Numerical simulation of droplet formation in a Co-flow microchannel capillary device

Numerical simulation of droplet formation in a Co-flow microchannel capillary device | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Numerical simulation of droplet formation in a Co-flow microchannel capillary device Peihua Zhang, Yongbiao Ma, Bao Song, Dengke Chen, Dengying Zhang, and 3 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3676725/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In this article, a numerical simulation of the droplet formation in a Co-flow microchannel capillary device, and the influencing factors of the formation of droplets are studied. The level set method is used to track the two-phase interface and droplet formation. In the Co-flow focusing device, we explored the influencing factors of the size of the generated droplets. The results show that as the ratio of the dispersed phase velocity to the continuous phase velocity increases, the volume of the generated droplets decreases significantly, the droplet generation frequency increases significantly, and the pressure of the droplets at the centerline decreases significantly. As the viscosity of continuous phase increases, the volume of generated droplets decreases significantly, the frequency of droplet generation increases significantly, and the pressure of droplets at the centerline decreases significantly. As the contact angle between the continuous phase and the wall increases, the volume of the generated droplets increases, but the volume increase is not obvious enough, the droplet generation frequency becomes smaller, and the droplet pressure at the centerline decreases. As the increase of interfacial tension, the volume of droplet generation increases significantly, the frequency of droplet generation decreases significantly, and the pressure of droplet at the centerline increases significantly. Microfluidic Numerical simulation Droplet Co-flow Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 1 Introduction Fluid is an important form of matter, and fluid flow is one of the most basic phenomena in nature. Microfluidic technology, also known as lab-on-a-chip technology, is a science and technology with the main feature of manipulating fluids in micron-scale space, minimizing the basic functions of biological and chemical laboratories onto a chip of a few square centimeters. The microfluidic device has the advantages of small sample volume, high sensitivity, good portability, short analysis time, and low power consumption. Microfluidic devices are also widely used in analytical chemistry, basic biological research, and material synthesis [ 1 , 2 ]. However, it is difficult to prepare droplets of a few microns in size. The generation of droplets originates from the instability of the fluid [ 3 , 4 ]. Droplets can be divided into water-in-oil (W/O), oil-in-water (O/W), water-in-oil-in-water (W/O/W), and oil-in-water-in-oil (O/W/O) [ 5 ]. Most droplets are generated by passive methods. The common passive method is the T-junction [ 6 ], flow-focusing method(MFFD), Co-flow focusing method. Among them, the T-junction method is the most common. It is the continuous phase under pressure to push the dispersed phase to break to produce droplets [ 7 – 11 ]. In MFFD, the continuous phase flows from both sides of the dispersed phase, which has the effect of interleaving the dispersed phase. The dispersed phase is subjected to the symmetrical shear force from the continuous phases on both sides and breaks into droplets [ 12 – 14 ]. Compared with the T-junction, the droplet generation process is more stable, the generated droplet size is controllable, and smaller droplets can be generated. Similar to MFFD, the Co-flow focusing method can produce smaller droplets [ 15 ] and the size of the generated droplets has a great relationship with the ratio of two-phase flow rate, viscosity, interfacial tension, wall effect and capillary surface [ 16 ]. The change in the concentration of the dispersed phase will determine whether or not satellite droplets are generated during the droplet generation process [ 17 ]. The microbubbles are generated by the Co-flow focusing device. The pressure of the gas, the flow rate of the liquid, and the width of the dispersed phase channel will all affect the volume of the bubble and the frequency of generation [ 18 ]. Using opeofoam's C + + program library to study droplet generation is also a very good approach. The droplet formation process is discussed by varying the viscosity of the dispersed phase and the velocity of the continuous phase [ 19 ]. There are two types of droplets generated by the co-flow device, dripping and jetting, and factors such as capillary number, Weber number, viscosity ratio, and density ratio also have an effect on the droplet diameter [ 20 ]. Of course we cannot ignore the influence of geometry on the droplet generation process, such as the inner diameter of the dispersed phase nozzle, the distance between the tip of the nozzle and the continuous phase passageway, the length of the platform, and the characteristics of the wall, all of which affect the droplet generation process [ 21 , 22 ]. Viscosity and interfacial tension are also important influencing factors, but droplet size strongly depends on the flow rate. Droplet size in microfluidic systems can be controlled by adjusting the input flow rate in such a way that it can also influence the frequency, size and shape of the droplets [ 23 , 24 ]. Differential pressure in microfluidic systems should also be of interest, and a device for measuring differential pressure was designed, the Capillary Laplace Gauge (CLG), a method that not only monitors pressure fluctuation trends in single droplets and large droplet groups in the dispersed phase, but also clearly identifies the transition between dripping and jetting [ 25 ]. In recent years, the preparation of double emulsion drops is also very important, and has great application value in food processing, medicine, chemistry, energy and other fields [ 26 – 28 ]. The active droplet generation is to control the droplet generation through electric field, magnetic field, light field, sound field, etc [ 29 – 31 ]. The authors of the above-mentioned literature have made great contributions to the field of microfluidics, and together they have promoted the development of the field of microfluidics. Unlike the above, in this paper, we mainly design a two-dimensional axisymmetric model, and use this model to numerically simulate the droplet generation to observe the droplet generation process more realistically, and to clearly see the more realistic state of the droplet at different times. In this paper, two flow types, dripping and jetting, are simulated, and the effects of two-phase flow rate ratio, continuous phase viscosity, contact angle and interfacial tension on droplet volume and generation frequency are investigated. The variation of pressure on the centerline of the model is investigated by varying the ratio of two-phase flow velocity, continuous-phase viscosity, contact angle, and interfacial tension. 2 Numercial methods This research is based on the software COMSOL Multiphysics 5.6 and the level-set method,neglecting fluid gravity on the microscopic scale, the key governing equations include the incompressible Navier–Stokes equation, the continuity equation and the level set equation. The following equations are included in the model. 2.1 Governing equations for the fluid flow (1) Incompressible Navier-Stokes equation. The incompressible Navier-Stokes equation (including interfacial tension) describes the mass and momentum transfer. As long as the fluid velocity is lower than the speed of sound, two-phase flow can be regarded as an incompressible fluid. In the above equation, ρ represents density (kg/m³), µ represents dynamic viscosity (N·s/m 2 ), u represents speed (m/s), p prepresents pressure (Pa), F st represents interfacial tension. (2) Continuity equation. (3) Level-set equation. The laminar two-phase flow, level-set interface uses the reinitialized traditional level set method to describe the fluid interface and its convection. The level set function φ is 0 in the continuous phase fluid and 1 in the dispersed phase fluid. In the transition layer near the interface, φ smoothly transitions from 0 to 1. The interface moves at fluid velocity u . ε is proportional to the thickness of the transition layer. It is proportional to the thickness of the transition layer. γ is a parameter, which determines the amount of reinitialization, and the appropriate γ value is the maximum amplitude that appears in the velocity field. In addition to defining the fluid interface, the level set function is also used to smooth the sudden changes in density and viscosity on the interface through the following definitions. In the above formula, ρ 1 represents the continuous phase density, ρ 2 represents the continuous phase density, µ 1 represents the continuous phase viscosity, and µ 2 represents the dispersed phase viscosity. The solution uses the COMSOL Multiphysics 5.6 two-phase flow module. In this work, the key output is the droplet volume,When the droplet is generated 0.7mm from the origin, the volume of the droplet is tracked, and the volume of the dispersed phase is calculated using an integrated operator. 2.2 Simulation setup In this research, we designed a Co-flow focusing device to simulate the generation of droplets. The device is a two-dimensional axisymmetric structure, as shown in Fig. 1 . We meshed it, and to ensure the accuracy of numerical simulationthe model divided a total of 41216 elements. Regarding the geometric model, we designed a two-dimensional model, and then rotated the rectangle on the left side of the two-dimensional model by 360° along the sideline. In order to facilitate the observation of droplet generation, we rotated the rectangle on the right side by 180° along the sideline. As shown in Fig. 2 . The geometric parameters of the model are shown in the figure. The inlet and outlet of the model are circular. The green arrow on the left is the inlet of the dispersed phase, the red arrow on the left is the continuous phase inlet, and the orange arrow on the right is the outlet. The geometric size of the model L 1 is 2000 µm, L 2 is 1900 µm, R 1 is 50 µm, and R 2 is 300 µm. Use the laminar two-phase flow level-set module in the COMSOL Multiphysics software to perform numerical simulation and analysis of the two-phase flow in the microchannel. To ensure that the two-phase fluid flows stably from the inlet in laminar flow, a smooth rectangular pulse is inserted into the numerical model function rect(t). The boundary conditions of the model are set as follows. Inlet 1: laminar flow inflow, velocity is 0.07m/s. Inlet 2: laminar flow inflow, velocity is 0.05m/s. Outlet: pressure outlet, P= 0 Pa. Wall surface: all wall surfaces are set as wetting walls. Contact angle θ w is set to 135°. Slip length is set to 5μm. We take a 2 wt% poly (vinyl alcohol) aqueous as the continuous phase and 1,6-hexanediol diacrylate as the dispersed phase. The properties of the two phases set in the simulation are listed in Table 1 . Fluid 1 is set as the continuous phase,and Fluid 2 is set as the dispersed phase in this numerical simulation. Table 1 Physical properties of continuous and dispersed phases The physical parameters Fluid 1 (water) Fluid 2 (oil) Dynamic viscosity (Pa·s) 1.95×10 − 3 6.71×10 − 3 Density(kg/m³) 1×10 3 1×10 3 Interfacial tension(N/m) 5×10 − 3 5×10 − 3 3 Results and discussion 3.1 Two different flow patterns This order value simulation adopts a two-position axisymmetric model. The continuous phase and the dispersed phase enter from the pipe on the right side respectively, where the dispersed phase enters in the middle yellow pipe, and the yellow pipe surrounds the continuous phase inlet. We observed the droplet generation mechanism through simulation. When the relative velocity of the dispersed phase is low, the dispersed phase droplets will form droplets at the mouth of the pipe, and the generation speed will be slow, showing a dripping shape. When the velocity of the dispersed phase becomes larger, the outlet of the dispersed phase will no longer produce droplets, but a jet will form. The jet will become unstable after a certain distance from the pipe mouth, and will eventually produce jetting. If you continue to increase the velocity of the dispersed phase, the distance of the jet will get farther and farther, and finally it will become a parallel fiow in the calculation area. 3.2 Effect of the flow rate ratio on the volume of the droplet This work studies the effect of the flow rate ratio on the droplet size and generation frequency. The ratio of the two-phase flow rate is the flow rate of the dispersed phase (Qd) divided by the flow rate of the continuous phase (Qc), which is Qd/Qc. Study the effect of the ratio of the two-phase flow rate on the formation of droplets. The flow velocity of the dispersed phase (Qd) remains unchanged at 0.07m/s, only the flow velocity of the continuous phase is changed, and the flow velocity (Qc) of the continuous phase is taken as 0.05m/s, 0.07m/s, 0.09m/s, 0.11m/s, 0.13m/s. That is, the ratio of the two-phase flow rate Qd/Qc is 1.4, 1, 0.778, 0.636, 0.538. Other parameters are fixed, the continuous phase viscosity is 1.95×10 − 3 Pa·s, the dispersed phase viscosity is 6.71×10 − 3 Pa·s, the interfacial tension between the two phases is 5×10 − 3 N/m, and the contact angle is 135°. As shown in Fig. 4 , the time from 0.002s to 0.012s, the state of droplet generation at different flow rate ratios. From the numerical simulation results, it can be found that when the flow velocity of the dispersed phase is fixed, that is, the flow velocity of the continuous phase increases from 0.05 m/s to 0.013 m/s, the volume of the droplets decreases and the frequency of droplets generation becomes larger. Due to the increase of the continuous phase velocity, the droplets are broken prematurely in the process of droplet generation, and droplets of smaller volume are generated. Because the droplets are broken prematurely, the frequency of droplets generation becomes higher. It can be seen from the Fig. 5 that when the flow ratio increases from 0.538 to 1.4, the volume of the droplet increases from 1.60×10 − 4 µL to 5.73×10 − 4 µL, an increase of 258.13%, and the frequency decreases from 322.58Hz to 125.79Hz. When the time is fixed at 0.02s, the pressure on the centerline of the model is as shown in the Fig. 6 . When a droplet is generated, as the droplet is farther from the inlet of the dispersed phase, the pressure on the centerline of the droplet decreases. As the continuous phase velocity increases, the pressure on the centerline of the droplets increases. Because the interfacial tension, the viscosity of the continuous phase, the viscosity of the dispersed phase and the flow rate are fixed, reducing the flow rate ratio means increasing the flow rate of the continuous phase. Numerical simulation results show that when the flow rate ratio decreases, the volume of micro-droplets decreases and the frequency of micro-droplet generation increases. 3.3 Effect of the continuous phase viscosity on the volume of the droplet The purpose of this section is to study the effect of continuous phase viscosity on the formation of micro-droplets. The viscosity of the continuous phase was set to 0.5×10 − 3 Pa·s, 1.0×10 − 3 Pa·s, 1.95×10 − 3 Pa·s, 3.0×10 − 3 Pa·s, and 4.0×10 − 3 Pa·s, respectively. The flow velocity of the dispersed phase is 0.07m/s, the flow velocity of the continuous phase is 0.05m/s, the interfacial tension is 5×10 − 3 N/m, and the contact angle is 135°. As shown in Fig. 7 , the time from 0.006s to 0.036s,the droplet generation states at different continuous phase viscosities. From the numerical simulation results, it can be found that when the viscosity of the continuous phase increases from 0.5×10 − 3 Pa·s to 4.0×10 − 3 Pa·s, the volume of the droplets decreases and the frequency of droplets generation increases. This means that as the viscous force of the continuous phase received on the dispersed phase increases, the generation process of micro-droplets is shortened, leading to premature breakage of the droplets in the process of droplet generation, and the generation of smaller droplets, because the droplet breaks prematurely, the generation frequency of droplets becomes higher. It can be seen from the Fig. 8 that when the viscosity of the continuous phase increases from 0.5×10 − 3 Pa·s to 4.0×10 − 3 Pa·s, the volume of the droplet decreases from 1.96×10 − 3 µL to 3.28×10 − 4 µL, It is decreased by 83.27%, and the frequency is increased from 35.78 Hz to 178.57 Hz. When the time is fixed at 0.05s, the pressure on the centerline of the model is shown in Fig. 9 . After the droplet is generated, as the droplet is farther from the inlet of the dispersed phase, the pressure on the centerline of the droplet is smaller. When the flow rates of the dispersed and continuous phases remain constant, and the interfacial tension remains constant, the shortening of the droplet generation time means that smaller micro-droplets are produced. A larger continuous phase viscosity (corresponding to a larger drag force) will force the micro-droplets to break up earlier. Numerical simulation results show that when the viscosity of the continuous phase increases, the volume of micro-droplets decreases and the frequency of micro-droplets increases when the time is 0.05s. 3.4 Effect of the contact angle on the volume of the droplet This section investigates the influence of channel wall wettability in microchannels on the size of micro-droplets. The wettability of the channel wall can be studied by changing the contact angle. In order to study the influence of wettability on the formation of micro-droplets, the formation process of micro-droplets was simulated under five different contact angles of 75˚, 90˚, 105˚, 120˚ and 135˚. The other parameters are the same as in the previous section. The flow velocity of the dispersed phase is 0.07 m/s, and the flow velocity of the continuous phase is 0.05 m/s. The continuous phase viscosity is 1.95×10 − 3 Pa·s, the dispersed phase viscosity is 6.71×10 − 3 Pa·s, and the interfacial tension is 5×10 − 3 N/m. As shown in Fig. 10 , the time from 0.003s to 0.018s,the droplet generation states at different contact angles. From the numerical simulation results, it can be found that when the contact angle increases from 75° to 135°, the volume of the droplet becomes larger and the frequency of droplet generation becomes smaller. This means that the breaking time of the droplet is longer, and the droplet volume increases. It can be seen from the Fig. 11 that when the contact angle increases from 75° to 135°, the volume of the droplet increases from 5.32×10 − 4 µL to 5.76×10 − 4 µL, an increase of 8.27%, and the frequency decreases from 124.46Hz to 123.15Hz. When the time is fixed at 0.02s, the pressure on the centerline of the model is shown in Fig. 12 . The change in contact angle has a very small effect on the pressure on the centerline. Under the condition that the flow rates of the dispersed phase and the continuous phase remain constant, the viscosity of the continuous phase remains constant, and the interfacial tension between the two phases remains constant. As the contact angle increases, the internal wall of the channel where the continuous phase of this model is located decreases the resistance of the continuous phase, which causes the continuous phase liquid to have a high velocity near the wall. Relatively speaking, the velocity at the center of the pipe decreases, the breakage time of the droplets increases, and the volume of the generated droplets becomes larger. The numerical simulation results show that the change in the volume of microdroplets is not obvious enough when the contact angle increases and the frequency of microdroplet generation decreases. Controlling the volume of droplets by changing the contact angle is not an effective method. 3.5 Effect of the Two-phase interfacial tension on the volume of the droplet In this section, the study analyzes the influence of the interfacial tension in the microchannel on the size of the micro-droplets. Keeps velocity and viscosity constant, the flow velocity of the dispersed phase is 0.07 m/s, and the flow velocity of the continuous phase is 0.05 m/s. The continuous phase viscosity is 1.95×10 − 3 Pa·s, the dispersed phase viscosity is 6.71×10 − 3 Pa·s, and the contact angle is 135°. For the numerical model established in this section, five different interfacial tensions were used for numerical simulation. The interfacial tensions were 5.0×10 − 3 N/m, 5.2×10 − 3 N/m, 5.4×10 − 3 N/m, 5.6 ×10 − 3 N/m, 5.8×10 − 3 N/m. As shown in Fig. 13 , the time from 0.003s to 0.018s,the droplet generation states at different interfacial tensions. From the numerical simulation results, it can be found that when the interfacial tension increases from 5.0×10 − 3 N/m to 5.8×10 − 3 N/m, the droplet volume becomes larger and the droplet generation frequency becomes smaller. This means that the breakup time of the droplet becomes longer, and the volume of the droplet increases. It can be seen from the Fig. 14 that when the interfacial tension increases from 5×10 − 3 N/m to 5.8×10 − 3 N/m, the volume of the droplet increases from 5.77×10 − 4 µL to 6.62×10 − 4 µL, an increase of 14.73%, The frequency is reduced from 123Hz to 109.29Hz. When the time is fixed at 0.02s, the pressure on the centerline of the model is as shown in the Fig. 15 . When a droplet is generated, as the droplet is farther from the inlet of the dispersed phase, the pressure on the centerline of the droplet decreases. With the increase of the interfacial tension, the Young Laplace pressure inside the droplet becomes larger, the micro-droplets are difficult to break from the dispersed phase, the breakage time of the droplet becomes longer, and the volume of droplets increases. The generation time of micro-droplets increases, and the frequency of generating droplets is smaller. Numerical simulation results show that when the interfacial tension, the volume of micro-droplets increases and the frequency of micro-droplet generation decreases. 4 Conclusions In this article, we designed a two-dimensional axisymmetric structure to numerically simulation of the droplet formation in a Co-flow microchannel capillary device. 1. By changing the velocity of the dispersed phase, we observed tow flow patterns. When the relative velocity of the dispersed phase is low, the dispersed phase droplets will form droplets at the pipe mouth, and the generation speed will be slow, showing a droplet shape（dripping）. When the velocity of the dispersed phase increases, the outlet of the dispersed phase will no longer produce droplets, but a jet will be formed. The jet will become unstable after a certain distance from the pipe mouth, and finally droplets will be produced（jetting）. If you continue to increase the velocity of the dispersed phase, the distance of the jet will become farther and farther, and finally it will become a parallel jet in the calculation area（parallel flow）. 2. When the flow rate ratio increases from 0.538 to 1.4, the volume of the droplet increases from 1.60×10 -4 μL to 5.73×10 -4 μL, an increase of 258.13%, and the frequency decreases from 322.58Hz to 125.79Hz. 3. When the viscosity of the continuous phase increases from 0.5×10 -3 Pa·s to 4.0×10 -3 Pa·s, the volume of the droplet decreases from 1.96×10 -3 μL to 3.28×10 -4 μL, a decrease of 83.27% , the frequency is increased from 35.78Hz to 178.57Hz. 4. When the contact angle increases from 75° to 135°, the volume of the droplet increases from 5.32×10 -4 μL to 5.76×10 -4 μL, an increase of 8.27%, and the frequency decreases from 124.46Hz to 123.15Hz. Controlling the volume of droplets by changing the contact angle is not an effective method. 5. When the two-phase interfacial tension increases from 5×10 -3 N/m to 5.8×10 -3 N/m, the volume of the droplet increases from 5.77×10 -4 μL to 6.62×10 -4 μL, an increase of 14.73%, and the frequency decreases from 123Hz to 109.29Hz. 6. As the droplet is farther away from the inlet of the dispersed phase, the pressure of the droplet on the centerline gradually decreases. As the velocity of the continuous phase increases, the pressure of the droplets on the centerline increases. As the viscosity of the continuous phase increases, the pressure on the centerline of the droplets increases. As the interfacial tension increases, the pressure of the droplet on the centerline increases. The change of contact angle has little effect on the pressure at the centerline. Declarations Acknowledgements This work was supported by Yantai Science and Technology Innovation Development Plan Key Basic Research Projects(2023JCYJ048), Special Supporting Funds for Leading Talents above Provincial Level in Yantai(220-20230002),Young Taishan Scholars Program of Shandong Province of China(tsqn202103091), Ludong University introduced talents and started funding project(LD22065). References Chen Y, Chen X (2019) An improved design for passive micromixer based on topology optimization method. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3676725","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":265205812,"identity":"a8328b6c-93c2-4846-a42a-9450b0dd2083","order_by":0,"name":"Peihua Zhang","email":"","orcid":"","institution":"Ludong University","correspondingAuthor":false,"prefix":"","firstName":"Peihua","middleName":"","lastName":"Zhang","suffix":""},{"id":265205814,"identity":"ddbfce30-e2e5-41d0-bccc-689ce711ab0d","order_by":1,"name":"Yongbiao Ma","email":"","orcid":"","institution":"Ludong University","correspondingAuthor":false,"prefix":"","firstName":"Yongbiao","middleName":"","lastName":"Ma","suffix":""},{"id":265205816,"identity":"70b84564-afb2-4c32-89ed-62eae82f3983","order_by":2,"name":"Bao Song","email":"","orcid":"","institution":"Ludong University","correspondingAuthor":false,"prefix":"","firstName":"Bao","middleName":"","lastName":"Song","suffix":""},{"id":265205817,"identity":"0009336d-457d-461c-b3d9-9e23a46c4452","order_by":3,"name":"Dengke Chen","email":"","orcid":"","institution":"Ludong University","correspondingAuthor":false,"prefix":"","firstName":"Dengke","middleName":"","lastName":"Chen","suffix":""},{"id":265205818,"identity":"a2732d96-8f86-44db-8e20-0463c12dd45f","order_by":4,"name":"Dengying Zhang","email":"","orcid":"","institution":"Ludong University","correspondingAuthor":false,"prefix":"","firstName":"Dengying","middleName":"","lastName":"Zhang","suffix":""},{"id":265205819,"identity":"647cae6e-7d24-4e6b-90ff-7a8a4ef0052e","order_by":5,"name":"Jinliang Yuan","email":"","orcid":"","institution":"Ludong University","correspondingAuthor":false,"prefix":"","firstName":"Jinliang","middleName":"","lastName":"Yuan","suffix":""},{"id":265205820,"identity":"6db679bd-0c5b-4392-9bdf-04692e75f451","order_by":6,"name":"Meichun Wang","email":"","orcid":"","institution":"Weifang University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Meichun","middleName":"","lastName":"Wang","suffix":""},{"id":265205822,"identity":"b465574f-372e-4cba-949a-8dbad0f0bce1","order_by":7,"name":"Xueye Chen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVRIiWNgGAWjYBCDBAYG5gMHPlSQpoUt8eCMM6Rp4TE+zNtChFKDG8nHJD4w2OXxS/d8OMDbwCDPL3YAvxbJGWlpkjMYkosl55zdcEByB4PhzNkJ+LXwS+SYSfMwHEjccCN3wwHDMwwJBrcJaGGTyP8m/QeoZf+NnAcHEtuI0AK0hU2aAWSLRA7DgYPEaJHseWZs2cOQnDjjRprBwYYzEoT9YnA8+eGNHwx2if0zkh9//lNhI88vTUALELBIMP6DcyQIKgcB5g9EKRsFo2AUjIKRCwAZrkd8V4bEHgAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-2377-352X","institution":"Ludong University","correspondingAuthor":true,"prefix":"","firstName":"Xueye","middleName":"","lastName":"Chen","suffix":""}],"badges":[],"createdAt":"2023-11-28 13:21:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3676725/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3676725/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":49312542,"identity":"51e56cc0-abfb-451b-bd1d-a60e05eee9f8","added_by":"auto","created_at":"2024-01-08 14:04:44","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":54715,"visible":true,"origin":"","legend":"\u003cp\u003eMesh element.\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/0538f598b3980ebd78ffdeba.jpeg"},{"id":49312002,"identity":"b8ce146b-e61b-4343-bbb8-ee5d4b31fe6a","added_by":"auto","created_at":"2024-01-08 13:56:44","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":34578,"visible":true,"origin":"","legend":"\u003cp\u003eThe geometry and dimensions of the microfluidic Co-flow capillary device used in the simulation.\u003c/p\u003e","description":"","filename":"floatimage2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/848a538168fb7f829e0215d0.jpeg"},{"id":49311745,"identity":"0db63586-f5b8-455e-b2c3-1c05084298a9","added_by":"auto","created_at":"2024-01-08 13:48:44","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":113171,"visible":true,"origin":"","legend":"\u003cp\u003eThe simulation of droplet generation process by different flow patterns.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/5d7773591959d3ed45eaf427.png"},{"id":49311749,"identity":"d22ef789-baa0-48fb-98f6-b60a5be577ed","added_by":"auto","created_at":"2024-01-08 13:48:44","extension":"jpeg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":135617,"visible":true,"origin":"","legend":"\u003cp\u003eThe simulations of the droplet generation process by five different flow rate ratios.\u003c/p\u003e","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/4625848dcde570832b04f5e3.jpeg"},{"id":49312000,"identity":"d48f78b9-f1df-4698-bc91-f6269cd03c06","added_by":"auto","created_at":"2024-01-08 13:56:44","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":72601,"visible":true,"origin":"","legend":"\u003cp\u003eVolume of droplets and generation frequency versus the flow rate ratio.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/595796d582ccf87dcf99161f.png"},{"id":49311746,"identity":"908ad9a3-1988-4a40-841a-2961badc4e25","added_by":"auto","created_at":"2024-01-08 13:48:44","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":91733,"visible":true,"origin":"","legend":"\u003cp\u003eThe pressure on the centerline of five different flow rate ratios when the time is 0.02s.\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/5f7d9e835c062c7215db256e.png"},{"id":49311754,"identity":"bb9e7e6e-f232-4e8d-a0a7-d263d403e7ad","added_by":"auto","created_at":"2024-01-08 13:48:44","extension":"jpeg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":142427,"visible":true,"origin":"","legend":"\u003cp\u003eThe simulations of the droplet generation process by five different continuous phase viscosities.\u003c/p\u003e","description":"","filename":"floatimage7.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/5e2ed11058e72c0254721f37.jpeg"},{"id":49312543,"identity":"5921ee52-9d43-41c7-aad1-b14e54da49fc","added_by":"auto","created_at":"2024-01-08 14:04:44","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":77541,"visible":true,"origin":"","legend":"\u003cp\u003eVolume of droplets and generation frequency versus the viscosity of the continuous phase.\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/97e7b708e8b77a01f37595f8.png"},{"id":49312001,"identity":"f7d08940-98b8-48c3-a9d9-10effa119361","added_by":"auto","created_at":"2024-01-08 13:56:44","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":82808,"visible":true,"origin":"","legend":"\u003cp\u003eThe pressure on the centerline of five different five different continuous phase viscosities when the time is 0.05s.\u003c/p\u003e","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/0cb6b28c791f411699f18dec.png"},{"id":49311751,"identity":"b406f57b-d787-4585-be32-a13b54f7641f","added_by":"auto","created_at":"2024-01-08 13:48:44","extension":"jpeg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":130684,"visible":true,"origin":"","legend":"\u003cp\u003eThe simulations of the droplet generation process by five different contact angles.\u003c/p\u003e","description":"","filename":"floatimage10.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/6e5b6b28bae0801d3421e3cc.jpeg"},{"id":49311755,"identity":"dcc84194-44aa-4c46-a9f8-aea90e2defea","added_by":"auto","created_at":"2024-01-08 13:48:44","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":71117,"visible":true,"origin":"","legend":"\u003cp\u003eVolume of droplets and generation frequency versus the contact angle.\u003c/p\u003e","description":"","filename":"floatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/5a5cfb609cfde5debee64a89.png"},{"id":49311756,"identity":"acea122a-d00f-480d-8f13-da5a8fc822f3","added_by":"auto","created_at":"2024-01-08 13:48:44","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":73512,"visible":true,"origin":"","legend":"\u003cp\u003eThe pressure on the centerline of five different contact angles when the time is 0.02s.\u003c/p\u003e","description":"","filename":"floatimage12.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/7ce7d8d18dccf587a58c61ba.png"},{"id":49312544,"identity":"c0d13ef1-29c5-4202-bffc-5856ef31cfc0","added_by":"auto","created_at":"2024-01-08 14:04:44","extension":"jpeg","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":135101,"visible":true,"origin":"","legend":"\u003cp\u003eThe simulations of the droplet generation process by five different interfacial tensions.\u003c/p\u003e","description":"","filename":"floatimage13.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/80134d5bfb61c567de29169c.jpeg"},{"id":49311757,"identity":"654ce467-ec81-484a-b8ea-df54d826fc8c","added_by":"auto","created_at":"2024-01-08 13:48:44","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":82025,"visible":true,"origin":"","legend":"\u003cp\u003eVolume of droplets and generation frequency versus the interfacial tension.\u003c/p\u003e","description":"","filename":"floatimage14.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/6e42e7046004f7756a0287b4.png"},{"id":49312004,"identity":"7b5f6d4a-9350-448d-a84d-f030edfd52c6","added_by":"auto","created_at":"2024-01-08 13:56:44","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":81872,"visible":true,"origin":"","legend":"\u003cp\u003eThe pressure on the centerline of five different interfacial tensions when the time is 0.02s.\u003c/p\u003e","description":"","filename":"floatimage15.png","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/764c84174fc6d14e77f84375.png"},{"id":50857278,"identity":"8570d879-e680-4fe0-93c2-c29ce327556b","added_by":"auto","created_at":"2024-02-08 12:30:27","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2116943,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3676725/v1/aa77c78e-5ebc-4cf0-8c11-7c8e0bd8bf06.pdf"}],"financialInterests":"","formattedTitle":"Numerical simulation of droplet formation in a Co-flow microchannel capillary device","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eFluid is an important form of matter, and fluid flow is one of the most basic phenomena in nature. Microfluidic technology, also known as lab-on-a-chip technology, is a science and technology with the main feature of manipulating fluids in micron-scale space, minimizing the basic functions of biological and chemical laboratories onto a chip of a few square centimeters. The microfluidic device has the advantages of small sample volume, high sensitivity, good portability, short analysis time, and low power consumption. Microfluidic devices are also widely used in analytical chemistry, basic biological research, and material synthesis [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. However, it is difficult to prepare droplets of a few microns in size. The generation of droplets originates from the instability of the fluid [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Droplets can be divided into water-in-oil (W/O), oil-in-water (O/W), water-in-oil-in-water (W/O/W), and oil-in-water-in-oil (O/W/O) [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Most droplets are generated by passive methods. The common passive method is the T-junction [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], flow-focusing method(MFFD), Co-flow focusing method. Among them, the T-junction method is the most common. It is the continuous phase under pressure to push the dispersed phase to break to produce droplets [\u003cspan additionalcitationids=\"CR8 CR9 CR10\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. In MFFD, the continuous phase flows from both sides of the dispersed phase, which has the effect of interleaving the dispersed phase. The dispersed phase is subjected to the symmetrical shear force from the continuous phases on both sides and breaks into droplets [\u003cspan additionalcitationids=\"CR13\" citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. Compared with the T-junction, the droplet generation process is more stable, the generated droplet size is controllable, and smaller droplets can be generated. Similar to MFFD, the Co-flow focusing method can produce smaller droplets [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] and the size of the generated droplets has a great relationship with the ratio of two-phase flow rate, viscosity, interfacial tension, wall effect and capillary surface [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. The change in the concentration of the dispersed phase will determine whether or not satellite droplets are generated during the droplet generation process [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. The microbubbles are generated by the Co-flow focusing device. The pressure of the gas, the flow rate of the liquid, and the width of the dispersed phase channel will all affect the volume of the bubble and the frequency of generation [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. Using opeofoam's C\u0026thinsp;+\u0026thinsp;+\u0026thinsp;program library to study droplet generation is also a very good approach. The droplet formation process is discussed by varying the viscosity of the dispersed phase and the velocity of the continuous phase [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. There are two types of droplets generated by the co-flow device, dripping and jetting, and factors such as capillary number, Weber number, viscosity ratio, and density ratio also have an effect on the droplet diameter [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. Of course we cannot ignore the influence of geometry on the droplet generation process, such as the inner diameter of the dispersed phase nozzle, the distance between the tip of the nozzle and the continuous phase passageway, the length of the platform, and the characteristics of the wall, all of which affect the droplet generation process [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. Viscosity and interfacial tension are also important influencing factors, but droplet size strongly depends on the flow rate. Droplet size in microfluidic systems can be controlled by adjusting the input flow rate in such a way that it can also influence the frequency, size and shape of the droplets [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. Differential pressure in microfluidic systems should also be of interest, and a device for measuring differential pressure was designed, the Capillary Laplace Gauge (CLG), a method that not only monitors pressure fluctuation trends in single droplets and large droplet groups in the dispersed phase, but also clearly identifies the transition between dripping and jetting [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. In recent years, the preparation of double emulsion drops is also very important, and has great application value in food processing, medicine, chemistry, energy and other fields [\u003cspan additionalcitationids=\"CR27\" citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]. The active droplet generation is to control the droplet generation through electric field, magnetic field, light field, sound field, etc [\u003cspan additionalcitationids=\"CR30\" citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe authors of the above-mentioned literature have made great contributions to the field of microfluidics, and together they have promoted the development of the field of microfluidics. Unlike the above, in this paper, we mainly design a two-dimensional axisymmetric model, and use this model to numerically simulate the droplet generation to observe the droplet generation process more realistically, and to clearly see the more realistic state of the droplet at different times. In this paper, two flow types, dripping and jetting, are simulated, and the effects of two-phase flow rate ratio, continuous phase viscosity, contact angle and interfacial tension on droplet volume and generation frequency are investigated. The variation of pressure on the centerline of the model is investigated by varying the ratio of two-phase flow velocity, continuous-phase viscosity, contact angle, and interfacial tension.\u003c/p\u003e"},{"header":"2 Numercial methods","content":"\u003cp\u003eThis research is based on the software \u003cem\u003eCOMSOL Multiphysics 5.6\u003c/em\u003e and the level-set method,neglecting fluid gravity on the microscopic scale, the key governing equations include the incompressible Navier\u0026ndash;Stokes equation, the continuity equation and the level set equation. The following equations are included in the model.\u003c/p\u003e\n\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n \u003ch2\u003e2.1 Governing equations for the fluid flow\u003c/h2\u003e\n \u003cp\u003e(1) Incompressible Navier-Stokes equation.\u003c/p\u003e\n \u003cp\u003eThe incompressible Navier-Stokes equation (including interfacial tension) describes the mass and momentum transfer. As long as the fluid velocity is lower than the speed of sound, two-phase flow can be regarded as an incompressible fluid.\u003c/p\u003e\n \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"EquationNumber\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"430\" height=\"44\"\u003e\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eIn the above equation, \u003cem\u003e\u0026rho;\u003c/em\u003e represents density (kg/m\u0026sup3;), \u003cem\u003e\u0026micro;\u003c/em\u003e represents dynamic viscosity (N\u0026middot;s/m\u003csup\u003e2\u003c/sup\u003e), \u003cstrong\u003eu\u003c/strong\u003e represents speed (m/s), p prepresents pressure (Pa), \u003cstrong\u003eF\u003c/strong\u003e\u003csub\u003est\u003c/sub\u003e represents interfacial tension.\u003c/p\u003e\n \u003cp\u003e(2) Continuity equation.\u003c/p\u003e\n \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n \u003cdiv class=\"EquationNumber\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAS8AAAAaCAYAAADmHtpyAAADPklEQVR4Ae2azY3qMBRG6YEOQKIRkOiCHUh0wRaJDmBPAZTAng7Y0gAF+OlD7yLHuTHjGYaJpWMpSvBPfDmBo2vDIFAgAAEIVEhgUGHMhAwBCEAgIC8+BBCAQJUEkFeVj42gIQAB5MVnAAIQqJIA8qrysRE0BCCAvPgMQAACVRJAXlU+NoKGAAQa8rpcLmE4HIbBYJA9drtdr8gpHov5dDr1KjaCgQAEmgTG43G4Xq+PyvV6/fzuzufzZ8fj8RjUlisNeanjYrEIo9Eo3G631jjVzWYzt63V+UMVkpXFq/h0jcA+BJ9pIFBAQMJSknE+nx+jttttkKRUrC0WmNoluq7Skpe++JrAE4AynD5lXff7PUyn00asilt1aqNAAAL9ISARmbgUlYnLIpSs5J64qK4rA2v2DOHxpdeXPxVAH7MuLXMnk0nQ2YpXZ22cIQCBvyEgCcVZlReFZJbKS/1S6dnYlrzUoOxKe1+xFEqzLvVXIDoUlGSoa1viWQA/OSvLSu/H0vEnRBkLgd8hIAGlmVY6k9o9wSnz8rIvV14mAO1/qWgJtlwui/a67B4SliSj45W8NJ/6dB0Wj71p5GUkOEOgvwRsPyteMnrRSly2kR+3S2re3pcrLw2Msy9JQq9LynfkVXJ/9fXkpWxRWaPaKBCAwN8TkLSUkHhisugkKC0tvaI2jU9Lu+Z/D5PAZrPJZl0mqTQrsnpNKpHo0HW6zEsDKnnt7W95dSX3pC8EIPBeAq/kJal5y0KLolheGmjLuFRMdtPc+TvysvkkOe9I4/B+bVSGmP7YkIuTNghA4HcJvFo2ekvCOKLiZaMGK1tKN+7jm+auTSyWedkG/jszL4vRZGXCZMmYezK0QeDzBLo27OWHuHhZWNGGvd1MAjocDvay+Lzf7x/ykwBXq9UzmyrdP3s1sYnRRPmqP+0QgMBnCaR/lbBszFthqS0uEp+32d/UXjyCawhAAAJvJNAlodwUWjJ6f5/QGOSVI0cbBCDwVgLKtCSkrxRla7n9MOT1FYr0gQAE3kag6/9c8QQSXO4XSPVFXjExriEAgWoIIK9qHhWBQgACMQHkFdPgGgIQqIYA8qrmUREoBCAQE/gHsbK4lpdonwkAAAAASUVORK5CYII=\" width=\"303\" height=\"26\"\u003e\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e(3) Level-set equation.\u003c/p\u003e\n \u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n \u003cdiv class=\"EquationNumber\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"427\" height=\"60\"\u003e\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eThe laminar two-phase flow, level-set interface uses the reinitialized traditional level set method to describe the fluid interface and its convection. The level set function \u003cem\u003e\u0026phi;\u003c/em\u003e is 0 in the continuous phase fluid and 1 in the dispersed phase fluid. In the transition layer near the interface, \u003cem\u003e\u0026phi;\u003c/em\u003e smoothly transitions from 0 to 1. The interface moves at fluid velocity \u003cstrong\u003eu\u003c/strong\u003e. \u0026epsilon; is proportional to the thickness of the transition layer. It is proportional to the thickness of the transition layer. \u0026gamma; is a parameter, which determines the amount of reinitialization, and the appropriate \u0026gamma; value is the maximum amplitude that appears in the velocity field. In addition to defining the fluid interface, the level set function is also used to smooth the sudden changes in density and viscosity on the interface through the following definitions.\u003c/p\u003e\n \u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n \u003cdiv class=\"EquationNumber\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"350\" height=\"70\"\u003e\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cp\u003eIn the above formula, \u003cem\u003e\u0026rho;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e represents the continuous phase density, \u003cem\u003e\u0026rho;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e represents the continuous phase density, \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e represents the continuous phase viscosity, and \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e represents the dispersed phase viscosity. The solution uses the \u003cem\u003eCOMSOL Multiphysics 5.6\u003c/em\u003e two-phase flow module. In this work, the key output is the droplet volume,When the droplet is generated 0.7mm from the origin, the volume of the droplet is tracked, and the volume of the dispersed phase is calculated using an integrated operator.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n \u003ch2\u003e2.2 Simulation setup\u003c/h2\u003e\n \u003cp\u003eIn this research, we designed a Co-flow focusing device to simulate the generation of droplets. The device is a two-dimensional axisymmetric structure, as shown in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. We meshed it, and to ensure the accuracy of numerical simulationthe model divided a total of 41216 elements.\u003c/p\u003e\n \u003cp\u003eRegarding the geometric model, we designed a two-dimensional model, and then rotated the rectangle on the left side of the two-dimensional model by 360\u0026deg; along the sideline. In order to facilitate the observation of droplet generation, we rotated the rectangle on the right side by 180\u0026deg; along the sideline. As shown in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. The geometric parameters of the model are shown in the figure. The inlet and outlet of the model are circular. The green arrow on the left is the inlet of the dispersed phase, the red arrow on the left is the continuous phase inlet, and the orange arrow on the right is the outlet. The geometric size of the model \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e is 2000 \u0026micro;m, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e is 1900 \u0026micro;m, \u003cem\u003eR\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e is 50 \u0026micro;m, and \u003cem\u003eR\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e is 300 \u0026micro;m.\u003c/p\u003e\n \u003cp\u003eUse the laminar two-phase flow level-set module in the \u003cem\u003eCOMSOL Multiphysics\u003c/em\u003e software to perform numerical simulation and analysis of the two-phase flow in the microchannel. To ensure that the two-phase fluid flows stably from the inlet in laminar flow, a smooth rectangular pulse is inserted into the numerical model function rect(t). The boundary conditions of the model are set as follows.\u003c/p\u003e\n \u003col class=\"decimal_type\"\u003e\n \u003cli\u003eInlet 1: laminar flow inflow, velocity is 0.07m/s.\u003c/li\u003e\n \u003cli\u003eInlet 2: laminar flow inflow, velocity is 0.05m/s.\u003c/li\u003e\n \u003cli\u003eOutlet: pressure outlet, P= 0 Pa.\u003c/li\u003e\n \u003cli\u003eWall surface: all wall surfaces are set as wetting walls.\u003c/li\u003e\n \u003cli\u003eContact angle \u003cem\u003e\u0026theta;\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e is set to 135\u0026deg;.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eSlip length is set to 5\u0026mu;m.\u003c/li\u003e\n \u003c/ol\u003e\n \u003cp\u003eWe take a 2 wt% poly (vinyl alcohol) aqueous as the continuous phase and 1,6-hexanediol diacrylate as the dispersed phase. The properties of the two phases set in the simulation are listed in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. Fluid 1 is set as the continuous phase,and Fluid 2 is set as the dispersed phase in this numerical simulation.\u003c/p\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003ePhysical properties of continuous and dispersed phases\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eThe physical parameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eFluid 1 (water)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eFluid 2 (oil)\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDynamic viscosity (Pa\u0026middot;s)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.95\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6.71\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDensity(kg/m\u0026sup3;)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u0026times;10\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u0026times;10\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eInterfacial tension(N/m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"},{"header":"3 Results and discussion","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Two different flow patterns\u003c/h2\u003e \u003cp\u003eThis order value simulation adopts a two-position axisymmetric model. The continuous phase and the dispersed phase enter from the pipe on the right side respectively, where the dispersed phase enters in the middle yellow pipe, and the yellow pipe surrounds the continuous phase inlet. We observed the droplet generation mechanism through simulation. When the relative velocity of the dispersed phase is low, the dispersed phase droplets will form droplets at the mouth of the pipe, and the generation speed will be slow, showing a dripping shape. When the velocity of the dispersed phase becomes larger, the outlet of the dispersed phase will no longer produce droplets, but a jet will form. The jet will become unstable after a certain distance from the pipe mouth, and will eventually produce jetting. If you continue to increase the velocity of the dispersed phase, the distance of the jet will get farther and farther, and finally it will become a parallel fiow in the calculation area.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Effect of the flow rate ratio on the volume of the droplet\u003c/h2\u003e \u003cp\u003eThis work studies the effect of the flow rate ratio on the droplet size and generation frequency. The ratio of the two-phase flow rate is the flow rate of the dispersed phase (Qd) divided by the flow rate of the continuous phase (Qc), which is Qd/Qc. Study the effect of the ratio of the two-phase flow rate on the formation of droplets. The flow velocity of the dispersed phase (Qd) remains unchanged at 0.07m/s, only the flow velocity of the continuous phase is changed, and the flow velocity (Qc) of the continuous phase is taken as 0.05m/s, 0.07m/s, 0.09m/s, 0.11m/s, 0.13m/s. That is, the ratio of the two-phase flow rate Qd/Qc is 1.4, 1, 0.778, 0.636, 0.538. Other parameters are fixed, the continuous phase viscosity is 1.95\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, the dispersed phase viscosity is 6.71\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, the interfacial tension between the two phases is 5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m, and the contact angle is 135\u0026deg;.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, the time from 0.002s to 0.012s, the state of droplet generation at different flow rate ratios. From the numerical simulation results, it can be found that when the flow velocity of the dispersed phase is fixed, that is, the flow velocity of the continuous phase increases from 0.05 m/s to 0.013 m/s, the volume of the droplets decreases and the frequency of droplets generation becomes larger. Due to the increase of the continuous phase velocity, the droplets are broken prematurely in the process of droplet generation, and droplets of smaller volume are generated. Because the droplets are broken prematurely, the frequency of droplets generation becomes higher. It can be seen from the Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e that when the flow ratio increases from 0.538 to 1.4, the volume of the droplet increases from 1.60\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e\u0026micro;L to 5.73\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e\u0026micro;L, an increase of 258.13%, and the frequency decreases from 322.58Hz to 125.79Hz. When the time is fixed at 0.02s, the pressure on the centerline of the model is as shown in the Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. When a droplet is generated, as the droplet is farther from the inlet of the dispersed phase, the pressure on the centerline of the droplet decreases. As the continuous phase velocity increases, the pressure on the centerline of the droplets increases. Because the interfacial tension, the viscosity of the continuous phase, the viscosity of the dispersed phase and the flow rate are fixed, reducing the flow rate ratio means increasing the flow rate of the continuous phase. Numerical simulation results show that when the flow rate ratio decreases, the volume of micro-droplets decreases and the frequency of micro-droplet generation increases.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Effect of the continuous phase viscosity on the volume of the droplet\u003c/h2\u003e \u003cp\u003eThe purpose of this section is to study the effect of continuous phase viscosity on the formation of micro-droplets. The viscosity of the continuous phase was set to 0.5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, 1.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, 1.95\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, 3.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, and 4.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, respectively. The flow velocity of the dispersed phase is 0.07m/s, the flow velocity of the continuous phase is 0.05m/s, the interfacial tension is 5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m, and the contact angle is 135\u0026deg;.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, the time from 0.006s to 0.036s,the droplet generation states at different continuous phase viscosities. From the numerical simulation results, it can be found that when the viscosity of the continuous phase increases from 0.5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s to 4.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, the volume of the droplets decreases and the frequency of droplets generation increases. This means that as the viscous force of the continuous phase received on the dispersed phase increases, the generation process of micro-droplets is shortened, leading to premature breakage of the droplets in the process of droplet generation, and the generation of smaller droplets, because the droplet breaks prematurely, the generation frequency of droplets becomes higher. It can be seen from the Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e that when the viscosity of the continuous phase increases from 0.5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s to 4.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, the volume of the droplet decreases from 1.96\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u0026micro;L to 3.28\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e\u0026micro;L, It is decreased by 83.27%, and the frequency is increased from 35.78 Hz to 178.57 Hz. When the time is fixed at 0.05s, the pressure on the centerline of the model is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. After the droplet is generated, as the droplet is farther from the inlet of the dispersed phase, the pressure on the centerline of the droplet is smaller. When the flow rates of the dispersed and continuous phases remain constant, and the interfacial tension remains constant, the shortening of the droplet generation time means that smaller micro-droplets are produced. A larger continuous phase viscosity (corresponding to a larger drag force) will force the micro-droplets to break up earlier. Numerical simulation results show that when the viscosity of the continuous phase increases, the volume of micro-droplets decreases and the frequency of micro-droplets increases\u003c/p\u003e \u003cp\u003ewhen the time is 0.05s.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Effect of the contact angle on the volume of the droplet\u003c/h2\u003e \u003cp\u003eThis section investigates the influence of channel wall wettability in microchannels on the size of micro-droplets. The wettability of the channel wall can be studied by changing the contact angle. In order to study the influence of wettability on the formation of micro-droplets, the formation process of micro-droplets was simulated under five different contact angles of 75˚, 90˚, 105˚, 120˚ and 135˚. The other parameters are the same as in the previous section. The flow velocity of the dispersed phase is 0.07 m/s, and the flow velocity of the continuous phase is 0.05 m/s. The continuous phase viscosity is 1.95\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, the dispersed phase viscosity is 6.71\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, and the interfacial tension is 5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e, the time from 0.003s to 0.018s,the droplet generation states at different contact angles. From the numerical simulation results, it can be found that when the contact angle increases from 75\u0026deg; to 135\u0026deg;, the volume of the droplet becomes larger and the frequency of droplet generation becomes smaller. This means that the breaking time of the droplet is longer, and the droplet volume increases. It can be seen from the Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e that when the contact angle increases from 75\u0026deg; to 135\u0026deg;, the volume of the droplet increases from 5.32\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e\u0026micro;L to 5.76\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e\u0026micro;L, an increase of 8.27%, and the frequency decreases from 124.46Hz to 123.15Hz. When the time is fixed at 0.02s, the pressure on the centerline of the model is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e. The change in contact angle has a very small effect on the pressure on the centerline. Under the condition that the flow rates of the dispersed phase and the continuous phase remain constant, the viscosity of the continuous phase remains constant, and the interfacial tension between the two phases remains constant. As the contact angle increases, the internal wall of the channel where the continuous phase of this model is located decreases the resistance of the continuous phase, which causes the continuous phase liquid to have a high velocity near the wall. Relatively speaking, the velocity at the center of the pipe decreases, the breakage time of the droplets increases, and the volume of the generated droplets becomes larger. The numerical simulation results show that the change in the volume of microdroplets is not obvious enough when the contact angle increases and the frequency of microdroplet generation decreases. Controlling the volume of droplets by changing the contact angle is not an effective method.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Effect of the Two-phase interfacial tension on the volume of the droplet\u003c/h2\u003e \u003cp\u003eIn this section, the study analyzes the influence of the interfacial tension in the microchannel on the size of the micro-droplets. Keeps velocity and viscosity constant, the flow velocity of the dispersed phase is 0.07 m/s, and the flow velocity of the continuous phase is 0.05 m/s. The continuous phase viscosity is 1.95\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, the dispersed phase viscosity is 6.71\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003ePa\u0026middot;s, and the contact angle is 135\u0026deg;. For the numerical model established in this section, five different interfacial tensions were used for numerical simulation. The interfacial tensions were 5.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m, 5.2\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m, 5.4\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m, 5.6 \u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m, 5.8\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e, the time from 0.003s to 0.018s,the droplet generation states at different interfacial tensions. From the numerical simulation results, it can be found that when the interfacial tension increases from 5.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m to 5.8\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m, the droplet volume becomes larger and the droplet generation frequency becomes smaller. This means that the breakup time of the droplet becomes longer, and the volume of the droplet increases. It can be seen from the Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e that when the interfacial tension increases from 5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m to 5.8\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eN/m, the volume of the droplet increases from 5.77\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e\u0026micro;L to 6.62\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e\u0026micro;L, an increase of 14.73%, The frequency is reduced from 123Hz to 109.29Hz. When the time is fixed at 0.02s, the pressure on the centerline of the model is as shown in the Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e. When a droplet is generated, as the droplet is farther from the inlet of the dispersed phase, the pressure on the centerline of the droplet decreases. With the increase of the interfacial tension, the Young Laplace pressure inside the droplet becomes larger, the micro-droplets are difficult to break from the dispersed phase, the breakage time of the droplet becomes longer, and the volume of droplets increases. The generation time of micro-droplets increases, and the frequency of generating droplets is smaller. Numerical simulation results show that when the interfacial tension, the volume of micro-droplets increases and the frequency of micro-droplet generation decreases.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Conclusions","content":"\u003cp\u003eIn this article, we designed a two-dimensional axisymmetric structure to numerically\u0026nbsp;simulation of the\u0026nbsp;droplet formation in a Co-flow microchannel\u0026nbsp;capillary device.\u003c/p\u003e\n\u003cp\u003e1. By changing the velocity of the dispersed phase, we observed tow flow patterns. When the relative velocity of the dispersed phase is low, the dispersed phase droplets will form droplets at the pipe mouth, and the generation speed will be slow, showing a droplet shape（dripping）. When the velocity of the dispersed phase increases, the outlet of the dispersed phase will no longer produce droplets, but a jet will be formed. The jet will become unstable after a certain distance from the pipe mouth, and finally droplets will be produced（jetting）. If you continue to increase the velocity of the dispersed phase, the distance of the jet will become farther and farther, and finally it will become a parallel jet in the calculation area（parallel flow）.\u003c/p\u003e\n\u003cp\u003e2. When the flow rate ratio increases from 0.538 to 1.4, the volume of the droplet increases from 1.60\u0026times;10\u003csup\u003e-4\u003c/sup\u003e\u0026mu;L to 5.73\u0026times;10\u003csup\u003e-4\u003c/sup\u003e\u0026mu;L, an increase of 258.13%, and the frequency decreases from 322.58Hz to 125.79Hz.\u003c/p\u003e\n\u003cp\u003e3. When the viscosity of the continuous phase increases from 0.5\u0026times;10\u003csup\u003e-3\u003c/sup\u003ePa\u0026middot;s to 4.0\u0026times;10\u003csup\u003e-3\u003c/sup\u003ePa\u0026middot;s, the volume of the droplet decreases from 1.96\u0026times;10\u003csup\u003e-3\u003c/sup\u003e\u0026mu;L to 3.28\u0026times;10\u003csup\u003e-4\u003c/sup\u003e\u0026mu;L, a\u0026nbsp;decrease of 83.27% , the frequency is increased from 35.78Hz to 178.57Hz.\u003c/p\u003e\n\u003cp\u003e4. When the contact angle increases from 75\u0026deg; to 135\u0026deg;, the volume of the droplet increases from 5.32\u0026times;10\u003csup\u003e-4\u003c/sup\u003e\u0026mu;L to 5.76\u0026times;10\u003csup\u003e-4\u003c/sup\u003e\u0026mu;L, an increase of 8.27%, and the frequency decreases from 124.46Hz to 123.15Hz. Controlling the volume of droplets by changing the contact angle is not an effective method.\u003c/p\u003e\n\u003cp\u003e5. When the two-phase interfacial tension increases from 5\u0026times;10\u003csup\u003e-3\u003c/sup\u003eN/m to 5.8\u0026times;10\u003csup\u003e-3\u003c/sup\u003eN/m, the volume of the droplet increases from 5.77\u0026times;10\u003csup\u003e-4\u003c/sup\u003e\u0026mu;L to 6.62\u0026times;10\u003csup\u003e-4\u003c/sup\u003e\u0026mu;L, an increase of 14.73%, and the frequency decreases from 123Hz to 109.29Hz.\u003c/p\u003e\n\u003cp\u003e6. As the droplet is farther away from the inlet of the dispersed phase, the pressure of the droplet on the centerline gradually decreases. As the velocity of the continuous phase increases, the pressure of the droplets on the centerline increases. As the viscosity of the continuous phase increases, the pressure on the centerline of the droplets increases. As the interfacial tension increases, the pressure of the droplet on the centerline increases. The change of contact angle has little effect on the pressure at the centerline.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eThis work was supported by Yantai Science and Technology Innovation Development Plan Key Basic Research Projects(2023JCYJ048), Special Supporting Funds for Leading Talents above Provincial Level in Yantai(220-20230002),Young Taishan Scholars Program of Shandong Province of China(tsqn202103091), Ludong University introduced talents and started funding project(LD22065).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eChen Y, Chen X (2019) An improved design for passive micromixer based on topology optimization method. Chem Phys Lett 734:136706\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eChen Y, Chen X (2020) Numerical and experimental investigations of novel passive micromixers with fractal-like tree structures. 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Microchim Acta 186(2):55\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Microfluidic, Numerical simulation, Droplet, Co-flow","lastPublishedDoi":"10.21203/rs.3.rs-3676725/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3676725/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this article, a numerical simulation of the droplet formation in a Co-flow microchannel capillary device, and the influencing factors of the formation of droplets are studied. The level set method is used to track the two-phase interface and droplet formation. In the Co-flow focusing device, we explored the influencing factors of the size of the generated droplets. The results show that as the ratio of the dispersed phase velocity to the continuous phase velocity increases, the volume of the generated droplets decreases significantly, the droplet generation frequency increases significantly, and the pressure of the droplets at the centerline decreases significantly. As the viscosity of continuous phase increases, the volume of generated droplets decreases significantly, the frequency of droplet generation increases significantly, and the pressure of droplets at the centerline decreases significantly. As the contact angle between the continuous phase and the wall increases, the volume of the generated droplets increases, but the volume increase is not obvious enough, the droplet generation frequency becomes smaller, and the droplet pressure at the centerline decreases. As the increase of interfacial tension, the volume of droplet generation increases significantly, the frequency of droplet generation decreases significantly, and the pressure of droplet at the centerline increases significantly.\u003c/p\u003e","manuscriptTitle":"Numerical simulation of droplet formation in a Co-flow microchannel capillary device","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-08 13:48:39","doi":"10.21203/rs.3.rs-3676725/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b7a53023-fd5f-4974-bf5c-92a559312b9d","owner":[],"postedDate":"January 8th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-02-08T12:22:19+00:00","versionOfRecord":[],"versionCreatedAt":"2024-01-08 13:48:39","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3676725","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3676725","identity":"rs-3676725","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}