Numerical Study of Droplet Formation in a Double T-junction Microchannel

Abstract

The present work deals with the study of oil–water flow inside a double T-junction microchannel employing the volume of fluid (VOF) model available in ANSYS FLUENT™. The simulations were carried out for water fraction, w_f = 0.2, with capillary number, Ca = 0.0004−0.0075. A total number of ~29,624 computational cells were generated to achieve grid independent solutions, and a time step size of 10^–6–10^–5 s was set to achieve converge solutions. The result depicts that the formation of droplets could take place by merging and alternating mode, respectively. The study of local pressure characteristics uncovered the inter-competing nature of local pressure at the inlets leading to the formation of droplets in alternate mode. The mode of droplet formation at Ca < 0.016 (Ca = 0.032−0.16) was observed to be in merging (alternating). Moreover, the flow morphologies with variation in contact angle, junction angle, inlet geometry, and viscosity contrast were explored. The results of the present work can be of significance for generating miniaturized alternating droplets for reaction, synthesis, and detection.

Introduction

The creation of miniaturized discrete droplets employing a microfluidic system has been extensively explored owing to its numerous targeted applications in the fields of reactions, synthesis of nanomaterials, point-of-care diagnostics, energy harvesting, and many more. The possibility of numerous applications in such systems is due to the ability of performing certain specific task such as fusion, splitting, and routing. These microfluidic systems provide a flexible platform directed towards the possible miniaturization of many existing unit operations and unit processes, finding applications in many field of science and technology. Usually, two approaches are utilized for developing an emulsion of fine droplets in microfluidic systems. The first approach is based on the generation of droplets at the channel junction by simply tuning of the flow velocities, fluid properties (density, viscosity, and interfacial tension), and wettability [1, 2]. While the second approach utilizes the splitting of a fully developed large size flow morphologies into miniature droplets by (i) externally applied field [3] and (ii) geometrical modifications [4] of the microfluidic system. Thus, in the past few decades, numerous research activities have been witnessed which focus on generating droplets inside a microfluidic system and utilizing them for various target applications. However, these studies were essentially engaged toward developing droplets with single composition in a single T-junction microchannel. While the generation of alternating droplets in a double T-junction microchannel has been less explored in comparison to the ones in a single T-junction microchannel and needs further in-depth understanding.

The formation of droplets with alternating compositions (also referred as alternate droplets) in microfluidic systems is a relatively a new concept and has been utilized for protein crystallization, indexing the concentration of solutes in droplet, synthesis of nanomaterials, and other applications. Such systems offer the advantage of handling reduced volume of protein samples, parallel operation, and higher control over accuracy. In this direction, some experimental and numerical studies have been performed with the objective of understanding the phenomenon of alternate droplets formation. A pioneering work by Zheng et al. [5] demonstrated the generation of alternating droplets in a double T-junction microchannel that consisted of a inlet for the continuous phase along with two vertical inlets for disperse phase. At constant water fraction, w_f = 0.2, they showed the transition of flow morphologies from merging to alternate to small droplets to laminar by variation in capillary number, (Ca). Here, water fraction is the ratio of sum of velocities of the disperse phase to the total velocity of both disperse and continuous phases. While the capillary number is defined as the ratio of viscous force to force due to interfacial tension. It could be observed that increasing the w_f led to increase in the size of droplets and decrease in the distance between two adjacent droplets. Similarly, a work by Surya et al. [6] showed the generation of alternating droplets in a double T-junction microfluidic channel. In addition, the results of numerical simulations were utilized to understand the mechanism of formation of alternating droplets by monitoring the fluctuations of pressure of the discrete phases. It was observed that the pressure build up in the discrete phase along the two vertical inlets took place alternately. Due to this, it resulted in the formation of droplet from one of the vertical inlets and caused retraction in the other inlet. Jin et al. [7] showed the formation of alternating droplets of different sizes by variation in the viscosity and flow rates of the disperse phase. They showed that the difference in droplet velocities could lead to droplet merging at the downstream section of the microfluidic channel. A numerical study on the formation of alternate droplets was conducted by Ngo et al. [8], where they could mimic the flow transitions observed by Zheng et al. [5], due to change in Ca. In addition, a number of parametric studies were carried out in order to investigate the influence of channel geometry, Ca and w_f, on the behavior of forming alternate droplets. With decrease in the λ (= width of disperse phase inlet to the width of continuous phase inlet) from 1.5 to 0.5, the parametric window of forming alternate droplets shifted toward the lower Ca, and the increase of viscosity ratio, µ_r (ratio of viscosity of disperse phase to viscosity of continuous phase) from 0.01 to 2 led to transition from alternating droplets to alternating-jetting droplets. An interesting work by Saqib et al. [9] showed the application of tapered inlets in a double T-junction channel for generating a periodic sequence of alternating droplets. It was observed that the tapering angle could influence the Laplace pressure, which depends on the radius of curvature of the dispersed phase and could be utilized to control the sequence of pattern repetition as well as uniformity of the trains of droplet. These studies suggest that the dynamics of formation of alternate droplets is rather a complex phenomena and pose many challenges in investigating its combine effect. They also present the scope of further research and the need of further study to understand the parametric effect on the mechanism and pressure characteristics. In the present work, we perform a numerical study of oil–water flow morphologies in a double T-junction microchannel using volume of fluid (VOF) model. We perform various numerical simulations to study the influence of interfacial tension, viscosity contrast, contact angle, and inlet geometries, on the oil–water flow inside a double T-junction microchannel. The formation of alternate droplets through the tapered inlet was correlated with the variation of local pressure characteristics.

Computational Methodology

The fluids (water and oil) are assumed to be incompressible and Newtonian. A two-dimensional (2D) computational domain consisting of a double T-junction microhannel was chosen for the study, as shown in Fig. 73.1. The effect of force due to gravity was considered along the negative direction of y-axis. Here, the image in Fig. 73.1 shows that the water (dispersed) phase flows through vertical inlets, while the flow of oil (continuous) phase flows through horizontal inlet. When the two phases come in contact at the junction, the two-phase flow structures (such as slug, plug, and droplet flow) are formed which flow through the downstream of the channel toward the outlet. The physical properties of fluid phases incorporated in the study are mentioned in the captions of Fig. 73.1. In this study, the volume of fluid (VOF) model was utilized to simulate the droplet dynamics in a 2D microchannel. The VOF method is based on the volume fraction function (α) that is used to identify the presence of fluid phases in the computational domain. Here, α_q = 1 indicates that the computational cell is completely occupied with phase q, while α_q = 0 indicates that the computational cell has no phase q and the value α_q = 0−1 denotes that the cell contains the interface between phases p and q. In order to describe the motion of oil–water flow in the computational domain, both continuity and momentum balance equations were employed.

$$\frac{\partial }{\partial t}(\rho ) + \nabla .(\rho \overline{v}) = 0$$

(73.1)

$$\frac{\partial }{\partial t}(\rho \vec{v}) + \nabla .(\rho \vec{v}\vec{v}) = - \nabla p + \nabla .\left[ {\mu \left\{ {\nabla \vec{v} + (\nabla \vec{v})^{T} } \right\}} \right] + \rho \vec{g} + \vec{F}$$

(73.2)

Fig. 73.1

a Schematics of the computational domain consisting of a double T-junction microchannel with channel diameter, D = 250 μm, b Wall adhesion configuration of oil–water system inside microchannel. Here, θ = 130° is the static contact angle of a water droplet pinning on wall surrounded by oil phase, with the interfacial tension, γ = 0.0365 N/m. The density of water (oil) phase is set at 1000 kg/m³ (970kg/m³), while the viscosity for both oil and water phase is set at 0.016 Pa s

Here, ρ, µ, p, $\vec{v}$, $\vec{g}$, and $\vec{F}$ represent the density, viscosity, pressure, velocity vector, vector for acceleration due to gravity, and surface force vector, respectively, while ρ and µ are calculated by taking the weighted average based on volume fraction of each phase.

$$\rho = \alpha_{q} \rho_{q} + (1 - \alpha_{q} )\rho_{p}$$

(73.3)

$$\mu = \alpha_{q} \mu_{q} + (1 - \alpha_{q} )\mu_{p}$$

(73.4)

The interfacial tension force ($\vec{F}$) used in Eq. (73.2) is calculated by using the continuum surface force (CSF) model [10] that is available in ANSYS FLUENT™.

$$\overrightarrow {F} = \gamma \left[ {\frac{{\rho K\hat{n}}}{{(1/2)(\rho_{p} + \rho_{q} )}}} \right]$$

(73.5)

where γ is the interfacial tension, $\hat{n}$ is the surface normal, and κ is the local surface curvature, which is calculated as

$$\kappa = \nabla \bullet \hat{n}$$

(73.6)

Alongside, $\hat{n} = n/\left| n \right|$ and $n = \nabla \bullet \alpha_{q}$. Moreover, the VOF model relies on the reconstructed oil–water interface by solving an advection equation for α_q. For the phase q, the advection equation is given as,

$$\frac{\partial }{\partial t}(\alpha_{q} \rho_{q} ) + \nabla .(\alpha_{q} \rho_{q} \vec{v}_{q} ) = 0$$

(73.7)

$$\hat{n} = \hat{n}_{w} \cos \theta_{w} + \hat{t}_{w} \sin \theta_{w}$$

(73.8)

The wall wettability was accounted by specifying the contact angle (θ). Here, $\hat{n}_{w}$ and $\hat{t}_{w}$ are the unit normal and tangential vectors w.r.t. the wall, respectively. The value of θ was used to determine the local curvature of the oil–water interface that is utilized for calculation of $\vec{F}$. Moreover, the transient governing equations were solved using the CFD solver of ANSYS FLUENT™ that uses a finite volume method to discretize the governing equations to an algebraic form. A time step size of 10^–6–10^–5 s (s) was found to provide a converged solution for each time step. Here, we employed the PRESTO scheme for pressure interpolation, SIMPLE for the pressure–velocity coupling, and QUICK scheme for upwinding the convective terms of the momentum equation.

Model Validation

In order to verify the formulation and computational execution, the results obtained in the present study are compared with that of literature [5], which is shown in Fig. 73.2. The images in Figs. 73.2a–c show the simulation results obtained from the present study for water fraction w_f = 0.2 and capillary number Ca = 0.0004−0.16, while images in Figs. 73.2d–f show the experimental results obtained by Zheng et al. [5], for similar values of w_f and Ca. It could be observed that the results obtained from simulation could well mimic the experimental results of Zheng et al. [5]. Although there exists a minute mismatching in the droplet length and jetting length, the simulation results provide good qualitative agreement and could be employed to predict the hydrodynamics of droplet formation in a double T-junction.

Fig. 73.2

Validation of the results obtained from present study to that of literature. Here, images a–c show the flow transition obtained from the present study at w_f = 0.2 and Ca = 0.0004 ‒ 0.16. Here, the velocity of oil and water phases was set to 9.125 × 10^–4 m/s and 1.14062 × 10^–4 m/s, respectively, while images d–f show the flow transition obtained by Zheng et al. [2] (Images are reprinted with permission from Zheng et al. [2], Copyright (2004), American Chemical Society)

Results and Discussions

The two-phase flow transition in microchannel is mainly influenced by the inter-competing forces, namely force due to gravity, interfacial tension, inertial, and viscous forces, respectively. The relative dominance of these competing forces is characterized by capillary number (Ca), Weber number (We), Bond number (Bo), and Reynolds number (Re). Moreover, the effect of these competing forces on the flow transitions could be analyzed by varying one of the parameters and fixing other parameters at a constant value. For example, in Figs. 73.2a–c, the value of Ca = 0.015−0.16 was varied by varying the interfacial tension γ = 0.01066−0.001 and setting constant for viscosity of oil µ_o = 0.016 Pa s and velocity of oil phase v_o = 0.01 m/s.

Effect of Interfacial Tension (γ)

The oil–water flow transition due to decrease in interfacial tension is shown in Fig. 73.3. At low Ca < 0.016, the formation of a droplet takes place by merging mode, as shown in Fig. 73.3a, b along with Fig. 73.2a. While at higher Ca = 0.032 ‒ 0.16, the formation of droplets takes place by alternating mode with the presence of jet formation [images (c–f)]. Alongside, it is observed that the jet length increases with decrease (increase) in interfacial tension, γ (w_f). Similar observations of increase in jet length have been observed by Timung et al. [1]. Moreover, the droplet length decreases with increase in Ca = 0.0004–0.16. The decrease in droplet length is associated with increase in surface to volume ratio of the flow structure, which can enhance momentum, heat, and mass transfer rates.

Fig. 73.3

Effect of flow transition due to increase in Ca = 0.015−0.16 by varying the γ = 0.01066−0.001 N/m with w_f = 0.2 and µ_o = 0.016 Pa s. Here, the velocity of oil and water phases was set to 0.01 m/s and 1.25 × 10^–3 m/s, respectively

Effect of Viscosity Contrast (μ_r)

The images (a–c) in Fig. 73.4 show the influence viscosity contrast μ_r = 0.1−1 (µ_w/µ_o) on the oil–water flow. Here, µ_o = 0.016 Pa s, w_f = 0.2, v_o = 0.01 m/s, and Ca = 0.04, respectively. It could be observed that with increase in μ_r the mechanism of formation of droplets changed from dripping to jetting mode which is in agreement with Ngo et al. [11]. Alongside, at lower μ_r = 0.1−0.2, there was a marginal variation in the size of each consecutive droplets which become uniform at µ_r = 1.

Fig. 73.4

Effect of flow transition due to increase in viscosity contrast µ_r (= µ_w/µ_o) = 0.1−1 by varying the viscosity of water µ_w = 0.0016−0.016 Pa s. Here, Ca and w_f were maintained constant at 0.04 and 0.2, respectively. Here, the velocity of oil and water phases was fixed at 0.01 m/s and 1.25 × 10^–3 m/s, respectively

Effect of Contact Angle (θ)

Here, the images (a–d) in Fig. 73.5 show the oil–water flow at various values of contact angle, θ = 0°−180°, while the flow parameters are maintained at w_f = 0.2, Ca = 0.04. The contact angle θ is set in accordance to Fig. 73.1b. At θ = 0° (hydrophilic condition), the flow observed is fully core-annular with oil (water) phase flowing through the core (annulus) region of the microchannel. This observation also continues for flow at θ = 90°. Moreover, at θ = 135° (hydrophobic condition), the formation of stable water droplets was observed from both water inlets, which were formed through jetting mode. The formation of stable water droplets continues with decrease in jet length and at θ = 180°, and the droplets are formed by dripping mode. The results obtained from the present work show the variation of flow morphology from core-annular to jetting mode droplets to dripping mode droplets.

Fig. 73.5

a–d Effect of flow transition due to variation in the wall wettability by varying the contact angle, θ = 0°–180°. e–h Effect of angle (λ = 90°–150°) of water inlet (s) on the flow morphology of oil–water flow at Ca = 0.04, w_f = 0.2, and µ_r = 1

Effect of Junction Angle (λ)

The oil–water flow morphologies are also influenced by the variation in the geometry of the microchannel. For example, Raj et al. [12] studied the influence of three junction configurations and concluded that the flow morphologies are independent of the type of junction used. However, for a constant wf and Ca, the Ngo et al. [11] showed that the size of the flow structures decreases with junction angle λ = 30°−60° and is almost unaffected for λ = 60°−90°. Both studies show the flow morphologies observed for λ≤ 90°. It would be interesting to investigate the same for λ ≥ 90°. The images (e–h) in Fig. 73.5 show the flow morphologies observed because of variation in junction angle, λ = 90°−150°. Here, the oil–water flow is maintained at wf = 0.2 and Ca = 0.04. At λ = 90°, the formation of stable droplets is observed with equal distance between the consecutive droplets. At λ = 105°, the size of the droplets reduces and the relative distance (l1 and l2) between consecutive droplets decreases. This phenomenon was also observed by Ngo et al. [8], for λ ≤ 90°. However, the frequency of the droplet formation from both water inlets remains constant. At λ = 135° and 150°, the flow morphology shows similar to that of λ = 90°, both in terms of droplet size and distance between consecutive droplets.

Effect of Inlet Tapering

The images (a–h) in Fig. 73.6 show the mechanism of formation of alternate droplets through tapered inlets at various time steps (t) in seconds (s). Here, Ca, w_f, and µ_r are set at 0.04, 0.3, and 1, respectively, while image (i) shows the variation of local pressures (P_A and P_B) at locations A and B, as shown in image (a). The points in the pressure curves are represented by the numbers, 1–17. Initially at t = 0 s, the water from both inlets starts to penetrate into the main channel stream, and the filling of water phase takes place, which allows the increase of fraction of water at the T-junctions. During the growth phase, the local pressures at both the inlets increase (point 1), as shown in image (i). At t = 0.04 s, the maximum growth of the water phase takes place, and the filling of water decreases to its minimum. This also results in reduction in the local pressures (point 2), as shown in image (i). As the continuous oil phase drags the oil–water interface along the channel downstream, the pressure P_A begins to exceed that of P_B, as shown by points 3 and 4, in image (i). Due to this, the oil–water interface along inlet A stretches longer than the interface along inlet B. At t = 0.10 s, the necking of the oil–water interface (A) takes place, and the pressure P_A starts to reduce, as shown by point 5 of image (i). Beyond point 5, the interface (A) retracts (t = 0.12 s), and the pressure reduces further. The points 6 and 7 show the increase in pressure P_B during the stretching of oil–water interface at inlet B. At point 8, the pressure of P_B reaches its maximum value leading to necking of interface at inlet B (t = 0.16 s). Here, images (a–f) show the mechanism of formation of alternate droplets from both inlets, while the profile of the local pressures also shows that the formation of droplet takes place through A (B), and the variation of inter-competing local pressure at A and B shows the reason for the formation of droplets in alternating mode instead of merging.

Fig. 73.6

Mechanism of formation of alternate droplets in a microchannel with tapered inlets and its pressure characteristic curves. Here, images a–h show the stages of droplet formation at various instant of time, t in seconds (s), while image (i) shows the variation in local pressure (P) in Pascal (Pa) at positions A and B, as indicated in image (a). Here, Ca, w_f, and µ_r are set at 0.04, 0.2, and 1, respectively

Conclusions

In the present work, we performed various numerical simulations to study the influence of interfacial tension, viscosity contrast, contact angle, and inlet geometries, on the oil–water flow inside a double T-junction microchannel. Here, we found the formation of droplet in merging (alternate) mode for lower (higher) capillary number, Ca < 0.016 (Ca = 0.032−0.16). Alongside, the increase in viscosity contrast, µ_r = 0.1−1, showed higher uniformity in the droplet size formation. The increase of contact angle, θ = 0°−180°, increases the hydrophobic property of the wall which showed the transition from annular (oil-core) flow to droplet in jetting mode to droplet in dripping mode. Moreover, by increasing the inlet angle, λ = 90°−150°, we observe a marginal reduction in the droplet size with variation in distance between consecutive droplets. A numerical simulation was performed to study the local pressure drop characteristics at the tapered inlet during the formation of droplets. The pressure drop curves show the inter-competing nature of local pressure in both tapered inlets leading to the alternating nature of droplet formation. The findings of the present study can be utilized to design and fabricate a micro-device that is more compact and poses the additional advantages such as high surface-to-volume ratio, ease in handling hazardous materials, controlling exothermic reactions, creation of a digitized micro-devices. Such miniaturized devices will contribute toward cost effectives and ease in accessibility to people who are located at remote areas. Many of such devices include the water monitoring kits, amylase detecting kit, rapid testing devices for COVID-19, and other pathogens.