A novel pressure-fluctuation-based method is proposed for measuring the size of microdroplets without the need for visualization through a microscope. In the present work, numerical simulations are carried out in a co-flow geometry to verify this concept. First, the droplet formation frequency is determined by applying the fast Fourier transform to measured pressure fluctuation data with respect to time at any point on the outer wall. Then, the size of dispersed phase microdroplets is determined using a relationship between dispersed-phase flow rate and the droplet formation frequency. The droplet size obtained using the pressure fluctuation method is compared with that from the volume fraction method, and it is found that the error is less than 5%. The deviation is attributed to the formation of satellite droplets in the simulations. The relationship between the nondimensional parameters flow-rate ratio, capillary number, and normalized droplet diameter is investigated systematically, and empirical relations are obtained through power-law regression. The effects of interfacial tension, flow-rate ratio, and viscosity ratio on the magnitude of pressure oscillations and the corresponding droplet size are studied. All the parameters are found to have significant effects on droplet size. The ability of the proposed method to predict microdroplet size is significant with regard to potential applications to biomedical systems and drug delivery.

## I. INTRODUCTION

Generating microdroplets has become an essential task for applications such as drug delivery,^{1} chemical engineering,^{2} biomedicine,^{3} micro-mixing,^{4} and research in multiple disciplines.^{5–7} In drug delivery, maintaining a balance between speed of droplet production and precision of droplet volume is essential for releasing the required amount of drug, and extensive research is going on in this area to improve treatment efficiency.^{8,9} Conventional techniques combining high-resolution imaging with advanced image processing enable precise analysis of droplet behavior and properties within microfluidic systems for various applications in fields such as cosmetics,^{10} flow cytometry,^{11} and other areas of biology and medicine.^{12}

Even though microfluidic experiments are effective in the biomedical field, detecting droplet size is the most time-consuming process. Therefore, researchers now prefer to employ manual measurements together with software such as Image J because of its simplicity of use.^{13} In the case of a ferrofluid, the effects of an applied magnetic field on the fluid flow field during droplet formation and on the final droplet size have been investigated both numerically and experimentally.^{14} A method for detection and quantification of droplets has been proposed based on measurements of impedance changes induced by droplets in microstrip transmission lines or thin film electrodes.^{15,16}

A deep-learning method, Segfit, based on a convolutional neural network (CNN), has been proposed as an efficient method for rapid and accurate analysis of microfluidic droplet size.^{17} Another deep-learning application, DropTrack, provides rapid and automated monitoring of the size distributions and structure of emulsions in microfluidic systems by tracking droplets in a fast-moving series of images.^{18} Through the use of an adaptive neural fuzzy inference system (ANFIS), it is possible to analyze complex correlations between input and output parameters to predict droplet sizes.^{19} In this application, the ANFIS model is trained and validated using experimental data obtained under different conditions to determine the droplet size inside a microdevice. A combination of numerical techniques and artificial neural networks has been utilized for the generation and analysis of droplet size.^{20} A machine learning algorithm has also been developed to increase droplet production efficiency by applying an automated droplet tracking technique to microfluidics videos, thereby improving the accuracy in achieving desired droplet sizes.^{21,22}

Determining emulsion droplet size is crucial in a number of biomedical applications, since droplet size has a direct influence on kinetics and dynamics, thus affecting the efficiency of biological interactions and drug delivery.^{23} The size of the droplets in a microemulsion system determines its stability and effectiveness for a wide range of applications.^{24}

A method has been proposed for determining the pressure drop across droplets using an optical tweezer to monitor the displacement of beads.^{25} The squeezing mechanism of droplets in a T-junction was investigated experimentally by analyzing pressure oscillations created in the continuous-phase fluid due to blockage of the channel by dispersed-phase fluid during droplet formation.^{26} Similar experiments were performed in a flow-focusing channel, and it was found that the pressure fluctuations were nearly identical to those in the T-junction.^{27} Directly investigation of pressure fluctuations during drop production in a co-flow microfluidic system has provided essential insights with important implications for microfluidic device design in various domains, including drug delivery, chemical synthesis, and biotechnology.^{28}

In the present work, we simulate the behavior of uniform-sized droplets in a co-flow geometry, and we explore a pressure-fluctuation-based method for determining droplet size and the effects of parameters including capillary number, flow-rate ratio, density ratio, and viscosity ratio.

The remainder of the paper is structured as follows. First, we establish the geometry for the numerical simulations together with a mathematical model. A mesh-independence study is performed, followed by the simulations themselves. The droplet size determined by our pressure-fluctuation method is then compared with that from the volume fraction method.

## II. GEOMETRY

As shown in Fig. 1, we construct a co-flow 2D axisymmetric geometry in COMSOL Multiphysics software. The lengths and radii of the outer and inner tubes are denoted by *L*_{1}, *L*_{2}, *r*_{1}, and *r*_{2}, respectively. The outer and inner tubes contain continuous-phase (CP) and dispersed-phase (DP) fluids, respectively. The flow is assumed to be isothermal and incompressible. The effect of gravity is considered insignificant, owing to the micrometer scale of the vertical gaps in these microchannels.

## III. MATHEMATICAL MODEL

### A. Governing equations

The governing equations for velocity and phase fields in the LSM are given below under the assumptions that the fluids are incompressible, Newtonian, and have a thin interface at the two-phase boundary. Thus, both the continuous and dispersed phases are considered here as immiscible and incompressible fluids.

^{32}

**u**is the velocity vector,

*ρ*(

*ϕ*) is the density,

*t*is the time,

*p*is the pressure,

**τ**is the shear stress,

*F*_{σ}is the interfacial tension force, and

*F*_{g}is the force due to gravity (which, as already mentioned, is neglected in the microfluidic channels). The shear stress

**is related to the shear rate, depending on the type of fluid. In this simulation, the fluids are considered to be Newtonian fluid, and so the shear stress is linearly proportional to the shear rate, with the proportionality constant being the viscosity. The interfacial force**

*τ*

*F*_{σ}depends on the surface tension and the radius of curvature of the interface. In this study, it is taken to be proportional to the surface tension and inversely proportional to the radius of curvature. The latter is determined from the second and first derivatives of the interface shape.

^{32}

*ɛ*

_{ls}is the interface thickness controlling parameter, which determines the width over which the level set function

*ϕ*transitions from one phase to another, and

*γ*is a numerical stabilization parameter used to control the stability and convergence properties of the numerical scheme.

*d*

_{2},

*v*

_{d},

*Q*

_{d}, and

*f*are the inner tube diameter, dispersed-phase velocity, dispersed-phase flow rate, and droplet formation frequency, respectively.

The fluid and flow properties used in the simulations are given in Table I.

Parameter . | Value . |
---|---|

Dynamic viscosity of continuous phase μ_{c} | 0.067 Pa s |

Dynamic viscosity of dispersed phase μ_{d} | 0.001 Pa s |

Density of continuous phase ρ_{c} | 916 kg/m^{3} |

Density of dispersed phase ρ_{d} | 998.2 kg/m^{3} |

Surface tension coefficient σ | 0.05 N/m |

Wetted angle θ_{w} | π/2 rad |

Velocity of continuous phase (constant) V_{c} | 0.1 m/s |

Velocity of dispersed phase V_{d} | 0.05, 0.1, 0.15, 0.2 m/s |

Radius of outer tube r_{1} | 50 µm |

Radius of inner tube r_{2} | 15 µm |

Length of outer tube L_{1} | 800 µm |

Length of inner tube L_{2} | 100 µm |

Parameter . | Value . |
---|---|

Dynamic viscosity of continuous phase μ_{c} | 0.067 Pa s |

Dynamic viscosity of dispersed phase μ_{d} | 0.001 Pa s |

Density of continuous phase ρ_{c} | 916 kg/m^{3} |

Density of dispersed phase ρ_{d} | 998.2 kg/m^{3} |

Surface tension coefficient σ | 0.05 N/m |

Wetted angle θ_{w} | π/2 rad |

Velocity of continuous phase (constant) V_{c} | 0.1 m/s |

Velocity of dispersed phase V_{d} | 0.05, 0.1, 0.15, 0.2 m/s |

Radius of outer tube r_{1} | 50 µm |

Radius of inner tube r_{2} | 15 µm |

Length of outer tube L_{1} | 800 µm |

Length of inner tube L_{2} | 100 µm |

### B. Mesh-independence study

After the geometry has been constructed in COMSOL Multiphysics, the various physical parameters, such as water as the dispersed-phase fluid and oil as the continuous-phase fluid, and the inlet and outlet boundary conditions are assigned as shown in Fig. 1. The mesh is then generated, as shown in Fig. 2. Before applying the finite element method to obtain the solution for the phase field *ϕ*, it is necessary to perform a mesh-independence study. A previous mesh independence study in the case of droplet generation was done using the droplet size as the key parameter.^{33}

The mesh size, which is the distance between two neighboring nodes, is varied from 8 to 4 *µ*m as shown on the horizontal axis of Fig. 3. It is found that the droplet size determined using the volume fraction method does not vary once the mesh size is 5 *µ*m or less. Hence, a 5 *µ*m mesh size is considered for all further simulations. The mesh used here is of tetrahedral shape, which offers better solution convergence with the COMSOL Multiphysics solver using the finite element method. To study 3D droplet formation, the simplest geometry is chosen, namely, a coaxial and axisymmetric one. When the solution is rotated about the central axis, this corresponds to 3D or axisymmetric droplet formation. Since parameters such as velocity and pressure variations can be considered insignificant in the *θ* direction of a circular channel, in the current work, a 2D axisymmetric geometry is adopted instead of a 3D geometry, to reduce computational time. In future work, fully 3D simulations will be carried out to examine droplet non-sphericity.

### C. Model validation

First, the 2D axisymmetric model is established, and parameters are assigned accordingly. The droplets are generated by the co-flow method at different capillary numbers of the continuous phase. The droplet formation frequency with respect to the continuous-phase capillary number is compared with the results of Wu *et al.*,^{34} as shown in Fig. 4. Wu *et al.* applied a constant dispersed-phase inlet velocity and varied the continuous-phase velocity in different simulation runs. In the present work, we vary the dispersed-phase inlet velocity and keep the continuous-phase inlet velocity constant. Nevertheless, the nondimensional numbers, such as capillary numbers, are matched between the studies. From the comparison, it can be concluded that out present approach of generating droplets in simulations using the level-set method (LSM) is valid and appropriate for further study with different parameters.

### D. Determination of droplet size

The flow chart in Fig. 5 gives an overview of the procedure employed to determine droplet size and empirical relations between flow-rate ratio, capillary number, and normalized droplet size.

## IV. RESULTS AND DISCUSSION

Droplets are generated for different flow-rate ratios, and a visual representation of this process is shown in Fig. 6, where the dispersed-phase fluid (gray) is surrounded by continuous-phase fluid (blue).

### A. Measurement of pressure fluctuations and droplet formation frequency

In the present work, since we are determining the droplet size on the basis of the pressure exerted on the wall, we will first examine the reason why pressure fluctuations occur on the wall during the passage of a droplet through the channel. As shown in the inset of Fig. 7, for each flow-rate ratio, there are two pressure dips, with the smaller dip corresponding to the passage of the droplet through the cross-section at the pressure measuring point and the larger dip corresponding to the pressure recorded at the same cross-section during the exit of the same droplet from the channel.

However, the passage of a single droplet from its formation point to the channel exit results in a set of smaller and larger dips in pressure. The peak phase of each pressure curve corresponds to the passage of continuous-phase fluid at the exit of the channel, while the trough corresponds to the presence of dispersed-phase droplets. The FFT of the pressure–time results provides the details of the number of droplets passing through the microchannel in unit time.

The inset of Fig. 7 shows that the number of pressure fluctuations occurring on the wall increases as the flow-rate ratio increases from *Q*_{1} to *Q*_{4}, indicating an increase in the droplet formation frequency. Pressure measurements in the upstream and downstream directions of the channel give the high and low magnitudes of the pressure fluctuations, which enables an experimentalist to adjust the pressure measuring point according to the operating range of the available pressure sensor.

An FFT is applied to the measured pressure–time data, and the droplet formation frequency is calculated. The droplet formation frequencies corresponding to the pressure oscillations are found to be 950, 1250, 1350, and 1400 s^{−1} for flow-rate ratios of 0.5, 1, 1.5, and 2, respectively, as shown in Fig. 8. It can be observed that an increase in the dispersed-phase flow rate at constant continuous-phase flow rate results in a higher droplet formation frequency.

In our approach, the pressure measured near the exit of the microchannel is recorded continuously with a very small time interval of the order of a fraction of a millisecond and then subjected to an FFT. The droplet formation frequency (DFF) can thus be obtained from measurements of the pressure signal itself, as the frequency of maximum amplitude in the FFT, and therefore it can be determined without any need to capture droplet images using a high-resolution microscope and high-speed camera.

### B. Analysis of droplet diameter

The pressure fluctuation method uses the relationship between the dispersed-phase flow rate and DFF given by Eq. (10) to accurately determine the size of the droplets produced in the microfluidic system. The droplet diameter is found to be 41.41, 47.62, 53.13, and 57.77 *μ*m for flow-rate ratios of 0.5, 1, 1.5, and 2, respectively. The droplet diameter is also determined by the volume fraction method in COMSOL Multiphysics. As shown in Fig. 9, the distance between the two opposite extreme points on the interface between dispersed- and continuous-phase volumes in any direction will give the actual diameter of the dispersed-phase volume. For flow-rate ratios of 0.5, 1, 1.5, and 2, the droplet diameter according to the volume fraction method is found to be 39.71, 45.44, 50.77, and 55.44 *μ*m, respectively.

A comparison between the droplet diameters obtained respectively by the pressure fluctuation method and the volume fraction method is presented in Fig. 10. It can be seen that that the droplet diameter determined by the pressure fluctuation method has slightly higher values than that from the volume fraction method. This discrepancy can be attributed to the formation of satellite droplets along with the primary droplets during droplet generation in the simulations. The droplet size determined by the pressure fluctuation method accounts for the overall volume of droplets, including satellite droplets. By contrast, in the volume fraction method, only the size of the primary droplets is measured. Except for this slight deviation, our analysis suggests that the proposed pressure fluctuation method for determining droplet size is valid.

The consistency between the two methods suggests that the pressure fluctuation technique can be adopted as a practical approach for experimentalist to use in determining droplet size.

### C. Analysis of normalized droplet diameter dependence on flow-rate ratio and capillary number of continuous phase

The effect of two nondimensional parameters, namely, the flow-rate ratio and the capillary number of the continuous phase, on the droplet size is analyzed. From Fig. 11(a), it can be observed that an increase in flow-rate ratio at constant capillary number leads to an increase in the normalized droplet size. This indicates a proportional relationship between normalized droplet size and flow-rate ratio. It is also noted that as the capillary number decreases, the droplet size increases because of an increase in interfacial tension between the immiscible fluids. The effect of the continuous-phase capillary number on normalized droplet size at a constant flow-rate ratio is shown in Fig. 11(b), from which it can be seen that there is an inverse relationship between normalized droplet size and continuous-phase capillary number, with the droplet size decreasing as the capillary number increases. It can again be noted that the droplet size increases with increasing flow-rate ratio.

A power-law regression analysis is conducted on the dataset shown in Fig. 11(a) to find the relationship between normalized droplet size and flow-rate ratio at constant capillary number. Figure 12 presents the results of this fit.

The regression fit constants are listed in Table II. The *R*^{2} values of the fit are all almost equal to one, which indicates a strong correlation between the normalized droplet size and flow-rate ratio at constant capillary number.

σ
. | Ca_{c}
. | m_{1}
. | $ec1$ . | R^{2}
. |
---|---|---|---|---|

0.05 | 0.134 | 0.46 | −0.87 | 0.9535 |

0.1 | 0.067 | 0.22 | −0.63 | 0.9999 |

0.15 | 0.045 | 0.17 | −0.54 | 0.9985 |

0.175 | 0.0383 | 0.18 | −0.50 | 0.9992 |

σ
. | Ca_{c}
. | m_{1}
. | $ec1$ . | R^{2}
. |
---|---|---|---|---|

0.05 | 0.134 | 0.46 | −0.87 | 0.9535 |

0.1 | 0.067 | 0.22 | −0.63 | 0.9999 |

0.15 | 0.045 | 0.17 | −0.54 | 0.9985 |

0.175 | 0.0383 | 0.18 | −0.50 | 0.9992 |

The regression fit constants are noted in Table III. The *R*^{2} values of the regression fit are again all almost unity.

Q_{r}
. | m_{2}
. | $ec2$ . | R^{2}
. |
---|---|---|---|

0.5 | −0.46 | −2.09 | 0.9143 |

1 | −0.28 | −1.39 | 0.9752 |

1.5 | −0.19 | −1.06 | 0.9960 |

2 | −0.21 | −1.07 | 0.9668 |

Q_{r}
. | m_{2}
. | $ec2$ . | R^{2}
. |
---|---|---|---|

0.5 | −0.46 | −2.09 | 0.9143 |

1 | −0.28 | −1.39 | 0.9752 |

1.5 | −0.19 | −1.06 | 0.9960 |

2 | −0.21 | −1.07 | 0.9668 |

The normalized droplet size is found to be in good agreement with the flow-rate ratio and capillary number with power-law regression constant values given by *m*_{3} = 0.208 94, *m*_{4} = −0.2374, and $ec3$ −1.2797. The *R*^{2} value of the regression is 0.960 01. Figure 14(b) presents a side view perpendicular to the axis representing the normalized droplet size parameter, where the plane looks like a line and offers a visual representation of how closely the data points are aligned with the regression plane.

### D. Effect of various parameters on pressure fluctuations and droplet size

Since our proposed method is based on pressure measurements on the wall of the outer tube, the influence of the various fluid and flow parameters on the pressure measurements is investigated. The effects of pressure fluctuations and their magnitude measured on the wall during the exit of the droplet from the channel are analyzed for different fluid and flow parameters.

#### 1. Effect of flow-rate ratio

Figure 15(a) shows the pressure exerted on the outer tube wall at a point 600 *μ*m from the entrance for different flow-rate ratios of 0.5, 1, 1.5, and 2. From Fig. 15(b), it can be observed that the amplitude of the pressure fluctuations increases with increasing flow-rate ratio. Figure 15(c) shows that the normalized droplet diameter increases with increasing dispersed-phase flow rate at a constant continuous-phase flow rate. Figure 15(d) depicts the relationship between pressure amplitude and normalized droplet diameter at different flow-rate ratios. Here, the increase in the amplitude of pressure fluctuations indicates an increase in the volume fraction of dispersed-phase fluid.

#### 2. Effect of viscosity ratio

Figure 16(a) depicts the pressure fluctuations experienced by the channel wall at a particular point for various viscosity ratios in the range of 0.25–2, and Fig. 16(b) shows the corresponding variations in pressure amplitude. From Fig. 16(c) it can be seen that the droplet size is smaller at lower viscosity ratios. The reason for this is that, up to a viscosity ratio of 1, the dispersed-phase fluid with low viscosity will experience less resistance, leading to more stretching and elongation. Under this condition, the continuous-phase fluid can easily break the dispersed-phase fluid into smaller droplets. At a viscosity ratio of 1, the droplet diameter reaches its maximum value, and a further increase in viscosity ratio results in a decrease in droplet size. Figure 16(d) illustrates the relationship between droplet size and pressure amplitude at various viscosity ratios.

#### 3. Effect of interfacial tension

The pressure variations transmitted to the channel wall at various interfacial tensions are shown in Fig. 17(a). The dependence of the pressure fluctuation amplitude on capillary number is shown in Fig. 17(b) for constant viscosity and flow-rate ratios. Figure 17(c) shows the variation of the normalized droplet diameter with the capillary number, and it can be seen that the droplet diameter decreases with increasing capillary number. Figure 17(d) illustrates the variation of the normalized droplet diameter with the pressure fluctuation amplitude.

These results suggest that the droplet diameter is primarily influenced by the flow-rate ratio, viscosity ratio, and interfacial tension.

## V. CONCLUSIONS

In the present work, we have generated micrometer-sized droplets in a 2D axisymmetric co-flow geometry using the LSM from COMSOL Multiphysics software in combination with MATLAB. We have proposed a new method to determine droplet size based on pressure fluctuations exerted on the outer wall of the channel and on the flow rate of the dispersed-phase fluid. This method has potential applications in the biomedical field, such as for drug delivery systems. It has been found that the pressure fluctuation method gives similar values for the droplet diameter as the volume fraction method, with a small deviation due to the formation of satellite droplets. The proposed method for droplet size measurement will be more useful for monodispersed droplets, emulsions, or other droplet size combinations. The effects of various flow and fluid parameters on the pressure fluctuations has been investigated. An empirical relation has been obtained to determine the droplet size on the basis of the input flow rate and capillary number. In summary, the amplitude of the pressure fluctuations is directly proportional to the flow-rate ratio, viscosity ratio, and interfacial tension between the immiscible fluids.

## ACKNOWLEDGMENTS

The authors acknowledge the IIT Tirupati for providing the COMSOL Multiphysics 6.2 software.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Babajan Bakthar Khan**: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **Sunil Kumar Thamida**: Conceptualization (equal); Methodology (equal); Software (equal); Supervision (lead); Validation (supporting); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal). **Anil B. Vir**: Conceptualization (supporting); Formal analysis (supporting); Methodology (supporting); Software (equal); Supervision (equal); Validation (supporting); Visualization (supporting); Writing – review & editing (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS)*(