# Effect of injection angle, density ratio, and viscosity on droplet formation in a microfluidic T-junction

M.Y.A. Jamalabadi, M. Daqiqshirazi, Ali Kosar, M.S. Shadloo

► To cite this version:

M.Y.A. Jamalabadi, M. Daqiqshirazi, Ali Kosar, M.S. Shadloo. Effect of injection angle, density ratio, and viscosity on droplet formation in a microfluidic T-junction. Theoretical and Applied Mechanics Letters, 2017, 7 (4), pp.243-251. 10.1016/j.taml.2017.06.002. hal-02127859

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<figure><img src="image_1.png" /></figure>

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# Effect of injection angle, density ratio, and viscosity on droplet formation in a microfluidic T-junction

M.Y.A. Jamalabadi, M. Daqiqshirazi, Ali Kosar, M.S. Shadloo

► To cite this version:

M.Y.A. Jamalabadi, M. Daqiqshirazi, Ali Kosar, M.S. Shadloo. Effect of injection angle, density ratio, and viscosity on droplet formation in a microfluidic T-junction. Theoretical and Applied Mechanics Letters, 2017, 7 (4), pp.243-251. 10.1016/j.taml.2017.06.002. hal-02127859

HAL Id: hal-02127859

https://hal.science/hal-02127859v1

Submitted on 29 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

Letter

# Effect of injection angle, density ratio, and viscosity on droplet formation in a microfluidic T-junction

Mohammad Yaghoub Abdollahzadeh Jamalabadi<sup>a,*</sup>, Mohammadreza DaqiqShirazi<sup>b</sup>, Ali Kosar<sup>c</sup>, Mostafa Safdari Shadloo<sup>d</sup>

<figure><img src="image_3.png" /></figure>

## HIGHLIGHTS

* A parameter study on droplet formation in T-junctions is performed.

* Perpendicular and parallel injection angles increase the diameter.

* Contact angle and slip length increase the diameter.

* Flow rate, density, and viscosity ratios decrease the diameter.

## ARTICLE INFO

**Article history:**

Received 30 January 2017

Received in revised form 15 May 2017

Accepted 31 May 2017

Available online 10 July 2017

*This article belongs to the Fluid Mechanics.

**Keywords:**

Microfluidics

Droplet formation

Flow rate ratio

Density ratio

## ABSTRACT

The T-junction microchannel device makes available a sharp edge to form micro-droplets from bio-material solutions. This article investigates the effects of injection angle, flow rate ratio, density ratio, viscosity ratio, contact angle, and slip length in the process of formation of uniform droplets in microfluidic T-junctions. The governing equations were solved by the commercial software. The results show that contact angle, slip length, and injection angles near the perpendicular and parallel conditions have an increasing effect on the diameter of generated droplets, while flow rate, density and viscosity ratios, and other injection angles had a decreasing effect on the diameter.

© 2017 The Author(s). Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Making micro-sized droplets in a controlled size and flow pattern has been recently one of the attractive research areas related to microfluidics in various applications such as consumer products of mono-dispersed micro particles, food processing, pharmaceuticals for drug delivery, cosmetics, tumor annihilation, synthesis of nanoparticles, enhancement of mixing reactions for chemical industries, creation of emulsions, crystallization of proteins, personal health care products and bubble generators among others [1–3]. Numerous research groups have experimentally investigated several features of droplet formation. There are a variety of devices that can be used to generate droplets in micro scale. There exist three major methods for droplet generation in microfluidic devices: hydrodynamic flow focusing in co-flowing fluids [4,5], T-junctions [6–13], and immiscible fluids mixing by the orifice [14–19]. Among these methods in microfluidic devices, cross-flow

shear in micro channels with T-junctions and flow-focusing configurations strongly depend upon the geometrical parameters controlling the break-up of droplets.

The cross-flow shear method at a T-junction utilizes shear force, imposed by an immiscible cross-flow fluid in a continuous fluid flow, to produce droplets or plugs at a constant rate. The first research focused on the droplet formation within a T-junction micro channel was conducted by Thorsen et al. [11]. They found that the produced droplet size decreases with the decrease in the interfacial tension $\sigma$, the increase in the continuous phase velocity $U$, and viscosity $\mu$. Using the scaling law by equating the Laplace pressure $4\sigma/d_d$ to the shear force $\mu U/d_m$, they observed that the dimensionless ratio of the drops diameter to the cross-section dimension of the micro channel ($d_d/d_m$) was proportional to the inverse of capillary number ($\mu U/\sigma$). For capillary number $Ca > 0.2$, this assumption was in good agreement with the experimental results. The effect of capillary number of the continuous phase on droplet formation was studied by Xu et al. [20]. They proved

<footnote><sup>a</sup> Department of Mechanical, Robotics and Energy Engineering, Dongguk University, Seoul 04620, Republic of Korea</footnote>

<footnote><sup>b</sup> School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran</footnote>

<footnote><sup>c</sup> Mechatronics Engineering Program, Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey</footnote>

<footnote><sup>d</sup> CORIA-UMR 6614, Normandie University, CNRS-University and INSA of Rouen, 76000 Rouen, France</footnote>

<footnote>* Corresponding author.
E-mail address: abdollahzadeh@dongguk.edu (M.Y.A. Jamalabadi).</footnote>
<figure><img src="image_2.png" /></figure>

Letter

# Effect of injection angle, density ratio, and viscosity on droplet formation in a microfluidic T-junction

Mohammad Yaghoub Abdollahzadeh Jamalabadi<sup>a,*</sup>, Mohammadreza DaqiqShirazi<sup>b</sup>, Ali Kosar<sup>c</sup>, Mostafa Safdari Shadloo<sup>d</sup>

<figure><img src="image_3.png" /></figure>

## HIGHLIGHTS

* A parameter study on droplet formation in T-junctions is performed.

* Perpendicular and parallel injection angles increase the diameter.

* Contact angle and slip length increase the diameter.

* Flow rate, density, and viscosity ratios decrease the diameter.

## ARTICLE INFO

**Article history:**

Received 30 January 2017

Received in revised form 15 May 2017

Accepted 31 May 2017

Available online 10 July 2017

*This article belongs to the Fluid Mechanics.

**Keywords:**

Microfluidics

Droplet formation

Flow rate ratio

Density ratio

## ABSTRACT

The T-junction microchannel device makes available a sharp edge to form micro-droplets from bio-material solutions. This article investigates the effects of injection angle, flow rate ratio, density ratio, viscosity ratio, contact angle, and slip length in the process of formation of uniform droplets in microfluidic T-junctions. The governing equations were solved by the commercial software. The results show that contact angle, slip length, and injection angles near the perpendicular and parallel conditions have an increasing effect on the diameter of generated droplets, while flow rate, density and viscosity ratios, and other injection angles had a decreasing effect on the diameter.

© 2017 The Author(s). Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/).

Making micro-sized droplets in a controlled size and flow pattern has been recently one of the attractive research areas related to microfluidics in various applications such as consumer products of mono-dispersed micro particles, food processing, pharmaceuticals for drug delivery, cosmetics, tumor annihilation, synthesis of nanoparticles, enhancement of mixing reactions for chemical industries, creation of emulsions, crystallization of proteins, personal health care products and bubble generators among others [1–3]. Numerous research groups have experimentally investigated several features of droplet formation. There are a variety of devices that can be used to generate droplets in micro scale. There exist three major methods for droplet generation in microfluidic devices: hydrodynamic flow focusing in co-flowing fluids [4,5], T-junctions [6–13], and immiscible fluids mixing by the orifice [14–19]. Among these methods in microfluidic devices, cross-flow

shear in micro channels with T-junctions and flow-focusing configurations strongly depend upon the geometrical parameters controlling the break-up of droplets.

The cross-flow shear method at a T-junction utilizes shear force, imposed by an immiscible cross-flow fluid in a continuous fluid flow, to produce droplets or plugs at a constant rate. The first research focused on the droplet formation within a T-junction micro channel was conducted by Thorsen et al. [11]. They found that the produced droplet size decreases with the decrease in the interfacial tension $\sigma$, the increase in the continuous phase velocity $U$, and viscosity $\mu$. Using the scaling law by equating the Laplace pressure $4\sigma/d_d$ to the shear force $\mu U/d_m$, they observed that the dimensionless ratio of the drops diameter to the cross-section dimension of the micro channel ($d_d/d_m$) was proportional to the inverse of capillary number ($\mu U/\sigma$). For capillary number $Ca > 0.2$, this assumption was in good agreement with the experimental results. The effect of capillary number of the continuous phase on droplet formation was studied by Xu et al. [20]. They proved

<footnote><sup>a</sup> Department of Mechanical, Robotics and Energy Engineering, Dongguk University, Seoul 04620, Republic of Korea</footnote>

<footnote><sup>b</sup> School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran</footnote>

<footnote><sup>c</sup> Mechatronics Engineering Program, Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey</footnote>

<footnote><sup>d</sup> CORIA-UMR 6614, Normandie University, CNRS-University and INSA of Rouen, 76000 Rouen, France</footnote>

<footnote>* Corresponding author.
E-mail address: abdollahzadeh@dongguk.edu (M.Y.A. Jamalabadi).</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 1. The schematic of the T-junction including inlet, outlet, and wall boundary conditions.</figcaption></figure>

that there were three regimes in the droplet formation based on the capillary number of the continuous phase: squeezing or plug flow ($Ca_c < 0.002$), transient or cobbles flow ($0.002 < Ca_c < 0.01$), and dripping or drop flow ($0.01 < Ca_c < 0.3$). Garstecki et al. [21] simplified the correlations of the Xu et al. [20]. They expressed the droplet size in terms of capillary number of continuous phase and ratio of flow rates. They found that there was a critical value of capillary number, which separated drop flow from plug flow dominated by the pressure drop across the plug. Some other groups observed that the size of droplets was controlled by thermophysical properties of the two phases and flow rates of the two phases [22–35].

Link et al. [36] found an experimental methodology to passive droplet formation with the accurate size. To precisely fit the droplet size, they used the extensional flow in the neighborhood of the stagnation point of a T-shaped junction. When the flow is split into two sidearms, the ratio of volumes of the two branches of droplets is proportional to the ratio of sidearm lengths. In addition, they found that drops with maximum extensions below unity never break [37], which is in agreement with the Rayleigh instability [38–40]. Rapid formation and mixing of droplets in T-junction were also investigated by Tice et al. [25,41], de Gennes et al. [40], Jullien et al. [42], Christopher et al. [43], Cubaud et al. [44], and a comprehensive review on droplet dynamics in complex flows is also present [45].

Gu and Liow [46] experimentally examined effects of injection angle on micro-droplet formation in microfluidic T-junctions with non-Newtonian solutions. They report that the maximum droplet size is obtained for injection angle of 90° (T-junction). Furthermore, they found that the behavior was quite different for acute contact angles (smaller than 90°) than for obtuse contact angles (larger than 90°).

Although numerous studies on generation of droplets in T-junction have been performed, the effect of the cross flow injection, which is an important parameter determining the droplet size, is still missing. The aim of the current investigation is to study the fluid flow and geometrical parameters affecting shear flow generation of droplets in microfluidic devices. The emphasis is on the effect of the angle between the entrance of the Newtonian fluid flow from the cross channel and the main channel, which has not been studied in the literature.

The schematic of the considered micro scale T-junction channel having two inlets and one outlet is displayed in Fig. 1. The fluid as the dispersed phase, Fluid 2, enters by injection through the lateral channel (to be dispersed into small droplets or make the plug), while the other fluid, Fluid 1, flows from inlet 1 through the main channel as the continuous phase. The channel length is chosen in such a way to provide computational efficiency as well as droplet generation effect. The two Newtonian fluids fill the defined micro-channel and laminar flow assumption is made inside the channel.

The governing equations, namely Continuity and Navier-Stokes equations are expressed as follows:

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0, \qquad (1) $$

$$ \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) - \frac{\partial p}{\partial x} - \frac{3\sqrt{2}\sigma}{4\epsilon} \left[ \varphi (\varphi^2 - 1) - \frac{\partial}{\partial x} \left( \epsilon^2 \frac{\partial \varphi}{\partial x} \right) - \frac{\partial}{\partial y} \left( \epsilon^2 \frac{\partial \varphi}{\partial y} \right) \right] \frac{\partial \varphi}{\partial x}, \qquad (2) $$

$$ \rho \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) - \frac{\partial p}{\partial y} - \frac{3\sqrt{2}\sigma}{4\epsilon} \left[ \varphi (\varphi^2 - 1) - \frac{\partial}{\partial x} \left( \epsilon^2 \frac{\partial \varphi}{\partial x} \right) - \frac{\partial}{\partial y} \left( \epsilon^2 \frac{\partial \varphi}{\partial y} \right) \right] \frac{\partial \varphi}{\partial y}, \qquad (3) $$

where $\rho$ is density (kg/m³), $u$ is velocity in x direction (m/s), $v$ is velocity in y direction (m/s), $t$ is time (s), $\mu$ is dynamic viscosity (Pa·s), $\rho$ is pressure (Pa), and $\sigma$ is the surface tension (N/m). The properties in Eqs. (1)–(3) are calculated using the phase field method. This method introduces a $\varphi$ variable such that the concentration of the two components or the volume fractions of the individual fluids are $(1 + \varphi)/2$ and $(1 - \varphi)/2$, respectively. The diffuse interface is defined as the region, where the dimensionless phase field variable goes from $-1$ to $1$. The Cahn-Hilliard equation reads as [47]:

$$ \frac{\partial \varphi}{\partial t} + u \frac{\partial \varphi}{\partial x} + v \frac{\partial \varphi}{\partial y} = \frac{\partial}{\partial x} \left\{ \frac{3\sqrt{2}\gamma\sigma}{4\epsilon} \frac{\partial}{\partial x} [\varphi(\varphi^2 - 1)] \right\} \\ + \frac{\partial}{\partial y} \left\{ \frac{3\sqrt{2}\gamma\sigma}{4\epsilon} \frac{\partial}{\partial y} [\varphi(\varphi^2 - 1)] \right\} \\ - \frac{\partial}{\partial x} \left\{ \frac{3\sqrt{2}\gamma\sigma}{4\epsilon} \frac{\partial}{\partial x} \left[ \frac{\partial}{\partial x} \left( \epsilon^2 \frac{\partial \varphi}{\partial x} \right) + \frac{\partial}{\partial y} \left( \epsilon^2 \frac{\partial \varphi}{\partial y} \right) \right] \right\} \\ - \frac{\partial}{\partial y} \left\{ \frac{3\sqrt{2}\gamma\sigma}{4\epsilon} \frac{\partial}{\partial y} \left[ \frac{\partial}{\partial x} \left( \epsilon^2 \frac{\partial \varphi}{\partial x} \right) + \frac{\partial}{\partial y} \left( \epsilon^2 \frac{\partial \varphi}{\partial y} \right) \right] \right\}, \qquad (4) $$

where $\gamma$ (m³·s·kg⁻¹) is the mobility, and $\epsilon$ (m) is a capillary width representative of the thickness of diffuse interface, which are numerical stabilization parameters. More specifically, 90% of variation in $\varphi$ occurs over a thickness of 4.1641$\epsilon$, while 99% of the variation corresponds to a thickness of 7.4859$\epsilon$. The right hand side of Eq. (4) originates from the mixing-energy density, which is a double-well potential of $\varphi$ (phobic effect) and its gradient (philic effect). The first term produces the classical sharp-interface, while the second one represents weakly non-local mixing interactions between the components. The parameter in first term is found from the relationship between the interfacial tension and capillary width [41]. It can typically set the interface thickness parameter to the half of the characteristic mesh size in the region covered by the interface. The parameter $\gamma\sigma/\epsilon$ on the right hand side of Eq. (4) determines the relaxation time of the interface, and has to be chosen in such a way that the interface will have enough resistance against thinning or thickening effect of fluid flow (to retain a constant interfacial thickness) as well as enough compatibility with fluid flows having no damping effect on convective terms. The term $\sigma\epsilon^3$ is usually the initial guess for the time scale of the Cahn-Hilliard diffusion to obtain the correct pressure variation over the interface. The density (kg/m³) and viscosity (Pa·s) in the phase
<figure><img src="image_4.png" /><figcaption>Fig. 1. The schematic of the T-junction including inlet, outlet, and wall boundary conditions.</figcaption></figure>

that there were three regimes in the droplet formation based on the capillary number of the continuous phase: squeezing or plug flow (Ca$_c$ < 0.002), transient or cobbles flow (0.002 < Ca$_c$ < 0.01), and dripping or drop flow (0.01 < Ca$_c$ < 0.3). Garstecki et al. [21] simplified the correlations of the Xu et al. [20]. They expressed the droplet size in terms of capillary number of continuous phase and ratio of flow rates. They found that there was a critical value of capillary number, which separated drop flow from plug flow dominated by the pressure drop across the plug. Some other groups observed that the size of droplets was controlled by thermophysical properties of the two phases and flow rates of the two phases [22–35].

Link et al. [36] found an experimental methodology to passive droplet formation with the accurate size. To precisely fit the droplet size, they used the extensional flow in the neighborhood of the stagnation point of a T-shaped junction. When the flow is split into two sidearms, the ratio of volumes of the two branches of droplets is proportional to the ratio of sidearm lengths. In addition, they found that drops with maximum extensions below unity never break [37], which is in agreement with the Rayleigh instability [38–40]. Rapid formation and mixing of droplets in T-junction were also investigated by Tice et al. [25,41], de Gennes et al. [40], Jullien et al. [42], Christopher et al. [43], Cubaud et al. [44], and a comprehensive review on droplet dynamics in complex flows is also present [45].

Gu and Liow [46] experimentally examined effects of injection angle on micro-droplet formation in microfluidic T-junctions with non-Newtonian solutions. They report that the maximum droplet size is obtained for injection angle of 90° (T-junction). Furthermore, they found that the behavior was quite different for acute contact angles (smaller than 90°) than for obtuse contact angles (larger than 90°).

Although numerous studies on generation of droplets in T-junction have been performed, the effect of the cross flow injection, which is an important parameter determining the droplet size, is still missing. The aim of the current investigation is to study the fluid flow and geometrical parameters affecting shear flow generation of droplets in microfluidic devices. The emphasis is on the effect of the angle between the entrance of the Newtonian fluid flow from the cross channel and the main channel, which has not been studied in the literature.

The schematic of the considered micro scale T-junction channel having two inlets and one outlet is displayed in Fig. 1. The fluid as the dispersed phase, Fluid 2, enters by injection through the lateral channel (to be dispersed into small droplets or make the plug), while the other fluid, Fluid 1, flows from inlet 1 through the main channel as the continuous phase. The channel length is chosen in such a way to provide computational efficiency as well as droplet generation effect. The two Newtonian fluids fill the defined micro-channel and laminar flow assumption is made inside the channel.

The governing equations, namely Continuity and Navier-Stokes equations are expressed as follows:

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0, \qquad (1) $$

$$ \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) - \frac{\partial p}{\partial x} - \frac{3\sqrt{2}\sigma}{4\epsilon} \left[ \varphi (\varphi^2 - 1) - \frac{\partial}{\partial x} \left( \epsilon^2 \frac{\partial \varphi}{\partial x} \right) - \frac{\partial}{\partial y} \left( \epsilon^2 \frac{\partial \varphi}{\partial y} \right) \right] \frac{\partial \varphi}{\partial x}, \qquad (2) $$

$$ \rho \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) - \frac{\partial p}{\partial y} - \frac{3\sqrt{2}\sigma}{4\epsilon} \left[ \varphi (\varphi^2 - 1) - \frac{\partial}{\partial x} \left( \epsilon^2 \frac{\partial \varphi}{\partial x} \right) - \frac{\partial}{\partial y} \left( \epsilon^2 \frac{\partial \varphi}{\partial y} \right) \right] \frac{\partial \varphi}{\partial y}, \qquad (3) $$

where $\rho$ is density (kg/m$^3$), $u$ is velocity in x direction (m/s), $v$ is velocity in y direction (m/s), $t$ is time (s), $\mu$ is dynamic viscosity (Pa·s), $\rho$ is pressure (Pa), and $\sigma$ is the surface tension (N/m). The properties in Eqs. (1)–(3) are calculated using the phase field method. This method introduces a $\varphi$ variable such that the concentration of the two components or the volume fractions of the individual fluids are $(1 + \varphi)/2$ and $(1 - \varphi)/2$, respectively. The diffuse interface is defined as the region, where the dimensionless phase field variable goes from $-1$ to $1$. The Cahn-Hilliard equation reads as [47]:

$$ \frac{\partial \varphi}{\partial t} + u \frac{\partial \varphi}{\partial x} + v \frac{\partial \varphi}{\partial y} = \frac{\partial}{\partial x} \left\{ \frac{3\sqrt{2}\gamma\sigma}{4\epsilon} \frac{\partial}{\partial x} [\varphi(\varphi^2 - 1)] \right\} \\ + \frac{\partial}{\partial y} \left\{ \frac{3\sqrt{2}\gamma\sigma}{4\epsilon} \frac{\partial}{\partial y} [\varphi(\varphi^2 - 1)] \right\} \\ - \frac{\partial}{\partial x} \left\{ \frac{3\sqrt{2}\gamma\sigma}{4\epsilon} \frac{\partial}{\partial x} \left[ \frac{\partial}{\partial x} \left( \epsilon^2 \frac{\partial \varphi}{\partial x} \right) + \frac{\partial}{\partial y} \left( \epsilon^2 \frac{\partial \varphi}{\partial y} \right) \right] \right\} \\ - \frac{\partial}{\partial y} \left\{ \frac{3\sqrt{2}\gamma\sigma}{4\epsilon} \frac{\partial}{\partial y} \left[ \frac{\partial}{\partial x} \left( \epsilon^2 \frac{\partial \varphi}{\partial x} \right) + \frac{\partial}{\partial y} \left( \epsilon^2 \frac{\partial \varphi}{\partial y} \right) \right] \right\}, \qquad (4) $$

where $\gamma$ (m$^3$·s·kg$^{-1}$) is the mobility, and $\epsilon$ (m) is a capillary width representative of the thickness of diffuse interface, which are numerical stabilization parameters. More specifically, 90% of variation in $\varphi$ occurs over a thickness of 4.1641$\epsilon$, while 99% of the variation corresponds to a thickness of 7.4859$\epsilon$. The right hand side of Eq. (4) originates from the mixing-energy density, which is a double-well potential of $\varphi$ (phobic effect) and its gradient (philic effect). The first term produces the classical sharp-interface, while the second one represents weakly non-local mixing interactions between the components. The parameter in first term is found from the relationship between the interfacial tension and capillary width [41]. It can typically set the interface thickness parameter to the half of the characteristic mesh size in the region covered by the interface. The parameter $\gamma\sigma/\epsilon$ on the right hand side of Eq. (4) determines the relaxation time of the interface, and has to be chosen in such a way that the interface will have enough resistance against thinning or thickening effect of fluid flow (to retain a constant interfacial thickness) as well as enough compatibility with fluid flows having no damping effect on convective terms. The term $\sigma\epsilon^3$ is usually the initial guess for the time scale of the Cahn-Hilliard diffusion to obtain the correct pressure variation over the interface. The density (kg/m$^3$) and viscosity (Pa·s) in the phase
<table><caption>**Table 1**
Fluid flow conditions.</caption><thead><tr><th>Case</th><th>Q<sub>c</sub> (mL·h<sup>-1</sup>)</th><th>Q<sub>d</sub> (mL·h<sup>-1</sup>)</th><th>ρ<sub>c</sub> (kg·m<sup>-3</sup>)</th><th>ρ<sub>d</sub> (kg·m<sup>-3</sup>)</th><th>μ<sub>c</sub> (Pa·s)</th><th>μ<sub>d</sub> (Pa·s)</th><th>θ<sub>w</sub></th><th>β (μm)</th></tr></thead><tbody><tr><td>1</td><td>2</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>-</td></tr><tr><td>2</td><td>1</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>5</td></tr><tr><td>3</td><td>4</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>5</td></tr><tr><td>4</td><td>1</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>6.71×10<sup>-2</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>5</td></tr><tr><td>5</td><td>1</td><td>0.2</td><td>1.02×10<sup>4</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>5</td></tr><tr><td>6</td><td>1</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>π/4</td><td>5</td></tr><tr><td>7</td><td>1</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>10</td></tr></tbody></table>

field are calculated from the mixed phase method and are defined as:

$$ \rho = \rho_1 + (\rho_2 - \rho_1) \phi, \quad (5) $$

$$ \mu = \mu_1 + (\mu_2 - \mu_1) \phi, \quad (6) $$

where physical properties $ρ_1$, $ρ_2$ are the densities and $μ_1$, $μ_2$ are the dynamic viscosities of Fluid 1 and Fluid 2, respectively. At both inlets, laminar flow conditions with prescribed volumetric flow rates are imposed. At the outflow boundary, a prescribed pressure and no viscous stress condition are set. The following expression is used for the walls of the channel:

$$ u_y = \beta u, \quad (7) $$

where $β$ is the wall slip length.

The definitions of the Reynolds number ($Re$), capillary number ($Ca$), density ratio ($ρ_r$), viscosity ratio ($ρ_t$), diameter ratio ($D_r$), angle ratio ($θ_r$), and slip ratio ($β_r$) are given below:

$$ Re = \frac{\rho v_{\max} w}{\mu}, \quad (8) $$

$$ Ca = \frac{\mu v_{\max}}{\sigma}, \quad (9) $$

$$ \rho_r = \frac{\rho_c}{\rho_d}, \quad (10) $$

$$ \mu_r = \frac{\mu_c}{\mu_d}, \quad (11) $$

$$ D_r = \frac{D}{w}, \quad (12) $$

$$ \beta_r = \frac{\beta}{\beta_0}, \quad (13) $$

$$ \theta_r = \frac{\theta}{\theta_0}, \quad (14) $$

where subscripts c and d denote continuous and discrete phases respectively. As well, parameters D and $β$ are the slip length and the equivalent droplet diameter which are calculated from the droplet surface ($A = πD^2/4$). Furthermore, the parameters in Eqs. (13) and (14) which are indicated by 0 are the values for experimental results used for making dimensionless parameters. Although the Reynolds number typically is defined in terms of average velocity, but here in order to maintain the similarity between experiment results and numerical modeling, the Reynolds number is defined in terms of maximum velocity ($v_{\max}$) and the channel width ($w$).

In order to understand the dynamics of droplet formation in a microfluidic T-junction many numerical methods are used such as two-phase level set method [13] and VOF model [46]. In this work, the commercial software is used to solve the governing equations. Flow conditions and values of parameters are tabulated in Table 1. Furthermore, dimensionless numbers of simulations corresponding to flow conditions are presented in Table 2. A collocated grid arrangement is adopted based on physically controlled meshing.

The impact of injection angle in non-Newtonian fluids has been shown through experimentation [46]. Although, the 2D geometry was susceptible to some errors, in which case, but it has a lower

<table><caption>**Table 2**
Dimensionless numbers of simulations corresponding to flow conditions.</caption><thead><tr><th>Case</th><th>Re<sub>c</sub></th><th>Ca<sub>c</sub></th><th>ρ<sub>r</sub></th><th>μ<sub>r</sub></th><th>θ<sub>r</sub></th><th>β<sub>r</sub></th><th>Q<sub>r</sub></th></tr></thead><tbody><tr><td>1</td><td>4.27</td><td>0.033</td><td>0.98</td><td>0.29</td><td>1</td><td>-</td><td>10</td></tr><tr><td>2</td><td>2.14</td><td>0.016</td><td>0.98</td><td>0.29</td><td>1</td><td>1</td><td>5</td></tr><tr><td>3</td><td>8.53</td><td>0.065</td><td>0.98</td><td>0.29</td><td>1</td><td>1</td><td>20</td></tr><tr><td>4</td><td>0.214</td><td>0.16</td><td>0.98</td><td>10</td><td>1</td><td>1</td><td>5</td></tr><tr><td>5</td><td>21.4</td><td>0.016</td><td>10</td><td>0.29</td><td>1</td><td>1</td><td>5</td></tr><tr><td>6</td><td>2.14</td><td>0.016</td><td>0.98</td><td>0.29</td><td>1/3</td><td>1</td><td>5</td></tr><tr><td>7</td><td>2.14</td><td>0.016</td><td>0.98</td><td>0.29</td><td>1</td><td>2</td><td>5</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Geometry of the T-junction.</figcaption></figure>

computational cost than a 3D model and in the literature just the 2D plot of the continuous versus the dispersed phase contours has been plotted [23] which shows the curvature in depth is negligible and the 2D assumption is sufficient to exclude the general characteristics of the system. The current 2D numerical method is validated by its comparison with the experimental results of the droplet formation in a 3D rectangular T-shaped microchannel junction presented by van der Graaf et al. [22]. Droplet generation experiments were achieved with 1,6-hexanediol diacrylate as the dispersed phase in the main stream of a 2 wt% poly (vinyl alcohol) aqueous solution [48]. The surface tension coefficient was taken as $5 \times 10^{-3}$ N/m. The no-slip boundary conditions and a contact angle of an oil droplet on a surface in a water phase of 135° were applied at the walls. A uniform grid with $Δx = Δy = 0.005$ mm was applied in numerical simulations. The difference in produced droplet size between this grid and a finer grid ($Δx = Δy = 0.001$ mm) was 2% so that sufficient grid refinement was performed. Figure 3 displays some images from droplet generation experiments along with numerical results at flow rates of the continuous phase and the to-be-dispersed phase of 2 mL/h and 0.2 mL/h, respectively. The droplet bends before separation from the stream of Fluid 2, and the Fluid 1 continuous phase imposes pressure on it (see Fig. 2). This applied pressure, which is a function of droplet shape and both fluid properties, is resisted by the interfacial tension in the triple point contact line. The shapes of the droplets at the similar time intervals are in agreement as shown in Fig. 3. Due to the 2D nature of the simulations, the numerical results under the above mentioned conditions overestimate the experimental results to some extent.

The resulting diameter for a droplet is calculated for various main stream flow rates (based on the milliliters per hour) and is compared with the experimental results of van der Graaf et al. [22]
<table><caption>**Table 1**
Fluid flow conditions.</caption><thead><tr><th>Case</th><th>Q<sub>c</sub> (mL·h<sup>-1</sup>)</th><th>Q<sub>d</sub> (mL·h<sup>-1</sup>)</th><th>ρ<sub>c</sub> (kg·m<sup>-3</sup>)</th><th>ρ<sub>d</sub> (kg·m<sup>-3</sup>)</th><th>μ<sub>c</sub> (Pa·s)</th><th>μ<sub>d</sub> (Pa·s)</th><th>θ<sub>w</sub></th><th>β (μm)</th></tr></thead><tbody><tr><td>1</td><td>2</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>-</td></tr><tr><td>2</td><td>1</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>5</td></tr><tr><td>3</td><td>4</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>5</td></tr><tr><td>4</td><td>1</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>6.71×10<sup>-2</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>5</td></tr><tr><td>5</td><td>1</td><td>0.2</td><td>1.02×10<sup>4</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>5</td></tr><tr><td>6</td><td>1</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>π/4</td><td>5</td></tr><tr><td>7</td><td>1</td><td>0.2</td><td>1.00×10<sup>3</sup></td><td>1.02×10<sup>3</sup></td><td>1.95×10<sup>-3</sup></td><td>6.71×10<sup>-3</sup></td><td>3π/4</td><td>10</td></tr></tbody></table>

field are calculated from the mixed phase method and are defined as:

$$ \rho = \rho_1 + (\rho_2 - \rho_1) \phi, \quad (5) $$

$$ \mu = \mu_1 + (\mu_2 - \mu_1) \phi, \quad (6) $$

where physical properties $ρ_1$, $ρ_2$ are the densities and $μ_1$, $μ_2$ are the dynamic viscosities of Fluid 1 and Fluid 2, respectively. At both inlets, laminar flow conditions with prescribed volumetric flow rates are imposed. At the outflow boundary, a prescribed pressure and no viscous stress condition are set. The following expression is used for the walls of the channel:

$$ u_y = \beta u, \quad (7) $$

where $β$ is the wall slip length.

The definitions of the Reynolds number (Re), capillary number (Ca), density ratio ($ρ_r$), viscosity ratio ($ρ_t$), diameter ratio ($D_r$), angle ratio ($θ_r$), and slip ratio ($β_r$) are given below:

$$ Re = \frac{\rho v_{\max} w}{\mu}, \quad (8) $$

$$ Ca = \frac{\mu v_{\max}}{\sigma}, \quad (9) $$

$$ \rho_r = \frac{\rho_c}{\rho_d}, \quad (10) $$

$$ \mu_r = \frac{\mu_c}{\mu_d}, \quad (11) $$

$$ D_r = \frac{D}{w}, \quad (12) $$

$$ \beta_r = \frac{\beta}{\beta_0}, \quad (13) $$

$$ \theta_r = \frac{\theta}{\theta_0}, \quad (14) $$

where subscripts c and d denote continuous and discrete phases respectively. As well, parameters D and $β$ are the slip length and the equivalent droplet diameter which are calculated from the droplet surface ($A = πD^2/4$). Furthermore, the parameters in Eqs. (13) and (14) which are indicated by 0 are the values for experimental results used for making dimensionless parameters. Although the Reynolds number typically is defined in terms of average velocity, but here in order to maintain the similarity between experiment results and numerical modeling, the Reynolds number is defined in terms of maximum velocity ($v_{\max}$) and the channel width ($w$).

In order to understand the dynamics of droplet formation in a microfluidic T-junction many numerical methods are used such as two-phase level set method [13] and VOF model [46]. In this work, the commercial software is used to solve the governing equations. Flow conditions and values of parameters are tabulated in Table 1. Furthermore, dimensionless numbers of simulations corresponding to flow conditions are presented in Table 2. A collocated grid arrangement is adopted based on physically controlled meshing.

The impact of injection angle in non-Newtonian fluids has been shown through experimentation [46]. Although, the 2D geometry was susceptible to some errors, in which case, but it has a lower

<table><caption>**Table 2**
Dimensionless numbers of simulations corresponding to flow conditions.</caption><thead><tr><th>Case</th><th>Re<sub>c</sub></th><th>Ca<sub>c</sub></th><th>ρ<sub>r</sub></th><th>μ<sub>r</sub></th><th>θ<sub>r</sub></th><th>β<sub>r</sub></th><th>Q<sub>r</sub></th></tr></thead><tbody><tr><td>1</td><td>4.27</td><td>0.033</td><td>0.98</td><td>0.29</td><td>1</td><td>-</td><td>10</td></tr><tr><td>2</td><td>2.14</td><td>0.016</td><td>0.98</td><td>0.29</td><td>1</td><td>1</td><td>5</td></tr><tr><td>3</td><td>8.53</td><td>0.065</td><td>0.98</td><td>0.29</td><td>1</td><td>1</td><td>20</td></tr><tr><td>4</td><td>0.214</td><td>0.16</td><td>0.98</td><td>10</td><td>1</td><td>1</td><td>5</td></tr><tr><td>5</td><td>21.4</td><td>0.016</td><td>10</td><td>0.29</td><td>1</td><td>1</td><td>5</td></tr><tr><td>6</td><td>2.14</td><td>0.016</td><td>0.98</td><td>0.29</td><td>1/3</td><td>1</td><td>5</td></tr><tr><td>7</td><td>2.14</td><td>0.016</td><td>0.98</td><td>0.29</td><td>1</td><td>2</td><td>5</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Geometry of the T-junction.</figcaption></figure>

computational cost than a 3D model and in the literature just the 2D plot of the continuous versus the dispersed phase contours has been plotted [23] which shows the curvature in depth is negligible and the 2D assumption is sufficient to exclude the general characteristics of the system. The current 2D numerical method is validated by its comparison with the experimental results of the droplet formation in a 3D rectangular T-shaped microchannel junction presented by van der Graaf et al. [22]. Droplet generation experiments were achieved with 1,6-hexanediol diacrylate as the dispersed phase in the main stream of a 2 wt% poly (vinyl alcohol) aqueous solution [48]. The surface tension coefficient was taken as $5 \times 10^{-3}$ N/m. The no-slip boundary conditions and a contact angle of an oil droplet on a surface in a water phase of 135° were applied at the walls. A uniform grid with $Δx = Δy = 0.005$ mm was applied in numerical simulations. The difference in produced droplet size between this grid and a finer grid ($Δx = Δy = 0.001$ mm) was 2% so that sufficient grid refinement was performed. Figure 3 displays some images from droplet generation experiments along with numerical results at flow rates of the continuous phase and the to-be-dispersed phase of 2 mL/h and 0.2 mL/h, respectively. The droplet bends before separation from the stream of Fluid 2, and the Fluid 1 continuous phase imposes pressure on it (see Fig. 2). This applied pressure, which is a function of droplet shape and both fluid properties, is resisted by the interfacial tension in the triple point contact line. The shapes of the droplets at the similar time intervals are in agreement as shown in Fig. 3. Due to the 2D nature of the simulations, the numerical results under the above mentioned conditions overestimate the experimental results to some extent.

The resulting diameter for a droplet is calculated for various main stream flow rates (based on the milliliters per hour) and is compared with the experimental results of van der Graaf et al. [22]
<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Images of numerical results (left) versus experimental results (right) for the droplet detachment at a rectangular T-junction with a 2.5 ms time step (Case 1).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Droplet diameter as a function of main stream maximum velocity for the contact angle of 3π/4.</figcaption></figure>

in Fig. 4. This benchmark problem has been extensively studied experimentally, computationally, and analytically. Although this problem has a 3D nature, but in order to capture the relevant physics, 2D simulations with lower numerical cost is used. By using 2D simulations, the qualitative information is aimed to study scaling, trends, etc. The first droplet is formed after approximately 5 ms. The effective droplet diameter can be found by equating the secondary phase total area ($\varphi > 0$) to the diameter of a cylindrical droplet with the same cross section area as the formed droplet. For the case of the problem shown in Fig. 2, the results show that the effective droplet diameter is about 0.13 mm. The results are in a reasonable agreement with experiments (see Fig. 4). It can be further seen that the numerical modeling results slightly overestimate the droplet diameter at low main stream velocities, and at higher main stream velocities, the simulation results slightly underestimate the droplet diameter.

Figure. 5 shows fluid interfaces at various times. The difference between Figs. 5 and 3 is that Fig. 5 corresponds to lower to-be-dispersed and continuous inlet velocity values. The main channel has one millimeter length and both inlets have a width of 100 µm. The injection angle in Fig. 5 is $\theta = \pi/2$. The distance of inlet 1 from the T-junction is 0.2 mm, and the vertical distance of the inlet 2 from that location is 0.4 mm. The parameter $\varepsilon$ is fixed at 5 µm, and the mobility is 0.05. Two steps contribute to the final dynamics of a droplet, as displayed in Fig. 5. The Fluid 2 is initially at rest at a vertical distance of 0.1 mm from the T-junction. After 5 ms, it takes its developed shape with the contact angle of 3π/4 and a slip length of 5 µm. At $t = 80$ ms, its front touches the main fluid steam and starts to change its symmetrical shape. At $t = 0.12$ s, the triple point reaches the junction, and the stream is ready to form the droplet. Although the tip of the dispersed phase

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Dynamic observation of break-up process of a plug at the perpendicular junction of the microchannels in the growing stage (Case 2).</figcaption></figure>

enters the main channel, the secondary flow does not block the main stream. At this moment, the “growth length” of the plug is equal to zero. As shown in Fig. 5, at 0.18 s, the right triple point is fixed at the previous location in the right corner of the T-junction, but the left triple point advances on the left wall. At $t = 0.24$ s, the Fluid 2 touches the bottom wall. With time, the blockage of the main stream is developed and results in a plug in the flow. Then, the increased pressure in the continuous phase fluid starts to “squeeze” the neck of the dispersed thread. The thickness of the neck (of a characteristic width of $d$) decreases at a rate, which is approximately equal to the mean speed of the continuous fluid. The contours $t = 0.5$ s and $t = 1$ s show that the Fluid 1 cannot trim the Fluid 2 stream, and creeping flow exists in parallel with it. Finally subsequent to $t = 1$ s the droplet is formed when the main stream of water trims the cross flow of oil.

Having confirmed the validity by comparison with the experimental results and giving insight into droplet formation, the next focus is on the effects of significant parameters on droplet formation. Figure 6 shows the effect of flow injection angle ($\theta = \pi/10$ to $9\pi/10$ by the step of $\pi/10$). All the images correspond to the same time ($t = 5$ ms). Droplet size is determined from this figure, and is displayed in Fig. 7. As shown in Fig. 7, the profile has a minimum at an angle of $\theta = \pi/3$.

Since $90^\circ$ is the standard for a T-junction, looking at the results, varying the angle at which the oil is driven into the water seems to have a slight effect on the droplet diameter, usually on the order of 10%. Overall it would appear that certain angles give smaller droplets; however, it seems that just as easy (or likely easier) to vary the flow rates to obtain the same result but for obtaining precisely regulated droplet sizes the current study can be used. Typically, according to their results, a $90^\circ$ channel tends to yield the maximum diameter but that is not the best result. Because based on the application maybe different diameter is needed for the experiments and the use of various angles plays the role of producing droplet with higher precision and finer control on the size will be achieved.

Capillary number of the continuous phase is an important parameter to classify two-phase flow patterns in T-junction microchannels. Xu et al. [20] found three typical flow regimes for the dispersed phase: squeezing regime ($A > 2W^2$), transient regime ($W^2 < A < 2W^2$), and dripping regime ($A < W^2$), where W
<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Images of numerical results (left) versus experimental results (right) for the droplet detachment at a rectangular T-junction with a 2.5 ms time step (Case 1).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Droplet diameter as a function of main stream maximum velocity for the contact angle of 3π/4.</figcaption></figure>

in Fig. 4. This benchmark problem has been extensively studied experimentally, computationally, and analytically. Although this problem has a 3D nature, but in order to capture the relevant physics, 2D simulations with lower numerical cost is used. By using 2D simulations, the qualitative information is aimed to study scaling, trends, etc. The first droplet is formed after approximately 5 ms. The effective droplet diameter can be found by equating the secondary phase total area ($\varphi > 0$) to the diameter of a cylindrical droplet with the same cross section area as the formed droplet. For the case of the problem shown in Fig. 2, the results show that the effective droplet diameter is about 0.13 mm. The results are in a reasonable agreement with experiments (see Fig. 4). It can be further seen that the numerical modeling results slightly overestimate the droplet diameter at low main stream velocities, and at higher main stream velocities, the simulation results slightly underestimate the droplet diameter.

Figure. 5 shows fluid interfaces at various times. The difference between Figs. 5 and 3 is that Fig. 5 corresponds to lower to-be-dispersed and continuous inlet velocity values. The main channel has one millimeter length and both inlets have a width of 100 µm. The injection angle in Fig. 5 is $\theta = \pi/2$. The distance of inlet 1 from the T-junction is 0.2 mm, and the vertical distance of the inlet 2 from that location is 0.4 mm. The parameter $\varepsilon$ is fixed at 5 µm, and the mobility is 0.05. Two steps contribute to the final dynamics of a droplet, as displayed in Fig. 5. The Fluid 2 is initially at rest at a vertical distance of 0.1 mm from the T-junction. After 5 ms, it takes its developed shape with the contact angle of 3π/4 and a slip length of 5 µm. At $t = 80$ ms, its front touches the main fluid steam and starts to change its symmetrical shape. At $t = 0.12$ s, the triple point reaches the junction, and the stream is ready to form the droplet. Although the tip of the dispersed phase

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Dynamic observation of break-up process of a plug at the perpendicular junction of the microchannels in the growing stage (Case 2).</figcaption></figure>

enters the main channel, the secondary flow does not block the main stream. At this moment, the “growth length” of the plug is equal to zero. As shown in Fig. 5, at 0.18 s, the right triple point is fixed at the previous location in the right corner of the T-junction, but the left triple point advances on the left wall. At $t = 0.24$ s, the Fluid 2 touches the bottom wall. With time, the blockage of the main stream is developed and results in a plug in the flow. Then, the increased pressure in the continuous phase fluid starts to “squeeze” the neck of the dispersed thread. The thickness of the neck (of a characteristic width of $d$) decreases at a rate, which is approximately equal to the mean speed of the continuous fluid. The contours $t = 0.5$ s and $t = 1$ s show that the Fluid 1 cannot trim the Fluid 2 stream, and creeping flow exists in parallel with it. Finally subsequent to $t = 1$ s the droplet is formed when the main stream of water trims the cross flow of oil.

Having confirmed the validity by comparison with the experimental results and giving insight into droplet formation, the next focus is on the effects of significant parameters on droplet formation. Figure 6 shows the effect of flow injection angle ($\theta = \pi/10$ to $9\pi/10$ by the step of $\pi/10$). All the images correspond to the same time ($t = 5$ ms). Droplet size is determined from this figure, and is displayed in Fig. 7. As shown in Fig. 7, the profile has a minimum at an angle of $\theta = \pi/3$.

Since $90^\circ$ is the standard for a T-junction, looking at the results, varying the angle at which the oil is driven into the water seems to have a slight effect on the droplet diameter, usually on the order of 10%. Overall it would appear that certain angles give smaller droplets; however, it seems that just as easy (or likely easier) to vary the flow rates to obtain the same result but for obtaining precisely regulated droplet sizes the current study can be used. Typically, according to their results, a $90^\circ$ channel tends to yield the maximum diameter but that is not the best result. Because based on the application maybe different diameter is needed for the experiments and the use of various angles plays the role of producing droplet with higher precision and finer control on the size will be achieved.

Capillary number of the continuous phase is an important parameter to classify two-phase flow patterns in T-junction microchannels. Xu et al. [20] found three typical flow regimes for the dispersed phase: squeezing regime ($A > 2W^2$), transient regime ($W^2 < A < 2W^2$), and dripping regime ($A < W^2$), where W
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Schematic illustration of the break-up process of a plug at the inclined junction of the microchannels in the growing stage (Case 2).</figcaption></figure>

is the width of the main channel, and A is the cross sectional area of the droplet. When the two-phase flow is in the squeezing regime, the tip of the dispersed fluid blocks almost the entire cross-section of the main duct, leads to hindering the flow of the continuous phase. As a result, a “plug-like” droplet is formed in the main channel. When the two-phase flow pattern falls in the transient regime, the dynamics of breakup is dominated by both the pressure drop across the droplet and the shear stress on the droplet. When the two-phase flow pattern falls in the dripping regime, the continuous phase flows faster through the gap between the droplet and the wall of the main channel, and the shear stress is sufficiently large to cause the dispersed phase to pinch off at the T-junction microchannel.

Figure 8 shows the process of the droplet formation at higher (compared to the conditions of cases 1 and 2 in Table 1) main stream flow rates. Bashir et al. [13] and Timgren et al. [49] showed droplet diameter depends on cross-flow velocity, but they have not considered the injection angle. As can be seen, the diameter of the droplet decreases with flow rate of the main stream phase at a fixed velocity of the continuous phase. Figure 9 shows calculated final diameter of the droplet as a function of injection angle. The $Q_c$ is changed, but $Q_d$ is constant for each of the data points. At a small injection angle of $\theta = \pi/10$, the difference in diameter of the droplet between low and high main stream flow rates is maximum.

Figure 3 of Bashir et al. [13] is very similar to Fig. 4 of the current paper; the only thing they did not do was the varying of the channel angle.

What has been shown is here that the angle at which the oil comes into the water channel smoothly varies the final droplet diameter by about 20 µm. Even at high flow rates, the injection angle has little effect on the size of the droplet.

The effect of the viscosity ratio is investigated in Fig. 10. The physical value of the dispersed phase viscosity is fixed at the

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Droplet size as a function of the injection angle (Case 2).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** The effect of the flow rate ratio on droplet formation process (Case 3).</figcaption></figure>

benchmark values of Fig. 5, and a different viscosity ratio of $\mu_r = 10$ is used in the simulations. Garstecki et al. [21] indicated that at low values of the capillary number, typically $Ca < 10^{-2}$, the shear stress exerted on the interface of the emerging droplet could be ignored, and the squeezing pressure plays a dominant role for droplet breakup. The droplet blocks almost the entire cross-section of the main channel, leading to an increasing pressure in the upstream of the emerging droplet. The neck of the droplet squeezes at a rate proportional to the flow rate of the continuous fluids, and this growth, in turn, precedes the droplet at a rate proportional to dispersed phase flow rate. The head of the dispersed phase is elon-gated into the main channel, until the neck of the dispersed phase
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Schematic illustration of the break-up process of a plug at the inclined junction of the microchannels in the growing stage (Case 2).</figcaption></figure>

is the width of the main channel, and A is the cross sectional area of the droplet. When the two-phase flow is in the squeezing regime, the tip of the dispersed fluid blocks almost the entire cross-section of the main duct, leads to hindering the flow of the continuous phase. As a result, a “plug-like” droplet is formed in the main channel. When the two-phase flow pattern falls in the transient regime, the dynamics of breakup is dominated by both the pressure drop across the droplet and the shear stress on the droplet. When the two-phase flow pattern falls in the dripping regime, the continuous phase flows faster through the gap between the droplet and the wall of the main channel, and the shear stress is sufficiently large to cause the dispersed phase to pinch off at the T-junction microchannel.

Figure 8 shows the process of the droplet formation at higher (compared to the conditions of cases 1 and 2 in Table 1) main stream flow rates. Bashir et al. [13] and Timgren et al. [49] showed droplet diameter depends on cross-flow velocity, but they have not considered the injection angle. As can be seen, the diameter of the droplet decreases with flow rate of the main stream phase at a fixed velocity of the continuous phase. Figure 9 shows calculated final diameter of the droplet as a function of injection angle. The $Q_c$ is changed, but $Q_d$ is constant for each of the data points. At a small injection angle of $\theta = \pi/10$, the difference in diameter of the droplet between low and high main stream flow rates is maximum.

Figure 3 of Bashir et al. [13] is very similar to Fig. 4 of the current paper; the only thing they did not do was the varying of the channel angle.

What has been shown is here that the angle at which the oil comes into the water channel smoothly varies the final droplet diameter by about 20 µm. Even at high flow rates, the injection angle has little effect on the size of the droplet.

The effect of the viscosity ratio is investigated in Fig. 10. The physical value of the dispersed phase viscosity is fixed at the

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Droplet size as a function of the injection angle (Case 2).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** The effect of the flow rate ratio on droplet formation process (Case 3).</figcaption></figure>

benchmark values of Fig. 5, and a different viscosity ratio of $\mu_r = 10$ is used in the simulations. Garstecki et al. [21] indicated that at low values of the capillary number, typically $Ca < 10^{-2}$, the shear stress exerted on the interface of the emerging droplet could be ignored, and the squeezing pressure plays a dominant role for droplet breakup. The droplet blocks almost the entire cross-section of the main channel, leading to an increasing pressure in the upstream of the emerging droplet. The neck of the droplet squeezes at a rate proportional to the flow rate of the continuous fluids, and this growth, in turn, precedes the droplet at a rate proportional to dispersed phase flow rate. The head of the dispersed phase is elon-gated into the main channel, until the neck of the dispersed phase
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Droplet size as a function of the injection angle for two different main stream flow rates (Case 3).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** The effect of viscosity ratio ($\mu_1/\mu_2 = 10$) on droplet formation process (Case 4).</figcaption></figure>

becomes thin enough and consequently breaks up into a droplet. However, with the increase in injection angle, the dispersed phase is pulled downstream to form a longer laminar segment before the droplet breaks up. Figure 11 displays the effective diameter of the generated droplet for two different viscosity ratios. As can be seen, the produced droplet size decreases with viscosity ratio. Furthermore, at this higher viscosity ratio there exists a maximum in the droplet size at the injection angle of $2\pi/180$.

As mentioned in the Introduction section, capillary number based on the continuous phase is one of the most important parameters in generation of isolated droplets, which make a trade-off between the ratio of viscous stresses and capillary pressure.

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Droplet size as the function of injection angle for two different viscosity ratios (Case 4).</figcaption></figure>

As the channel size shrinks, Reynolds number becomes smaller so that inertial effects are dominated by viscous stresses. As a result, the combination of injection angle and density is not expected to have any significant effect on droplet generation in small scale. Figure 12 shows the effect of injection angle on the case of higher main stream density on the droplet detachment point. For all the injection angles, the droplet is pinched off at the corner of the T-junction. In this case, the inertial and capillary pressures are more dominant over viscous stresses. This effect is more amplified for density ratios higher than one so that it is more appropriate to use Weber number ($Re \times Ca$) to parametrize droplet breakup processes in this case. Furthermore, in the studies, where the lighter fluid is at the bottom, the injected phase is heavier, and the buoyancy effect is important at the top, Bond number should be considered. On the other hand, for the current geometry and sizes, the Bond number is on order of $O(10^{-3})$ even for air bubbles in water. Figure 13 shows the effect of injection angle on the diameter of generated droplet for two different main stream densities. This figure shows the difference in produced droplet's diameters for cases 1 and 5 (Table 1). As shown, an increase in the density ratio leads to smaller droplet size, but the higher density provides inertia to the process, where the some injection angles do not generate any significant change in generated droplets.

The wall contact angle of the droplet acts an important role on the final diameter of the droplet at low capillary number (plug flow regime), where the size of the droplet is very large. As a result, a large portion of the droplet sticks to the micro-channel surface. As the hydrophilic surfaces attract more liquid phase than the hydrophobic surfaces, the droplet generated on the hydrophobic walls is longer than on the hydrophilic channel walls. The wetting conditions used for the simulations in Fig. 5 were based on a measured equilibrium contact angle of a droplet in equilibrium. As shown in Fig. 14, the decrease in contact angle from $3\pi/4$ to $\pi/4$ causes a significant difference in droplet formation process and leads to enhanced necking near the inlet and droplet formation at downward positions. However, in order to obtain approximate spherical droplets, the contact angle between the droplet and the channel walls should not be too small. When the contact angle between the droplet and channel wall reaches 45°, a proximate semi-ellipse attaching to the upper wall of the channel is observed in the microchannel. The longer length for the dispersed phase leads to smaller diameter droplets as shown in Fig. 15, which displays the difference in produced droplet diameters for cases 1 and 6 mentioned (Table 1).
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Droplet size as a function of the injection angle for two different main stream flow rates (Case 3).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** The effect of viscosity ratio ($\mu_1/\mu_2 = 10$) on droplet formation process (Case 4).</figcaption></figure>

becomes thin enough and consequently breaks up into a droplet. However, with the increase in injection angle, the dispersed phase is pulled downstream to form a longer laminar segment before the droplet breaks up. Figure 11 displays the effective diameter of the generated droplet for two different viscosity ratios. As can be seen, the produced droplet size decreases with viscosity ratio. Furthermore, at this higher viscosity ratio there exists a maximum in the droplet size at the injection angle of $2\pi/180$.

As mentioned in the Introduction section, capillary number based on the continuous phase is one of the most important parameters in generation of isolated droplets, which make a trade-off between the ratio of viscous stresses and capillary pressure.

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Droplet size as the function of injection angle for two different viscosity ratios (Case 4).</figcaption></figure>

As the channel size shrinks, Reynolds number becomes smaller so that inertial effects are dominated by viscous stresses. As a result, the combination of injection angle and density is not expected to have any significant effect on droplet generation in small scale. Figure 12 shows the effect of injection angle on the case of higher main stream density on the droplet detachment point. For all the injection angles, the droplet is pinched off at the corner of the T-junction. In this case, the inertial and capillary pressures are more dominant over viscous stresses. This effect is more amplified for density ratios higher than one so that it is more appropriate to use Weber number ($Re \times Ca$) to parametrize droplet breakup processes in this case. Furthermore, in the studies, where the lighter fluid is at the bottom, the injected phase is heavier, and the buoyancy effect is important at the top, Bond number should be considered. On the other hand, for the current geometry and sizes, the Bond number is on order of $O(10^{-3})$ even for air bubbles in water. Figure 13 shows the effect of injection angle on the diameter of generated droplet for two different main stream densities. This figure shows the difference in produced droplet's diameters for cases 1 and 5 (Table 1). As shown, an increase in the density ratio leads to smaller droplet size, but the higher density provides inertia to the process, where the some injection angles do not generate any significant change in generated droplets.

The wall contact angle of the droplet acts an important role on the final diameter of the droplet at low capillary number (plug flow regime), where the size of the droplet is very large. As a result, a large portion of the droplet sticks to the micro-channel surface. As the hydrophilic surfaces attract more liquid phase than the hydrophobic surfaces, the droplet generated on the hydrophobic walls is longer than on the hydrophilic channel walls. The wetting conditions used for the simulations in Fig. 5 were based on a measured equilibrium contact angle of a droplet in equilibrium. As shown in Fig. 14, the decrease in contact angle from $3\pi/4$ to $\pi/4$ causes a significant difference in droplet formation process and leads to enhanced necking near the inlet and droplet formation at downward positions. However, in order to obtain approximate spherical droplets, the contact angle between the droplet and the channel walls should not be too small. When the contact angle between the droplet and channel wall reaches 45°, a proximate semi-ellipse attaching to the upper wall of the channel is observed in the microchannel. The longer length for the dispersed phase leads to smaller diameter droplets as shown in Fig. 15, which displays the difference in produced droplet diameters for cases 1 and 6 mentioned (Table 1).
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** The effect of density ratio ($ρ_1/ρ_2 = 10$) on droplet formation process (Case 5).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Droplet size as a function of the injection angle for two different density ratios (Case 5).</figcaption></figure>

Furthermore, as known a liquid droplet sticks to a solid wall and deforms in the presence of the continuous phase flow to form an advancing and a receding dynamic contact angle until the drag overcomes the contact angle hysteresis force [50]. Since the advancing and receding contact angles affect the problem, the Hoffman–Jiang correlation [51] for advancing contact angle and Hoffman–Tan law [52] for the receding contact angle shows dependency of these angles on the capillary number. Therefore, the wall contact angle effects on droplet diameter change with capillary

<figure><img src="image_17.png" /><figcaption>**Fig. 14.** The effect of contact angle on droplet formation process ratio (Case 6).</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Droplet size as a function of injection angle for two different contact angles (Case 6).</figcaption></figure>

number, shear rates, adhesive, gravitational, and buoyancy forces of the drop [53].

The slip ratio of the channel walls affects the final diameter of the droplet. Changing interfacial characteristics of the microchannel may cause interfacial slip at the wall interface, which reduces shear and can allow the droplet to grow particularly for cylindrical-shaped droplets. Figure 16 reveals the effect of the increase in slip length ($β = 10 μm$) on droplet formation process. As shown in all the cases the droplet forms in nearer place to the injection place and the diameter becomes bigger with slip length. The droplet size
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** The effect of density ratio ($ρ_1/ρ_2 = 10$) on droplet formation process (Case 5).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Droplet size as a function of the injection angle for two different density ratios (Case 5).</figcaption></figure>

Furthermore, as known a liquid droplet sticks to a solid wall and deforms in the presence of the continuous phase flow to form an advancing and a receding dynamic contact angle until the drag overcomes the contact angle hysteresis force [50]. Since the advancing and receding contact angles affect the problem, the Hoffman–Jiang correlation [51] for advancing contact angle and Hoffman–Tan law [52] for the receding contact angle shows dependency of these angles on the capillary number. Therefore, the wall contact angle effects on droplet diameter change with capillary

<figure><img src="image_17.png" /><figcaption>**Fig. 14.** The effect of contact angle on droplet formation process ratio (Case 6).</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Droplet size as a function of injection angle for two different contact angles (Case 6).</figcaption></figure>

number, shear rates, adhesive, gravitational, and buoyancy forces of the drop [53].

The slip ratio of the channel walls affects the final diameter of the droplet. Changing interfacial characteristics of the microchannel may cause interfacial slip at the wall interface, which reduces shear and can allow the droplet to grow particularly for cylindrical-shaped droplets. Figure 16 reveals the effect of the increase in slip length ($β = 10 μm$) on droplet formation process. As shown in all the cases the droplet forms in nearer place to the injection place and the diameter becomes bigger with slip length. The droplet size
<figure><img src="image_19.png" /><figcaption>**Fig. 16.** The effect of the increase in slip length on droplet formation process (Case 7).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 17.** Droplet size as a function of the injection angle for two different slip lengths (Case 7).</figcaption></figure>

is found to increase with slip length. However, there is not much change with injection angle (Fig. 17).

Varying the slip length had an impact, which has not previously been examined and here the overall impact is shown.

Droplet formation in T-junction microchannel configurations was investigated in this study. Newtonian solutions, which are promising for application in biological and medical areas, were used to study the effect of the inlet angles. The results suggested that the effect of the Newtonian fluids were not as marked as expected although differences particularly with $\theta = 18^\circ$ was observed. Droplet formation was imaged at 3000 frames/second to

highlight the effect of acute and obtuse inlet angles on the droplet forming mechanism. The effects of injection angle and other parameters affecting the formation of the droplets including the flow rate ratio, viscosity ratio, density ratio, contact angle, and wetting properties of the channel walls were systematically analyzed. Fluid properties, flow conditions, and wetting property of channel walls had a significant effect on the effective diameter of the droplet. The variation in injection angle changes the shape of the droplet from circular to elliptical and even causes parallel flow without distinguishable droplets. Higher main streamflow rate ratio (continuous phase divided by to be dispersed phase) leads to smaller droplet diameters. With the increase in flow rate of continuous phase at a fixed velocity of the to-be-dispersed phase, the detachment point of the droplet moved from the corner of T-junction to the downstream of the corner, respectively. Based on the various injection angles of the geometry of the T-junction microchannel, the injection angle of $\theta = 9\pi/10$ has the minimum droplet size among different injection angles. Smaller droplets form with an increase in density ratio and viscosity ratio. A decrease in contact angle causes a significant decrease in droplet diameter. An increase in slip length accelerates the droplet formation process and leads to bigger droplet diameters. Injection angles near the perpendicular and parallel conditions have an increasing effect on the diameter of the generated droplet.

## References

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[2] H.A. Stone, A.D. Stroock, A. Ajdari, Engineering flows in small devices: microfluidics towards a lab-on-a-chip, Ann. Rev. Fluid Mech. 36 (2004) 381–411.

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[8] A. Gupta, R. Kumar, Effect of geometry on droplet formation in the squeezing regime in a microfluidic T-junction, Microfluid. Nanofluid. 8 (2010) 799–812.

[9] A. Gupta, R. Kumar, Flow regime transition at high Capillary numbers in a microfluidic T-junction: viscosity contrast and geometry effect, Phys. Fluids 22 (2010) 122001.

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[12] A. Bedram, A.E. Darabi, A. Moosavi, et al., Numerical investigation of an efficient method (T-junction with valve) for producing unequal-sized droplets in micro- and nano-fluidic systems, ASME J. Fluids Eng. 137 (2015) 031202.

[13] S. Bashir, J.M. Rees, W.B. Zimmerman, Simulations of microfluidic droplet formation using the two-phase level set method, Chem. Eng. Sci. 66 (2011) 4733–4741.

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[16] P. Garstecki, M.J. Fuerstman, G.M. Whitesides, Nonlinear dynamics of a flowfocusing bubble generator: an inverted dripping faucet, Phys. Rev. Lett. 94 (2005) 234502.

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[18] P.A. Romero, A.R. Abate, Flow focusing geometry generates droplets through a plug and squeeze mechanism, Lab Chip 12 (2012) 5130–5132.

[19] C. Zhou, P. Yue, J.J. Feng, Formation of simple and compound drops in microfluidic devices, Phys. Fluids 18 (2006) 092105.

[20] J.H. Xu, S.W. Li, J. Tan, et al., Correlations of droplet formation in T-junction microfluidic devices: from squeezing to dripping, Microfluid. Nanofluid. 5 (2008) 711–717.
<figure><img src="image_19.png" /><figcaption>**Fig. 16.** The effect of the increase in slip length on droplet formation process (Case 7).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 17.** Droplet size as a function of the injection angle for two different slip lengths (Case 7).</figcaption></figure>

is found to increase with slip length. However, there is not much change with injection angle (Fig. 17).

Varying the slip length had an impact, which has not previously been examined and here the overall impact is shown.

Droplet formation in T-junction microchannel configurations was investigated in this study. Newtonian solutions, which are promising for application in biological and medical areas, were used to study the effect of the inlet angles. The results suggested that the effect of the Newtonian fluids were not as marked as expected although differences particularly with $\theta = 18^\circ$ was observed. Droplet formation was imaged at 3000 frames/second to

highlight the effect of acute and obtuse inlet angles on the droplet forming mechanism. The effects of injection angle and other parameters affecting the formation of the droplets including the flow rate ratio, viscosity ratio, density ratio, contact angle, and wetting properties of the channel walls were systematically analyzed. Fluid properties, flow conditions, and wetting property of channel walls had a significant effect on the effective diameter of the droplet. The variation in injection angle changes the shape of the droplet from circular to elliptical and even causes parallel flow without distinguishable droplets. Higher main streamflow rate ratio (continuous phase divided by to be dispersed phase) leads to smaller droplet diameters. With the increase in flow rate of continuous phase at a fixed velocity of the to-be-dispersed phase, the detachment point of the droplet moved from the corner of T-junction to the downstream of the corner, respectively. Based on the various injection angles of the geometry of the T-junction microchannel, the injection angle of $\theta = 9\pi/10$ has the minimum droplet size among different injection angles. Smaller droplets form with an increase in density ratio and viscosity ratio. A decrease in contact angle causes a significant decrease in droplet diameter. An increase in slip length accelerates the droplet formation process and leads to bigger droplet diameters. Injection angles near the perpendicular and parallel conditions have an increasing effect on the diameter of the generated droplet.

References

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