Coalescence Patterns Research Articles

Influence of weak inclusions on the fracturing and fractal behavior of a jointed rock mass containing an opening: Experimental and numerical studies

To gain a better understanding of the interaction between weak inclusions and jointed rock masses, a conceptual model containing a joint set and an opening is prepared, and cases involving unfilled and filled openings are considered. The influence of weak inclusions on the fracturing and fractal behavior of these models is investigated by using laboratory experiments, the rock failure process analysis (RFPA) code and the fractal geometry. An overhanging beam model is proposed to explain the initiation mechanism of tensile cracks around the unfilled and filled openings. The RFPA simulations provide deep and quantitative insight into the fracturing behavior. The coalescence patterns of the surrounding rock mass obtained from physical and numerical tests are in reasonable agreement, and can be classified into two categories: shearing along the joint set and cutting through the rock bridges. The inclusion with a low strength exerts little influence on the failure pattern but has an appreciable reduction in the stress values and an appreciable increment in the mechanical properties. The crack coalescence exhibits fractal properties, and there are good correlations between the fractal dimension and mechanical properties of these jointed rock specimens containing unfilled and filled openings.

Failure Characteristics of Complicated Random Jointed Rock Mass Under Compressive-Shear Loading

Most natural rock masses contain a large number of random joints and fissures, and most of the rock masses at the rock engineering are commonly in both compression and shear stress environment. However, the research on the failure characteristics of complex random jointed rock mass under compressive-shear loading is still limited. To address this gap, this paper uses the particle flow code 2D to establish a discrete fractured rock mass model and carry out a series of numerical tests with different compressive-shear angles (α) and different joint geometric parameters. The effects of compressive-shear angle and joint geometric parameters on the strength and failure characteristics of fractured rock masses are studied. The results indicate that with the increase of α, the peak strength of the specimen decreases gradually, and the failure mode changes from the composite shear failure mode (Mode-I) to a plane shear failure mode (Mode-II) and then to intact shear failure mode (Mode-III). Specifically, the three failure modes occur in the specimens with α = 15°, 30° or 45°, 60°, respectively. The existence of joints affects stress distribution on rock mass during the loading process. Furthermore, the stress at the joint tip is relatively concentrated, while on both sides of the joint is smaller. Three kinds of crack coalescence patterns are observed: tensile, shear, and tensile-shear mixed coalescence. The inclination angle of the rock bridge between adjacent joints affects the specific type of coalescence.

Democratic backsliding in the European Union: the role of the Hungarian-Polish coalition

ABSTRACT Combining the insights of EU-specific research on backsliding and coalitions with the literature on the international collaboration of autocrats, we argue that right-wing political leadership in Hungary and Poland have coalesced to advance their respective projects of democratic backsliding. We identify three distinct but intertwined uses of the coalition: (1) mutual protection afforded within the supranational arena aimed at limiting the EU's sanctioning capacities; (2) learning in the form of transfer of democratic backsliding policies; and (3) domestic legitimation. Three factors have driven coalescence patterns: intersecting interests, ideological proximity, and the EU’s decision rules regarding sanctions.

Strain localization and cracking behavior of sandstone with two gypsum-infilled parallel flaws

The infilled flaws within rocks significantly influence the mechanical behavior of non-through flawed rocks. To investigate the influence of infilling on strain localization in the pre-cracking, cracking and cracking coalesce stages, experimental and numerical investigations were conducted on sandstone containing two gypsum-infilled parallel flaws, but of different flaw geometric configurations. The full-field strain evolution and progressive cracking process are analyzed using the acoustic-optical-mechanical methods, while the stress evolution and stress transfer by infilling were studied by the particle flow code modeling. The results show that the major principal strain of flawed sandstone with infilling changes from the diffused ellipse to highly localized linear strip with the increase of loading level, while the shear strain keeps a diffused ellipse shape invariable. The shear strain localization of infillings with a large flaw inclination initiates at a lower loading level than that of small one. The peak strength and Young’s modulus of flawed sandstone show a first decrease, then increase and finally decrease trend with the ligament angle increasing from 0° to 150°, and both achieve their minimums at 60°. Ten types of cracks were observed, and four patterns of crack coalescence were identified. By comparison with sandstone of open flaws, the flaw infilling simplifies the geometric pattern of produced crack networks, which is quantitatively characterized by fractal dimension analysis. The numerical modeling shows that the infilling enlarges the compressive bond force zone and decreases the tensile bond force zone in the surrounding areas of pre-existing flaw, thus increasing the reinforcement coefficient. Moreover, the flaw geometric configuration has a significant influence on the reinforcement coefficient. It is also found that the increment of reinforcement coefficient defined by crack initiation stress is greater than those defined by either crack coalescence or peak stresses.

Crack coalescence patterns and local strain behaviors near flaw tip for rock-like material containing two flaws subjected to biaxial compression

To understand crack coalescence patterns and local strain behaviors near flaw tip for flawed rock subjected to biaxial compression, a series of biaxial compression tests with a lateral stress of 2.0 MPa were carried out on rock-like specimens containing two flaws. The paper classified crack coalescence patterns, measured the local strain concentration near the flaw tip, and analyzed the stress–strain behavior related to crack initiation, propagation, and coalescence. Some interesting cracking behaviors of flawed specimens subjected to biaxial compression were observed. Seven basic crack types (wing crack, anti-wing crack, oblique and coplanar shear cracks, lateral crack, remote crack, and en echelon cracks) are identified and ten patterns of crack coalescences are observed subjected to biaxial compression. In general, the crack coalescence varies from the shear crack coalescence (S-mode) to the tensile wing crack coalescence (T-mode), and then to mixed shear/tensile crack coalescence (TS-mode), with the increase of the rock bridge ligament angle β, ranging from 25 to 120° subjected to biaxial compression. Crack initiation and coalescence usually can be reflected by a load stress rapid drop, a measured strain jump, or turning near flaw tip subjected to biaxial compression. The measured tensile or compression strains near flaw tip tend to increase almost simultaneously with higher axial stress before crack initiation; however, become very complex after crack initiation, due to the stress relaxation and adjustment near flaw tip induced by the combination of crack development and constant lateral stress. The existence of lateral stress results in many fragments in a shear failure mode with local surface spallings in flawed specimens containing two flaws.

Dynamic mechanical behaviors of interbedded marble subjected to multi-level uniaxial compressive cyclic loading conditions: An insight into fracture evolution analysis

Rocks in civil and mining engineering usually experience rather complex stress disturbance. Rock dynamic mechanical behaviors under constant stress amplitude condition have been widely studied. However, the rock fracture evolution characteristics subjected to multi-level cyclic loading conditions are not well understood. In this work, multi-level cyclic compressive loading experiments were performed on marble with interbed orientation of 0°, 30°, 60° and 90°. Anisotropic fracture evolution characteristics were revealed using dynamic stress strain descriptions and post-test CT scanning technique. Results show that rock fatigue deformation, strength, lifetime, dynamic elastic modulus and damping ratio are all impacted by rock structure. The interbed structure plays a dominant role in fracture evolution compared with natural fracture and pyrite band. Rock stiffness degrades and damping ratio increases with the increase of interbed orientation. In addition, a two-phase damage accumulation pattern was found for marble under multi-level cyclic loading condition. A damage evolution model was first established to model damage accumulation, the proposed model fits well to the experimental data. Moreover, post-test CT scanning reveals the internal crack coalescence pattern and failure mode is impacted by the interactions of interbeds, natural fracture and pyrite bands. The rock bridge structure in marble with 60° and 90° interbed orientation alters the crack propagation path. Although the crack pattern is relatively simple at rock bridge segment, much more energy is needed to drive crack propagation. The testing results are expected to improve the understanding of the influence of rock structure on fracture evolution when subjected to variable stress amplitude loading conditions.

Experimental and numerical investigations on crack development in 3D printed rock-like specimens with pre-existing flaws

Rock masses contain various discontinuities such as flaws, cracks, join sets and more, each significantly affect the crack initiation, propagation and coalescence patterns of new cracks which dominates the ultimate failure of geostructures. The interaction of the pre-existing flaws with each other and with the newly formed cracks is complicated which demands a comprehensive look into the phenomenon. This paper focuses on the applicability of the 3D printing technology coupled with the digital image correlation (DIC) and the bonded particle model (BPM) in replicating real behaviour of natural rocks containing pre-existing flaws. Systematic flaws configurations are considered including single, coplanar, partially overlapped and fully overlapped arrangements to enable a comprehensive coalescence analysis. The results show that the BPM is capable of precisely modelling the crack initiation location, type (tensile, shear or mixed-mode) and coalescence type detected by the DIC. Moreover, it is shown that the 3D printed specimens can reproduce the previously identified cracks and coalescence types. By the analysis of the displacement fields, a new coalescence type is detected and introduced and named as type X, which is a mix of wing crack and quasi coplanar secondary shear crack. The analysis of the displacement vectors revealed five types of cracks: shear opposite slip crack, tensile opposite crack, tensile compliant crack, shear compliant crack, and mixed tensile-shear crack. Furthermore, the comparisons made between the peak loads of the experiments and the simulations show a god agreement in terms of peak load magnitude and flaws inclination angle dependency.

Coalescence as autosegmental spreading and delinking

Phonological coalescence, understood as a type of synchronic alternation in which two phonological elements seem to fuse into one, presents aprima faciechallenge for versions of Optimality Theory that assume the principle of containment. If all underlying material has to be present in the output form, replacing two input elements with a single output element is not straightforward. I argue that, under the assumptions of Autosegmental Coloured Containment Theory, a distinct operation of coalescence is unnecessary, as all major types of coalescence patterns can be analysed in terms of (i) adding new association lines between some autosegmental nodes, and (ii) the underparsing of other nodes, leading to their phonetic non-realisation. The proposed analysis accurately reflects the heterogeneity of coalescence alternations, which are shown to fall into three different types.

An Experimental and Numerical Study on the Influence of Filling Materials on Double-Crack Propagation

Filling materials such as clay or sand widely exist in natural rock joints and work as weak bonds between the joint surfaces. The fillings affect rock deformation and failure behavior, and show different influences in terms of single crack or multiple cracks. While most of the literature have focused on unfilled cracks in brittle materials, this study aims to investigate various filling materials on the crack behavior, e.g., initiation, secondary cracks and peak strength. In this paper, the crack propagation in rock-like specimens with double-filled and unfilled cracks are investigated experimentally and numerically. Uniaxial compression tests were conducted and the experimental observations indicate that the peak stress and first crack initiation stress of the specimens vary with different geometries and different filling materials, while the crack initiation location and the pattern of crack coalescence show similar behavior between filled and unfilled cracks. In parallel to the experimental tests, numerical simulations were carried out using a modified phase field model (PFM) to complement the experiments and provide a new perspective. The PFM is found to produce consistent stress–strain curve, strength, and crack patterns with those observed in the experimental tests for both unfilled and filled cracks.

Mechanical behaviours of granite containing two flaws under uniaxial increasing‐amplitude fatigue loading conditions: An insight into fracture evolution analyses

AbstractThis work aims to investigate the fracture evolution of granite containing two pre‐existing flaws under uniaxial increasing‐amplitude fatigue conditions using macroscopic stress strain descriptions and posttest three‐dimensional (3D) computed tomography (CT) technique. The impacts of flaw arrangement (i.e., approach angle of 20°, 50° and 70°) on the stress strain responses, hysteresis loop shape, damage evolution and crack coalescence pattern at rock bridge segment were investigated. Results show that rock structure has an obvious impact on macroscopic stress strain responses, volumetric strain, dynamic elastic modulus and damping ratio. The sparse‐dense pattern of hysteresis loop is different at each loading stage caused by the differential accumulative damage. The damping ratio increases and dynamic elastic modulus decreases with the increasing fatigue loading stage. Posttest 3D CT visualization reveals a most striking finding that crack coalescence is easy for rock having low approach angle, and complex crack network forms for rock having high approach angle.

Short report - Usefulness of point-of-care ultrasound in pediatric SARS-CoV-2 infection.

SARS-CoV-2 infection in children is uncommon compared to adult population. However, some children required hospital and/or PICU admission. The aim of this short communication is to share our experience with Point-of-Care Ultrasound (POCUS) when managing these patients. Remarkably, all cases presented pleural and pericardial effusions, detected by POCUS, despite showing an adequate urinary output and prior to receiving any kind of fluid resuscitation. Effusions have been described as rare among SARS-CoV-2 infection in adult population. By performing portable chest X-Ray they would have gone unnoticed in our patients. Other POCUS findings consisted of all types of consolidations and coalescent B-line patterns. POCUS was also performed in order to optimize PEEP, checking adequate endotracheal intubation positioning (avoiding the risk of contagiousness related to auscultation in this framework), and to assess volemia status, cardiac performance, and brain neuro-monitoring. There was not cross-infection. In pediatric SARS-CoV-19 effusions are frequent but easily unnoticed unless lung and echo POCUS are performed.

Acoustic emission and CT investigation on fracture evolution of granite containing two flaws subjected to freeze–thaw and cyclic uniaxial increasing-amplitude loading conditions

The deterioration mechanism and fatigue fracturing evolution of granite with two pre-existing flaws experiencing freeze–thaw (F-T) treatment are investigated in this work. The flaws in the rock sample were prepared as a combination of a horizontal flaw with an upper inclined flaw above the horizontal flaw according to the joint characteristics in an open pit slope. In-situ acoustic emission monitoring combined with the post-test 3D computed tomography (CT) technique was employed to reveal the fracture evolution behaviors of rock treated with 0, 50, and 80 freeze–thaw cycles. Results show that the freeze–thaw damage impacts the frost heaving force, cyclic deformation, AE activates, crack coalescence pattern and fatigue life of the granite samples. The accumulative AE count/energy decreases with increasing number of freeze–thaw cycles, and the accumulated AE count/energy in a loading stage gradually grows faster. In addition, AE spectral frequency analysis reveals the impact of previous freeze–thaw damage on the formation of the crack scale, the sample is prone to producing large scale cracks under high freeze–thaw treatment. Moreover, 3D reconstructed CT images present an internal crack network pattern, and the most striking finding is that a simple crack network forms for a sample experiencing high F-T fatigue damage. It is suggested that deterioration of the rock bridge structure is strongly impacted by the accumulative freeze–thaw damage. The testing results are helpful to understand the influence of freeze–thaw and fatigue loading on the fracture evolution characteristics of rock in cold regions.

Influence of Two Cross-Flaws Geometry on the Strength and Crack Coalescence of Rock-Like Material Specimens under Uniaxial Compression

Cross-flaws are very common in natural rocks. To date, the understanding of the failure process of rocks with cross-flaws is very limited. In this research, we study the influence of the cross-flaws geometry on the rock strength and coalescence modes with rock-like specimens. In this work, three groups of specimens with pre-existing cross-flaws are investigated: two aligned cross-flaws, two step cross-flaws, and two collinear primary flaws. The crack propagation and strength of specimens containing two parallel flaws are also studied to compare the results with those of specimens with cross-flaws. The results demonstrate that the cross-flaw geometry influences the rock bridge coalescence patterns in rock-like specimens. Specimens with two aligned cross-flaws and two step cross-flaws coalesce with tensile cracks in the rock bridge areas, while specimens with two collinear primary flaws coalesce with shear cracks. Specimens containing two cross-flaws may have a higher uniaxial compressive strength than specimens containing two parallel flaws, and the number of specimens with a higher uniaxial compressive strength is influenced by the cross-flaws geometry in the different groups. The cross-flaws geometry influences the strength of the specimens. In this research, specimens in the group containing two aligned cross-flaws have the highest mean uniaxial compressive strength, and specimens in the group with two step cross-flaws have the lowest strength.

Rock bridge fracturing characteristics in granite induced by freeze-thaw and uniaxial deformation revealed by AE monitoring and post-test CT scanning

This work presents the impacts of freeze-thaw (F-T) fatigue damage on the crack coalescence behaviors at the rock bridge segment for granite samples containing two unparallel flaws. The pre-existing flaw geometry in the rock was a combination of a horizontal flaw and an upper inclined flaw forming a rock bridge structure. In-situ acoustic emission (AE) detection technique and post-test X-ray computed tomography (CT) scanning were used to reveal the rock bridge fracturing characteristics during the whole deformation. Results show that both the F-T cycle and flaw inclined angle influence the crack propagation path, sample strength, deformation and AE pattern and the rock bridge fracture behavior. The AE counts and energy curves present skip phenomenon and are in consistent with the stress drop points on the macroscopic stress-strain curves. The accumulative AE energy and count decrease with increasing F-T cycles. In addition, the cracks during sample failure can be classified into six types by AE spectral frequency analysis, and the high amplitude - low frequency signal was good evident to predict the brittle fracture of the rock bridge. By the post-test CT scanning visualization of the internal crack coalescence pattern at the rock bridge segment, the most striking finding is that the rock bridge structure easily deteriorates for rock subjected to high F-T damage, which suggests that repeated F-T damage drives the crack propagation at the flaw tip and can accelerate the communication of the pre-existing flaws. The testing results are expected to enhance the understanding of the F-T fatigue damage effect on the rock bridge fracturing and can be favorable for predicting the stability of rock structures and rock mass in cold regions.

Mechanical behavior around double circular openings in a jointed rock mass under uniaxial compression

To better understand the mechanical behavior in a jointed rock mass, a series of uniaxial compression tests were conducted on non-persistently jointed rock specimens with double circular holes. Acoustic emission (AE) and digital image correlation (DIC) techniques were applied to capture micro-crack events and real-time strain field evolution in the specimens. The results indicate that the existence of non-persistent joints has a significant influence on the strength characteristics of the specimens. Specifically, peak strength decreases at first and reaches a minimum at 30° then increases with increase in the joint dip angle. DIC technology has successfully monitored the development of surface strain fields. The fracture evolution process is comprehensively understood. Every sudden change in a strain field is usually accompanied by apparent AE events and stress–strain curves take the form of oscillations. The crack coalescence modes among joints can be summarized as six types and the crack coalescence patterns around holes and joints can be divided into three categories. These results are helpful to understanding further the mechanical properties and fracture mechanism of openings in non-persistently jointed rock masses.

Investigation on Crack Coalescence Behaviors for Granite Containing Two Flaws Induced by Cyclic Freeze-Thaw and Uniaxial Deformation in Beizhan Iron Mining, Xinjing, China

This work is aimed at investigating the effect of freeze-thaw (F-T) cycle on the crack coalescence behavior for granite samples containing two unparallel flaws under uniaxial compression. The flaw geometry in the samples was a combination of an upper inclined flaw with a horizontal flaw underneath. After the uniaxial compression experiments, macroscopic crack pattern description and the mesoscopic posttest CT imaging were used to reveal the effects of F-T cycle on the crack coalescence morphology at the rock bridge area. Results show that the stress–strain curves present a fluctuating growth trend and stress drop phenomenon becomes weaker with increasing F-T cycles. In addition, three different kinds of cracks (tensile-wing cracks, oblique shear cracks, and antiwing cracks) were observed, and the crack coalescence pattern was influenced by the F-T cycles and approach angle. A mix of tensile and shear failure occurs for the sample subjected to weak F-T treatment, and simple tensile failure occurs for the sample subjected to high F-T treatment. Moreover, CT imaging reveals a crack network pattern at the rock bridge area, and it is found that the fracture degree deceases with increasing F-T cycles and increases with the increasing approach angle. It suggests that the rock bridge area can be easily fractured for the sample subjected to high F-T cycles. Results of this study can provide theoretical foundation for the instability predication of fractured rock structures in cold regions.

Investigating the crack initiation and propagation mechanism in brittle rocks using grain-based finite-discrete element method

Fracturing process and possible factors influencing crack initiation, propagation and coalescence of granitic rocks are investigated using a grain-based finite-discrete element method (GB-FDEM). In contrast to conventional methods, the GB-FDEM used herein consists of dual-scale contact models ensuring grain breakage, and pre-processing scheme for reproducing the realistic micro heterogeneity of rocks. A standardised calibration procedure is proposed after an analysis of uncertain parameters. An optimized model is built according to calibration results of benchmark experiments including uniaxial compression test, confined compression test and Brazilian disc test in laboratory. The compression testing under different end friction, slenderness, loading rate and confining stress are systematically performed to investigate the external influences on rock fracturing. Boundary constraint is revealed in association with the macro failure pattern, in which the shearing slide leads to slight extension on ends and causes limited influence on the stress distribution in the far field. Effects on crack stresses are consistent with that of uniaxial compressive strength when the end friction takes effect. Slenderness affects the stress distribution and in turn changes the fracture pattern of rocks. Slight influence on the stress level of crack initiation and crack damage can be caused by the change of height-to-width ratio. Loading rate dramatically increase the rock strength based on two underlying mechanisms that the increase in overall number of micro cracks and the transition from intergranular fracturing to transgranular fracturing. These effects on crack initiation and crack damage stresses are inconsistent because the responses of normalized crack initiation and crack damage stresses subject to strain rate have inverse responses. The proportion of different micro cracks is shown to characterize the unchanged micro fracturing when the confining stress is increased. The enhancement of crack initiation and the change of the pattern of crack coalescence are attributed as the mechanism of confinement effect.

Experimental investigation on hydraulic fracturing of granite specimens with double flaws based on DIC

Hydraulic fracturing is a crucial technology for shale gas exploitation and enhanced geothermal systems. However, the fundamental mechanisms of hydraulic fracturing, such as the mechanism (tensile or shear) of cracks, physical process of crack initiation, propagation, and coalescence, are not comprehensively understood. Therefore, further direct observations and quantitative investigations are required. Accordingly, an experiment of hydraulic fracturing on double-flaw granite specimens was performed. The fracturing processes were captured using a high-speed camera, and a quantitatively study based on digital image correlation (DIC) was achieved. DIC was proved to be a powerful analysis tool, especially for visualising the crack evolution process. Results show that the crack types in the hydraulic fracturing tests were primarily Type I or Type Ib tensile cracks. The process of strain localisation zones (SLZs) revealed the physical process of the hydraulic fracturing: SLZs initiated at or near pre-existing flaw tips before fracturing, and then cracks initiated and propagated along the SLZs. Some cracks initiated previously may close with the decrease in fluid pressure after fracturing and evolved into closure cracks. The bridging angle was the main factor determining the coalescence patterns, and the specimens exhibited direct coalescence (Category 6 or 7) when β ≥ 90°. SLZs were initiated at both inner tips in Category 6, while SLZs could only be observed at one inner tip in Category 7. The ultimate fracturing pressure first decreased then increased as the bridging angle increased.

Failure mechanism of S-shaped fissure in brittle materials under uniaxial tension: Experimental and numerical analyses

The behavior and mechanism of crack initiation and coalescence of S-shaped fissure are investigated in the study to shed more light on the failure characteristics of non-straight fissures. The uniaxial tensile tests with 3D-printed samples and numerical simulations are carried out to study the influence of different geometric parameters of S-shaped fissure, including the inclination angle β and effective curvature A/c. The crack coalescence of an S-shaped fissure is observed to occur exclusively through tensile cracks, while the tip cracking and non-tip cracking patterns of S-shaped fissures can be identified based on the crack initiation position. It is found that the inclination angle and effective curvature have a substantial influence on the failure mode of the S-shaped fissure. Based on experimental results, extensive numerical simulations are performed to predict the cracking patterns, e.g., tip cracking/non-tip cracking. A reasonably effective, quick criteria approach for identifying the coalescence patterns in terms of parameters A/c-β for S-shaped fissures under uniaxial tension is derived based on vast numerical calculations. The current study yields a favorable demonstration of the application of 3D printing technology in adept arbitrary geometry formation.

Numerical study on electrohydrodynamic multiple droplet interactions.

We present a numerical study of inviscid multiple droplet coalescence and break-up under the action of electric forces. Using an embedded potential flow model for the droplet hydrodynamics, coupled with an unbounded exterior electrostatic problem, we are able to perform computations through various singular events and analyze the effects of the electrical field intensity on droplet interactions. Laboratory experiments on the electrodynamics of droplet pairs show a much richer, and sometimes unexpected, behavior than that of isolated droplets. For example, it has been found that opposite charged droplets tend to repel each other when the electric field intensity is above a certain critical value. Although the mathematical model employed in this work incorporates very simple flow and electric assumptions, many of the droplet coalescence patterns seen in laboratory experiments can be reproduced. In this model, the interaction pattern of two droplets of radii R_{0} separated a distance D_{0}, depends on the ratio X_{0}=D_{0}/R_{0} and the applied uniform electric field intensity, E_{∞}. By performing a vast number of numerical simulations we are able to characterize the coalescence modes before and after drop merging as a function of these two parameters. The simulations predict that droplet repulsion occurs within a narrow interval of E_{∞} values, different for each X_{0}. Surprisingly, in this E_{∞} interval, a sharp transition between two power-law precoalescence flow regimes is seen. The evolution of several flow characteristics before and after coalescence, and the shape of the deformed droplets at coalescing time and the double cone angle, are also addressed and analyzed in detail. Cone angles below 35^{∘} lead to droplet coalescence for any X_{0} value, which is in accordance with previously reported studies. Finally, it is shown that the model and algorithm can handle multiple droplet interactions. The simulations qualitatively match results from water in oil experiments in microchannels, despite the fact that the exterior fluid is not considered in the mathematical model.