Droplet Diameter Variability Induced by Flow Oscillations in a Micro Cross-Junction

Abstract

This study investigates the stochastic variation in droplet size generated within a microfluidic flow-focusing cross-junction. A commercial micro cross-junction was used to experimentally analyze droplet formation under fixed flow rate conditions. An in-house machine learning-based algorithm was developed to automatically detect and measure droplet dimensions from high-speed video recordings. Despite constant flow rates, the analysis revealed fluctuations in droplet size, attributed to velocity oscillations induced by syringe pumps. To explore this phenomenon, micro-Particle Image Velocimetry (micro-PIV) was employed to capture velocity profiles, which were then used to define time-dependent boundary conditions for numerical simulations. Simulations were conducted using the OpenFOAM solver interFoam and validated against experimental data. The results demonstrate good agreement and confirm that velocity fluctuations significantly influence droplet formation. This combined experimental and numerical approach provides an innovative, robust framework for understanding and predicting droplet behavior in microfluidic systems.

1. Introduction

Microfluidic technologies have revolutionized the way researchers manipulate fluids at the microscale, enabling precise control over droplet formation for applications in drug delivery, diagnostics, chemical synthesis, and biological assays. Among the various microfluidic architectures, the flow-focusing cross-junction has emerged as a particularly effective configuration for generating monodisperse droplets [1]. Its geometric simplicity and operational versatility allow for the production of droplets with controlled size and shape across a wide range of flow conditions [2]. Despite the apparent stability of droplet formation in flow-focusing junctions, subtle perturbations in the flow field can lead to significant variability in droplet size [3,4]. This variability is especially critical in applications where droplet uniformity directly impacts performance, such as in encapsulation processes or reaction kinetics. One often overlooked source of such perturbations is the mechanical operation of syringe pumps, which can introduce low-frequency oscillations in the flow rate, even under nominally constant conditions. These oscillations, though small, can propagate through the microchannel network and affect the dynamics of droplet pinch-off and formation. These oscillations alter the local shear and pressure conditions at the junction, which in turn affect the timing and dynamics of droplet pinch-off. We support this explanation with experimental micro-PIV measurements and show how these velocity profiles were used to define time-dependent boundary conditions in our CFD simulations. Previous studies have extensively characterized the behavior of droplets in microfluidic junctions under steady-state conditions. Anna et al. [5] and Dreyfus et al. [6] laid the foundation for understanding flow-focusing mechanisms, while subsequent works [7,8,9,10] explored the influence of capillary number, flow rate ratios, and fluid properties on droplet size and formation regimes (squeeze, drip, jetting). Rostami et al. [8] and Garstecki et al. [11] provided scaling laws and correlations that are widely used to predict droplet behavior. However, these models typically assume idealized, time-invariant flow conditions and do not account for the stochastic nature of real-world fluid delivery systems. Recent advances in experimental techniques and computational modeling have opened new avenues for investigating transient phenomena in microfluidics. Machine learning-based image analysis tools [12] have enabled high-throughput quantification of droplet characteristics, while micro-Particle Image Velocimetry (micro-PIV) has provided detailed insights into velocity fields within microchannels [13]. However, the integration of these tools in order to study the impact of flow oscillations on droplet formation remains limited. In this study, we present a comprehensive framework that combines high-speed imaging, automated droplet detection via a custom machine learning algorithm, micro-PIV velocity profiling, and time-dependent computational fluid dynamics (CFD) simulations using OpenFOAM. The detection and measurement of droplet dimensions from high-speed video recordings involve processing thousands of frames, each containing complex visual information. Manual analysis is not only time-consuming but is also prone to inconsistency and human error. To address this, we developed a custom machine learning-based image analysis algorithm capable of automatically identifying complete droplets and extracting key parameters such as radius, velocity, and count. This approach ensures high-throughput, reproducible, and accurate data extraction in varying lighting conditions and geometries. In addition, the ability of the algorithm to generate synthetic training datasets from experimental images enhances its adaptability and robustness, making it suitable for broader applications in microfluidic research. Although experimental observations provide valuable information about droplet formation, they are limited in their ability to isolate and quantify the influence of specific flow parameters, particularly transient ones such as velocity oscillations. Computational Fluid Dynamics (CFD) modeling, using the OpenFOAM v2212 platform, allows us to simulate the droplet formation process under controlled and customizable conditions [14,15,16,17,18]. By incorporating time-dependent boundary conditions derived from micro-PIV measurements, we can directly assess the impact of syringe pump-induced flow oscillations on droplet size variability. This dual approach of experimental validation coupled with numerical simulation offers a comprehensive understanding of the phenomenon and enables predictive modeling that would be difficult to achieve through experiments alone. This methodology allows for the quantification and modeling of droplet size variability induced by syringe pump-driven flow oscillations. By incorporating experimentally measured velocity fluctuations into the boundary conditions of our simulations, we demonstrate that even minor deviations in flow rate can lead to measurable changes in droplet dimensions. This work contributes to the field by highlighting the importance of accounting for dynamic flow instabilities in microfluidic systems. It provides a validated methodology for predicting droplet behavior under realistic operating conditions and offers insights that are crucial for the design and optimization of droplet-based microfluidic devices. Furthermore, the use of Latin Hypercube Sampling (LHS) to model flow rate variability introduces a statistically robust and computationally efficient strategy for exploring the parameter space of droplet formation.

2. Material and Methods

This section describes the experimental setup deployed for the generation and analysis of droplets in the micro-junction, the algorithm for droplet image analysis, and the numerical method developed for the simulation of droplet generation in the micro-junction.

2.1. Experimental Apparatus

The micro-junction employed in this study, illustrated in Figure 1a, is a commercial flow-focusing cross-junction produced by Dolomite (Royston, UK). It features stadium-shaped microchannels and a constriction at the junction, as shown in Figure 1b, which accelerates the continuous phase and focuses it on the dispersed phase. The junction is composed of three inlet branches where the fluid enters and one common outlet. This geometry is particularly effective in producing spherical droplets with a lower coefficient of variation compared to other junction types [9].

The width and depth of the microchannel are, respectively, $w_c=390.0\mathsf{\mu }$m and $h_c=190.0\mathsf{\mu }$m, while the width and depth of the junction are, respectively, $w_j=195.0\mathsf{\mu }$m and $h_j=190.0\mathsf{\mu }$m. The microfluidic chip is made of quartz to ensure excellent optical access, which is essential for accurate flow visualization and measurement. The experimental setup used for both micro-PIV and image analysis is shown in Figure 2 [12,15].

Two syringe pumps (Harvard Instruments PHD 400, Harvard Apparatus, Holliston, MA, USA) are used to drive the working fluids: demineralized water as the dispersed phase and mineral oil as the continuous phase. The microfluidic chip is mounted on an inverted microscope (Nikon Eclipse, Nikon, Tokyo, Japan), and a high-speed camera (Olympus i-Speed, Olympus, Tokyo, Japan) is connected to the microscope for image acquisition. Due to the high frame rate of the camera (2000 fps), an additional light source (a DC LED lamp) is employed to ensure adequate illumination. The high-speed camera is connected to a computer for real-time image acquisition and subsequent analysis. For velocity measurements in the water phase, a micro-PIV system is deployed, with polystyrene microspheres with a diameter of 1.19 $\mathsf{\mu }$m and a density of 1050 kg/m³ inserted at low concentration in the water and used as a passive tracer. Micro-PIV analysis is performed using the open-source DefocusTracker library in the MATLAB v2.0.0 environment [16].

2.2. Image Analysis Algorithm

This section presents an in-house algorithm developed for automated droplet detection and analysis. To evaluate how droplet dimensions vary over time under fixed flow rate conditions, about 10,000 images were processed. To streamline this task and minimize manual intervention, a machine learning-based tool was implemented. The algorithm automatically identifies frames containing complete droplets and extracts key parameters such as droplet dimensions (radius along the flow direction and perpendicular to it), droplet velocity, and droplet count. The algorithm is based on a classification model that employs logistic regression:

$$f\left(\varphi \right)={\displaystyle \frac{1}{1-e^{-\varphi }}}$$

(1)

where $\varphi$ is defined as

$$\varphi \left(\mathbf{I}\right)=\mathbf{w}·\mathbf{I}+b_0$$

(2)

This function can be used to classify the images containing or not containing a drop; in fact, I is a vector related to an image composed of the light intensity value of each pixel; w are the width values during the training, and $b_0$ is a bias set equal to 0. By applying this model to each image, the algorithm can immediately predict whether an entire droplet is present; if this is the case, the drop’s characteristics are evaluated. Once properly trained, the algorithm can operate effectively in different microchannel geometries and under varying environmental conditions, such as changes in lighting. To ensure robustness, the training process uses synthetic images generated by the algorithm itself. These are created from an initial pool of experimental images and a user-selected reference image that contains a droplet. From this, the algorithm creates an image of the empty channel and combines it with the reference image of the droplet, generating a dataset containing all the possible positions of the drop inside the channel. With this approach, it is possible to obtain a very good accuracy on the test dataset, ranging for the different cases from 0.992 to 0.998 and showing a log-loss ranging from 0.19 to 0.0001 The algorithm’s workflow is illustrated in Figure 3. As shown in the figure, for high-speed video recordings (a), the algorithm automatically selects frames containing full droplets and performs the necessary measurements. Training begins with a seed image (b), which is manually selected by the user, and an image of the empty channel (d), which is generated from the initial image pool (c). Using these inputs, the algorithm creates two synthetic datasets (e, f) for training purposes.

At this stage, the algorithm is capable of identifying only those frames that contain a complete droplet and performs all measurements exclusively on these selected images. The measurement process is illustrated in Figure 4.

The initial image is processed to remove the noise and background, and then it is converted to a binary image where at least one entire drop is present. Then, the eventual other drops are removed from the image. At the end of the process, the processed image is stored in the database to measure the radius and velocity of the droplet. The radius is obtained by the conversion of the contours of the droplet from pixels to meters, while the droplet velocity is obtained by analyzing the displacement of the droplet in a series of five subsequent images. From the analysis shown in this section, the distribution of the droplet sizes and the associated velocities are obtained.

2.3. Numerical Simulations

The numerical simulations were performed using OpenFOAM open-source computational fluid dynamics (CFD) software, with the interFoam solver [14,18], which is designed for isothermal, incompressible, and immiscible Newtonian fluids. It is based on the volume-of-liquid (VOF) method for interface tracking. The solver operates by solving five equations per iteration (two scalar equations and one vector equation). For pressure–velocity coupling, it employs the PIMPLE algorithm, a hybrid approach that combines the PISO (Pressure Implicit with Splitting of Operators) and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithms, offering both stability and efficiency for transient simulations. The time step was automatically selected at each iteration to maintain a Courant number of below 1. The total simulated time depends on the boundary conditions used. Each simulation was considered complete after the formation of 10 drops. The time needed for each simulation was approximately 4–5 h when executed in parallel on 20 cores. The equations solved are Navier–Stokes equations,

$${\displaystyle \frac{\partial \left(\rho \mathbf{u}\right)}{\partial t}}+\nabla ·\left(\rho \mathbf{u}\mathbf{u}\right)=-\nabla p+\nabla ·\mathbf{\tau }+\rho \mathbf{g}+{\mathbf{f}}_{\sigma }$$

(3)

together with the continuity equation

$$\nabla ·\mathbf{u}=0$$

(4)

where $\mathbf{u}$ (m/s) denotes the velocity vector, p (Pa) the pressure, $\mathbf{\tau }$ (Pa) the viscous stress tensor, $\mathbf{g}$ (m/s²) the gravitational acceleration vector, and $\rho$ (kg/m³) and $\mu$ (Pa·s) the fluid density and dynamic viscosity, respectively, defined as

$$\rho =\alpha {\rho }_d+(1-\alpha ){\rho }_c$$

(5)

$$\mu =\alpha {\mu }_d+(1-\alpha ){\mu }_c$$

(6)

An additional equation is solved to capture the interface between the fluids and the advection of the scalar quantity $\alpha$,

$${\displaystyle \frac{\partial \alpha }{\partial t}}+\nabla ·\left(\alpha \mathbf{u}\right)-\nabla ·\left[\alpha (1-\alpha ){\mathbf{u}}_r\right]=0$$

(7)

The field $\alpha$ is used to distinguish the two fluids and is defined as

$$\alpha =\left\{\begin{array}{cc}1\hfill & \mathrm{in}\mathrm{the}\mathrm{continuous}\mathrm{phase}\hfill \\ 0.5\hfill & \mathrm{at}\mathrm{the}\mathrm{interface}\hfill \\ 0\hfill & \mathrm{in}\mathrm{the}\mathrm{dispersed}\mathrm{phase}\hfill \end{array}\right.$$

(8)

To solve these equations, the MULES algorithm (a multidimensional limiter for explicit solutions) is used to guarantee the boundness of the solution and obtain more smeared interfaces. The last term of the right-hand side is the so-called compression term, and it is not negligible only at the interface between the two fluids. In this term, the relative velocity ${\mathbf{u}}_f$ is defined as

$${\mathbf{u}}_f=min(C_{\alpha }\left|\mathbf{u}\right|,max\left(\right|\mathbf{u}\left|\right)){\displaystyle \frac{\nabla \alpha }{|\nabla \alpha |}}$$

(9)

For microfluidic applications, $C_{\alpha }$ can be taken to be equal to 1 [14]. As can be seen from the momentum equations, the source term ${\mathbf{f}}_{\sigma }$ is used to estimate the surface tension forces; to calculate this contribution, the continuous surface force model CSF is used.

$${\mathbf{f}}_{\sigma }=\sigma \kappa \nabla \alpha$$

(10)

where $\kappa$ is the curvature (units: m⁻¹), which is evaluated by starting from the volume fraction $\alpha$

$$\kappa =-\nabla ·{\displaystyle \frac{\nabla \alpha }{|\nabla \alpha |}}$$

(11)

The first step involves validating the simulations by comparing the droplet shape obtained from simulations under constant input flow rate conditions with the experimental results. This validation ensures that the numerical model accurately captures the physical behavior of the system. Following validation, the velocity profiles obtained from micro-PIV measurements are used to estimate the corresponding variations in the mass flow rate. These time-dependent flow rates are then applied as boundary conditions in a series of simulations to numerically investigate the impact of velocity fluctuations on droplet formation.

2.4. Computational Grid and Boundary Conditions

In the simulations, a computational grid with about 3.5 million polyhedral elements was used. A sensitivity analysis of the number of elements was performed, as shown in Figure 5a, where the dimension of the drop (blue line) and the velocity of the drops (red line) are plotted as a function of the number of elements used in five grids. It can be observed that the results stabilize when using a grid with approximately 3.5 million elements. The corresponding computational mesh is illustrated in Figure 5b.

For this simulation, the mass flows were kept constant, the continuous phase (namely silicon oil) entered from the inlets along the z-axes with a total flow rate of 5 mL/h, and the dispersed phase (namely demineralized water) entered from the inlet along the x-axes with a flow rate of 0.6 mL/h (Figure 6). All walls were considered adiabatic and the no-slip and impermeability condition was set on this basis.The contact angle of the water was equal to 160°, and the surface tension of the water in the oil was set to 0.04244 N/m [8].

2.5. Validation of Simulation

The comparison between the numerical droplet surface and the experimental one, obtained under the same boundary conditions, is shown in Figure 7. This figure shows the final stages of the droplet formation process. The time interval between consecutive frames is 0.0002 s. The grayscale images compare numerical (top) and experimental (bottom) results. The additional images, obtained from numerical simulations, show the isosurface corresponding to $\alpha =0.5$ and the velocity field on a plane that intersects the junction in the middle. The figure shows very good agreement between the numerical results and the experiments. This form of simulation can capture the evolution of the drop profile during breakdown.

To validate the numerical simulations, a comparison was also made between the diameters of the drops obtained by the numerical simulations and the diameters measured in the experiments, as shown in Figure 8. To compare the numerical and experimental data, a Python (version 3.9.16) code was built that overlaps the numerical data obtained from the simulations with the recorded drop images.

The agreement between the drop diameters obtained numerically and experimentally was excellent.

3. Results and Discussion

This section presents the experimental droplet size distributions obtained at different mass flow rates. Subsequently, numerical simulations were carried out to reproduce these distributions. The comparison between the experimental and numerical results is then discussed along with the relevant considerations.

3.1. Drop Radius Evaluation

Using the experimental apparatus described in Figure 2 several trains of droplets were analyzed for a fixed continuous flow rate equal to 5.0 mL/h and different values of the dispersed flow rates, equal to 0.6, 1,2, 1.8, 2.4 and 3.2 mL/h, respectively. Using the trained algorithms, the dimensions of the drop and their distributions were obtained. The drop dimensions are in a normal distribution around an average value. Statistical analysis of droplet dimensions in the x direction along the flow direction and in the z direction in accordance with the flow direction shows different distributions in the two axes. Focusing on the first case, characterized by $Q_c$ = 5.0 mL/h and $Q_d$ = 0.6 mL/h, the analysis performed on a train of 200 droplets shows that the dimensions of the droplets (in the x and z directions) typically follow a normal distribution centered around a mean value. Defining the length of the drop along the x direction as the length measured along the flow direction and the length along the z direction as the length measured along the direction normal to the wall, it is possible to obtain the distribution described in Figure 9. From here, it is possible to see that the drop dimensions for fixed flow rates can vary by up to 7%; the difference between the largest and the smallest droplets obtained for fixed conditions is graphically depicted in Figure 10. In droplet-based microfluidic systems, the acceptable level of variability is highly dependent on the intended application [13]. For example, in biological assays or drug delivery, where droplet volume directly affects reagent concentration or dosage, even small deviations can be critical. In contrast, for screening applications or emulsion-based synthesis, a variability of up to 10% is often considered acceptable. Recent studies have reported droplet size variability ranging from 5% to 12% depending on the geometry, fluid properties, and control mechanisms used. In this context, the 7% variability observed in our system falls within the typical range for passive flow-focusing devices driven by syringe pumps, reflecting the inherent limitations of mechanical pumping systems and the sensitivity of droplet formation to transient flow fluctuations. Although our system demonstrates good overall reproducibility, we acknowledge that further reduction in variability could be achieved through active flow control (e.g., pressure-driven systems) or feedback mechanisms.

The phenomenon was observed for all the cases studied. These results are reported in Figure 11, where it is possible to observe not only the distributions of drop lengths for different dispersed phase flow rates but also how this parameter influences drop size. For a fixed capillary number ($Ca$), increasing the dispersed phase flow rate ($Q_d$) leads to an increase in the mean value of the drop size. The mean value, standard deviation, and confidence interval for the different values analyzed are reported in

Table 1. Statistical values of drop length along the x and z directions for different $Q_d$ values.

Qd	X Direction			Z Direction
	Mean [µm]	Std. Dev. [µm]	95% CI [µm]	Mean [µm]	Std. Dev. [µm]	95% CI [µm]
0.6	196.7689	1.8556	[196.4007, 197.1371]	193.9971	1.8580	[193.6285, 194.3658]
1.2	214.3839	3.4201	[213.6417, 215.1261]	209.6913	3.4418	[208.9444, 210.4383]
1.8	224.0132	3.1243	[223.3933, 224.6331]	214.5557	2.9090	[213.9785, 215.1329]
2.4	232.2032	3.1570	[231.5768, 232.8296]	223.1775	2.3745	[222.7064, 223.6486]
3.2	244.3690	2.5265	[243.8651, 244.8730]	231.8474	1.7378	[231.5008, 232.1940]

.

This variation may be related to the oscillation generated inside the microchannel by the syringe pumps used to move the fluids. To better investigate this, micro-PIV measurements were taken inside the channels using micro-particles with a diameter of 1.19 $\mathsf{\mu }$m as a passive tracker. The images obtained by the high-speed camera were recorded and analyzed to extract velocity profiles [15]. These measures confirm the presence of oscillations in the velocity field. By analyzing the velocity profile, it is possible to extrapolate a distribution of flow rates; this distribution is used to generate a series of new inlet boundary conditions for the CFD simulation. Starting from the cumulative distribution of the velocity profile, it is possible to extrapolate a series of samples using latin hypercubic sampling (LHS) analysis. LHS is a statistical method that is useful for analyzing variable space. With this sampling technique, it is possible to strongly decrease the computational times needed to analyze the entire search space; the distribution is divided into a series of equally probable segments, and for each of these segments, it is possible to randomly select a value ensuring a representative description of the entire space. Using this, it is possible to reduce the number of samples needed to obtain a correct description of the phenomena. Upon applying LHS to distributions of velocity fields, forty values of mass flow rates were obtained as a basis for the numerical simulations.

Figure 12 shows the flow rates obtained starting from the cumulative distribution (centered on 2.5 mL/h) obtained from the micro-PIV measurements used as input conditions for the numerical simulations.

In addition to the primary focus on syringe pump-induced velocity oscillations, we acknowledge that other sources of uncertainty may influence droplet formation in microfluidic systems. One such factor is geometric confinement, particularly the aspect ratio of the microchannels and junction. In microfluidic environments, where surface tension forces dominate over inertial effects, channel geometry plays a critical role in shaping the flow field and determining the droplet pinch-off dynamics. A high aspect ratio enhances lateral confinement, increases shear stress, and promotes faster and more uniform droplet detachment. In contrast, low aspect ratios may allow the dispersed phase to expand more freely, leading to larger and potentially less stable droplets. In addition, confinement can increase the capillary pressure at the interface, influencing the curvature and detachment threshold of droplets. These geometric effects, coupled with potential hydrodynamic instabilities, such as interfacial fluctuations and recirculation zones, can contribute to the variability in droplet size. We also recognize the presence of measurement uncertainties, particularly in image processing and micro-PIV analysis, including resolution limits, particle tracking accuracy, and frame selection criteria. By coupling micro-PIV measurements with numerical simulations, our objective is to elucidate the influence of input flow rate variability on droplet diameter, providing a deeper understanding of how transient velocity fluctuations affect droplet formation dynamics in microfluidic systems.

3.2. Numerical Results

The flow rates obtained by the cumulative distribution (centered on 2.5 mL/h) shown in the previous section were used as input conditions for the simulations. In this section, the results obtained by the simulations are shown and compared with the experiments. Figure 13 compares the distributions of the dimension of the drop (in the x- and z-directions) obtained from the numerical simulation and for the experimental measurement.

It is possible to see that for the z-direction, the predicted average value $\mu$ is the same as that obtained from the experimental results (with an error smaller than 0.1 %), and for the x-direction, we have a higher error (around 7%). The higher error in the x-direction can be related to the fact that the length of the drop in this direction is more affected by the velocity. Overall, the numerical simulation seems to predict the effects of the oscillations in the velocity field on droplet formation.

4. Conclusions

This study presents a comprehensive investigation into the stochastic behavior of droplet formation within a microfluidic flow-focusing cross-junction, highlighting the subtle but impactful role of flow instabilities in microscale fluid dynamics. By integrating high-speed imaging, machine learning-based image analysis, micro-PIV measurements, and CFD simulations using OpenFOAM, we demonstrate that even under nominally constant flow-rate conditions, droplet size exhibits measurable fluctuations. These variations, also influenced by geometric confinement, hydrodynamic instabilities, and measurement limitations, are primarily driven by low-frequency velocity oscillations induced by syringe pumps. These oscillations were experimentally confirmed through micro-PIV measurements and systematically incorporated into time-dependent boundary conditions for numerical simulations, enabling realistic and dynamic modeling of droplet formation.

The strong agreement between the simulated and experimental droplet size distributions, particularly in the transverse direction, validates the robustness of our combined approach and underscores the importance of accounting for transient flow phenomena in microfluidic design. The use of Latin Hypercube Sampling (LHS) to model flow rate variability further enhances the efficiency and statistical reliability of the simulations, enabling a representative exploration of the parameter space with reduced computational cost.

Beyond the core findings, this work contributes methodologically by demonstrating the value of coupling experimental velocity profiling with dynamic CFD modeling to capture real-world operating conditions.

In general, the insights gained from this study have direct implications for the design, optimization, and control of droplet-based microfluidic systems in both scientific and industrial contexts. Future work may extend this framework to explore non-Newtonian fluids, complex junction geometries, and active flow control strategies, further advancing the precision and reliability of microfluidic droplet generation.