayer, neither the droplet deformation nor the stability is significantly affected by the droplet viscosity. The transient state resulting from instability is fundamentally different for drops with and without surfactant. Tip streaming occurs only in the presence of surfactants. The critical eigenmode leading to tip streaming is qualitatively the same as that yielding the central pinching mode for a clean interface, which indicates that the small local scale characterizing tip streaming is set during the nonlinear droplet deformation. The viscous surface stress does not significantly affect the droplet deformation and the critical capillary number. However, the damping rate of the dominant mode considerably decreases for viscous surfactants. Interestingly, shear viscous surface stress considerably alters the tip streaming arising in the supercritical regime, even for very small surface viscosities. The viscous surface stresses alter the balance of normal interfacial stresses and affect the surfactant transport over the stretched interface.

Effect of the Molecular Structure Change of a Matrix Polymer (Nylon 6) on the Deformation of Dispersed Phase (a Thermotropic Liquid Crystalline Polymer) Droplets in Shear Flow.

In this work, we investigated the effect of a change in the molecular structure and ensuing molar mass change of a matrix polymer (polyamide 6, Ny 6) on droplet deformation of a dispersed thermotropic liquid crystalline polymer (TLCP, a poly(ester amide)) in shear flow. This study focuses on a total capillary number (the sum of the shear capillary number and the elasticity capillary number) and the viscosity ratio between the TLCP and Ny 6, for the morphological development and mechanical performance of TLCP/Ny 6 blends. In contrast to Ny 6, which has a lower melt viscosity than the TLCP melt, a modified Ny 6 (m-Ny 6) with ca. 2 orders higher melt viscosity than that of Ny 6 at a shear rate of 1 s–1 was found to facilitate the deformation of the TLCP phase. A total capillary number was defined to characterize the viscoelasticity effect on droplet deformation in the blend system. The first normal stress difference obtained from the viscosity curve using Steller’s method was used for the evaluation of the elasticity capillary number. The total capillary number for the Ny 6 blend was far less than the critical capillary number and was insufficient for the dispersed TLCP droplets to be deformed. The shear capillary number of the m-Ny 6 blend was greater than the critical capillary number but was still insufficient for droplet deformation into fibril shapes. The total capillary number, including the elastic capillary number, was sufficiently greater than the critical capillary number for deformation of the dispersed TLCP droplets. Morphological observations and a comparison with the theoretical work confirmed the importance of the viscoelasticity of the melt in the immiscible Ny 6/TLCP blends for in situ composite fabrication in shear flow. Both the high viscosity and the first normal stress difference of m-Ny 6 promote the deformation and fibrillation of the dispersed TLCP droplets.

Coalescence dynamics of two droplets in T-junction microchannel with a lantern-shaped expansion chamber

BackgroundTo improve the coalescence efficiency of droplets in microchannel, a lantern-shaped expansion chamber is proposed to facilitate droplets coalescence. MethodsA high-speed digital camera was used to study the coalescence process and the mechanism of droplet coalescence was analyzed based on the liquid film draining model. FindingsThe existence of shrinkage mouth could prolong the contact time of droplets and accordingly promote droplet coalescence. The drainage time of continuous phase liquid film decreases with the increase of superficial flow velocity and droplet size, but increases with increasing dispersed phase viscosity. The film drainage time for contact coalescence, squeeze coalescence and tail coalescence increases successively. There exists a minimum critical capillary number Camin and a maximum critical capillary number Camax, and coalescence can occur only when Camin < Ca < Camax. The correlations for predicting the minimum critical capillary number and the maximum critical capillary number are proposed.

Pore-scale investigation of immiscible displacement in rough fractures

Immiscible displacement in geological fractures is crucial for various subsurface processes. This paper aims to study the microscopic interaction between fluid and complex rough wall, and how roughness controls the immiscible displacement in fractures. The geometry model of rough surfaces is reconstructed embodying grooves with deviation depths following a Gaussian distribution. The immiscible flow and interface morphology are simulated by the Navier-Stokes equation coupled with a phase-field method. The effects of surface roughness, wall wettability and capillary number on immiscible displacement are systematically investigated through a series of displacement simulations. The results show that the pinning behavior of contact line appears at the rough surface, impelling the growth of interface length and finger formation. The surface roughness contributes to strengthening the wettability effect on interface deformation. The increasing surface roughness promotes the occurrence of contact line jumping, which causes fluid trapping in the grooves of rough surface. The amount of trapped fluid increases with capillary number in rough fractures and the critical capillary number of finger formation decreases with the increase of rough degree. The presented results can significantly improve our fundamental understanding on microscopic displacement process in geological fractures and assist to optimize the displacement scheme for enhanced oil recovery.

Nonlinear Transient and Steady State Stretching of Deflated Vesicles in Flow.

Membrane-bound vesicles and organelles exhibit a wide array of nonspherical shapes at equilibrium, including biconcave and tubular morphologies. Despite recent progress, the stretching dynamics of deflated vesicles is not fully understood, particularly far from equilibrium where complex nonspherical shapes undergo large deformations in flow. Here, we directly observe the transient and steady-state nonlinear stretching dynamics of deflated vesicles in extensional flow using a Stokes trap. Automated flow control is used to observe vesicle dynamics over a wide range of flow rates, shape anisotropy, and viscosity contrast. Our results show that deflated vesicle membranes stretch into highly deformed shapes in flow above a critical capillary number Cac1. We further identify a second critical capillary number Cac2, above which vesicle stretch diverges in flow. Vesicles are robust to multiple nonlinear stretch-relax cycles, evidenced by relaxation of dumbbell-shaped vesicles containing thin lipid tethers following flow cessation. An analytical model is developed for vesicle deformation in flow, which enables comparison of nonlinear steady-state stretching results with theories for different reduced volumes. Our results show that the model captures the steady-state stretching of moderately deflated vesicles; however, it underpredicts the steady-state nonlinear stretching of highly deflated vesicles. Overall, these results provide a new understanding of the nonlinear stretching dynamics and membrane mechanics of deflated vesicles in flow.

Dynamics of a viscous drop under an oscillatory uniaxial extensional Stokes flow

We quantify the transient deformation and breakup of a neutrally-buoyant drop with viscosity λμ∗ immersed in another Newtonian fluid with viscosity μ∗ undergoing oscillatory uniaxial extension at zero Reynolds number. The interfacial tension acting between drop phase and medium phase is γ∗. The drop is initially a sphere of radius a∗. Since the external flow oscillates harmonically with a frequency ω∗, the strength of the imposed flow is characterized by an instantaneous capillary number, Ca=Ca0cos(Det), where Ca0=μ∗ε̇a∗/γ∗ and De=ω∗μ∗a∗/γ∗ is the dimensionless frequency, or Deborah number. Here, ε̇ is the rate of extension in the imposed flow. We utilize boundary-integral computations to calculate the evolution of drop interface as a function of Ca0 and De, focusing primarily on the case where the drop and surrounding fluid have equal viscosities. The computations suggest two families of behavior for the drop deformation. First, below a critical Deborah number (which we determine to be in the interval 0.375<De<1.0), the drop breaks up in a finite time at a critical capillary number that is a function of De. At sufficiently small De the critical capillary number increases linearly with De and the breakup mode is that of “center-pinching”. On increasing De the break-up mode switches to “end-pinching”, and on further increasing De it appears that the critical capillary number diverges at a critical Deborah number between 0.375 and 1.0. This divergence signals the transition to the second family of behavior where the drop attains a long-time periodic state, or alternance, regardless of Ca0. However, at large Ca0 the drop dynamics exhibits a two time-scale behavior: the drop deforms instantaneously at a fast capillary relaxation time-scale τ∗=μ∗a∗/γ∗, whereas the maximum deformation attained during a cycle of the imposed flow grows at the slow time-scale τ∗Ca0=ε̇(μ∗a∗/γ∗)2. As such, the state of alternance is approached exceedingly slowly at large Ca0. Lastly, we perform calculations for different viscosity ratios and find the drop dynamics to be in qualitative agreement with above observations.

Impact of surface viscosity on the stability of a droplet translating through a stagnant fluid

This study examines the impact of interfacial viscosity on the stability of an initially deformed droplet translating through an unbounded quiescent fluid. The boundary-integral formulation is employed to investigate the time evolution of a droplet in the Stokes flow limit. The droplet interface is modelled using the Boussinesq–Scriven constitutive relationship having surface shear viscosity $\eta _\mu$ and surface dilatational viscosity $\eta _\kappa$ . We observe that, below a critical value of the capillary number, $Ca_C$ , the initially perturbed droplet reverts to its spherical shape. Above $Ca_C$ , the translating droplet deforms continuously, growing a tail at the rear end for initial prolate perturbations and a cavity for initial oblate perturbations. We find that surface shear viscosity inhibits the tail/cavity growth at the droplet's rear end and increases the $Ca_C$ compared with a clean droplet. In contrast, surface dilatational viscosity increases tail/cavity growth and lowers $Ca_C$ compared with a clean droplet. Surprisingly, both shear and dilatational surface viscosity appear to delay the time at which pinch off occurs, and hence satellite droplets form. Lastly, we explore the combined influence of surface viscosity and surfactant transport on droplet stability by assuming a linear dependence of surface tension on surfactant concentration and exponential dependence of interfacial viscosities on the surface pressure. We find that pressure-thinning/thickening effects significantly affect the droplet dynamics for surface shear viscosity but play a small role for surface dilatational viscosity. We lastly provide phase diagrams for the critical capillary number for different values of the droplet's viscosity ratio and initial Taylor deformation parameter.

Ternary phase-field simplified multiphase lattice Boltzmann method and its application to compound droplet dynamics on solid surface in shear flow

A ternary phase-field simplified multiphase lattice Boltzmann method (TPF-SMLBM) is developed in a numerical investigation of a compound droplet placed on solid substrate in shear flow at moderate Reynolds numbers. Three major kinematic modes are recovered: quasi-steady sliding (QSS), tumbling-sliding, and tumbling-detachment. Analysis of QSS dynamics explains the wetting length exponential shrinking rate in the early evolution stage. A new dimensionless parameter, the Tumbling number (Tu), is proposed to identify the mode transition towards tumbling. The dynamics of detachment is also investigated, showing that the critical Capillary number of detachments can be described by a scaling law.

Investigation of the dispersing characteristics of antral contraction wave flow in a simplified model of the distal stomach

The dispersing characteristics of antral contraction wave (ACW) flow in the antrum are investigated by reproducing the flow generated by an ACW and determining its effect on liquid drops. The goal is to gain information about the flow field and mechanical stresses, which are responsible for the food disintegration. Toward this end, a model antrum prototype was constructed, consisting of a cylinder that was closed at one end to represent the antrum and closed pylorus. A moving hollow piston with a parabolic inner contour was used to model an ACW. A computational model was developed that reflects this prototype. Experiments and simulations were first performed for fluids with different rheological properties, two relative occlusions (0.60 and 0.75), and several ACW speeds (1.0–7.5 mm/s). The simulations were validated with velocity measurements, and the characteristics of the retropulsive jet were quantified at different Reynolds numbers (0.5–105.3). Experiments were then performed in which liquid drops of different viscosity were placed in a highly viscous fluid with low interfacial tension, similar to conditions in a stomach. It was found that the viscosity ratio (0.001–0.1) influences the retraction dynamics of a drop's tail after stresses are relaxed. The flow and stress information from the simulations was used to analyze fluid transport in the antrum and to quantify drop breakup conditions. It was found that a drop broke up if both a critical capillary number of 0.51 was exceeded and the drop passed within a critical dimensionless distance of 0.3 to the wave apex.

Droplet breakup of a high-viscosity-ratio system in a nonuniform temperature field under laser irradiation

This study conducted experimental and simple numerical studies to investigate the effect of change in viscosity ratio on the dispersion progress in a two-phase immiscible fluid. The viscosity ratio of the fluid was successfully modified by supplying direct heat radiation from an infrared laser. In the experiment, polybutenes and polydimethylsiloxane silicone oils were used as the dispersed droplet and matrix phases, respectively, and the radiation from an infrared laser with an intensity ranging from 10.9 to 87.3 W/cm2 was applied. The results show that the selective radiation-heating method using different radiation absorption coefficients against the infrared laser wavelength caused significant deformation of the droplet phase, reaching even the breakup point of the droplet. We further performed a numerical simulation of three-dimensional thermal conduction, including radiation heating, to estimate the temperature changes in the droplet phase. The results show that the droplet size significantly affects the heat absorption and temperature distribution of the system. Finally, we discuss a suitable radiation intensity on a nondimensional chart using the modified viscosity ratio and critical capillary number.

Electric field mediated droplet spheroidizing in an extensional flow

A 3D mathematical model coupling the phase-field model and the electric current model is applied to describing the DC electric control of droplet deformation in an extensional flow field. Based on this model, electric field mediated droplet spheroidizing in an extensional flow is explored, and the underlying electro-hydrodynamics is clarified. Regime diagrams are plotted to quantitatively recognize the operating regimes for different droplet morphologies, from which the critical electro-hydrodynamic criteria for droplet spheroidizing are summarized. In addition, the influence of electrophysical parameters of fluids on electric field mediated droplet spheroidizing is analyzed. It is indicated that the hydrodynamic forces imposed on the droplet from the pure extensional flow can be completely counterweighted by imposing a proper electric field, so as to realize spheroidizing of the droplet. Within the scope of the current investigation, the critical electric capillary number (CaE) for droplet spheroidizing is found to have linear relationship with the hydrodynamic capillary number (Ca), which can be expressed as CaE = aCa. Specifically, the linear coefficient, a, decreases with increment of RS (i.e., the product of conductivity ratio and permittivity ratio between the droplet and continuous phase) when RS &amp;gt; 1, while it decreases with decreasing RS when RS &amp;lt; 1. Compared with RS &amp;gt; 1, the critical CaE for droplet spheroidizing is generally smaller under RS &amp;lt; 1 for a given Ca, suggesting less electric effort is required to realize droplet spheroidizing.

Large eddy simulation and experiment of shear breakup in liquid-liquid jet: Formation of ligaments and droplets

Understanding the shear breakup in jet flows and the formation of droplets from ligaments is important to determine the final droplet size distribution (DSD). The initial droplet size, which affects the final DSD, is considered to be generated by the shear breakup. Large eddy simulation (LES) was performed to investigate the shear breakup in liquid-liquid jet flows. The explicit Volume of Fluid (VOF) model with the geometric reconstruction scheme was used to capture the oil-water interface. The estimated oil distribution including wave peaks, ligaments, droplets and water streaks were compared to the experiments with a good agreement. The estimated DSD matched with the measurements favorably well. In the simulation, the formation of droplets with a smooth and curved surface from ligaments or sheet-like structures was obtained. Different mechanisms were observed along with the shear layer including the formation of droplets from ligament through the capillary forces, breakage of a droplet into smaller ones and attachment of a droplet to a ligament. The destructive shear forces and resisting surface tension forces were quantified on stretching and retracting ligaments. The influence of internal viscous force was found to be negligible due to low oil viscosity. The critical capillary number was found to be larger than 5.0 for ligaments breaking with the shear breakup. The capillary number was below unity for retracting ligaments. The coalescence of two equal-sized droplets was obtained in the shear breakup region. The shear stress magnitude at the contact region increased more than two folds. The total surface area decreased nearly 20% after the coalescence.

Coalescence dynamics of two droplets of different viscosities in T-junction microchannel with a funnel-typed expansion chamber

The coalescence behavior of two droplets with different viscosities in the funnel-typed expansion chamber in T-junction microchannel was investigated experimentally and compared with droplet coalescence of the same viscosity. Four types of coalescence regimes were observed: contact non-coalescence, squeeze non-coalescence, two-droplet coalescence and pinch-off coalescence. For droplet coalescence of different viscosities, the operating range of non-coalescence becomes narrowed compared to the droplet coalescence of same viscosity, and it shrinks with increasing viscosity ratio η of two droplets, indicating that the difference in the viscosity of two droplets is conducive to coalescence, especially when 1＜ η ＜6. Furthermore, the influences of viscosity ratio and droplet size on the film drainage time ( T dr ) and critical capillary number ( Ca c ) were studied systematically. It was found that the film drainage time declined with the increase of average droplet size, which abided by power-law relation with the size difference and viscosity ratio of the two droplets: T dr ~ ( l d ) 0.25±0.04 and T dr ~ ( η ) −0.1±0.02 . For droplet coalescence of same viscosity, the relation of critical capillary number with two-phase viscosity ratio and dimensionless droplet size is Ca c = 0.48 λ 0.26 l −2.64 , while for droplet coalescence of different viscosities, the scaling of critical capillary number with dimensionless average droplet size, dimensionless droplet size difference and viscosity ratio of two droplets is Ca c = 0.11 η −0.07 l s −2.23 l d 0.16 .

Modelling of damage of a liquid-core microcapsule in simple shear flow until rupture.

Capsules, composed of a liquid core protected by a thin deformable membrane, offer high-potential applications in many fields of industry such as bioengineering. One of their limitations comes from the absence of models of capsule damage and/or rupture when they are subjected to an external flow. To assess when rupture is initiated, we develop a fluid-structure interaction (FSI) numerical model of a capsule in Stokes flow that accounts for potential damage of the capsule membrane. We consider the framework of Continuum Damage Mechanics and model the membrane with an isotropic brittle damage model, in which the membrane damage state depends on the history of loading. The FSI problem is solved by coupling the finite element method, to solve for the membrane deformation, with the boundary integral method, to solve for the inner and outer fluid flows. The model is applied to an initially spherical capsule subjected to a simple shear flow. Damage initiates at a critical value of the capillary number, ratio of the fluid viscous forces to the membrane elastic forces, and rupture at a higher capillary number, when it reaches a threshold value. The material parameters introduced in the damage model do not influence the mode of damage but only the values of the critical and threshold capillary numbers. When the capillary number is larger than the critical value, damage develops in the two symmetric central regions containing the vorticity axis. It is indeed in these regions that the internal tensions are the highest on the membrane.

Film entrainment and microplastic particles retention during gas invasion in suspension-filled microchannels

Understanding of microplastics transport mechanism is highly important for soil contamination and remediation. The transport behaviors of microplastics in soils are complex and influenced by various factors including soil and particle properties, hydrodynamic conditions, and biota activities. Via a microfluidic experiments we study liquid film entrainment and microplastics transport and retention during two-phase displacement in microchannels with one end connected to the air and the other connected to the liquid with suspended particles. We discover three transport patterns of microplastic particles, ranging from no deposition to particle entrapment and to particle layering within liquid films, depending on the suspension withdrawal rates and the particle volume fraction in the suspension. The general behavior of particle motion is effectively captured by the film thickness evolution which is shown to be dependent on a modified capillary number Ca0 taking into account the effects of flow velocity, particle volume fraction, and channel shape. We also provide a theoretical prediction of the critical capillary number Ca0* for particle entrapment, consistent with the experimental results. In addition, the probability of microplastics being dragged into the trailing liquid film near the gas invading front is found to be proportional to both particle volume fraction and the capillary number. This work elucidates the microplastics transport mechanism during unsaturated flow, and therefore is of theoretical and practical importance to understand the contaminant migration in many natural and engineered systems spanning from groundwater sources to water treatment facilities.

Breakup a droplet passing through an obstacle in an orthogonal cross-section microchannel

In this work, the breakup of a droplet passing through an obstacle in an orthogonal cross section is numerically investigated. The relevant boundary data of the velocity field is numerically computed by solving the depth-averaged Brinkman equation via a self-consistent integral equation using the boundary element method. To study the dependence of the droplet breakup on the obstacle shape, two different shapes of obstacle, circular and elliptical, are considered in the present work. We investigate the effect of obstacle size, obstacle position, and capillary number on the breakup treatment of the droplet. Numerical results indicate that the critical capillary number depends on the obstacle shape, obstacle position and droplet size. In the elliptical obstacle, in addition, the results also show that the area ratio of daughter droplets depends on the capillary number. Results show that the area ratio of daughter droplets depends on the capillary number, obstacle shape, and obstacle position. Our results is in a good agreement with the previous studies.