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1. Simulating Paraffin Wax Droplets Using Mixed Finite Element Method

   Nathawani, Darsh K.; Knepley, Matthew G. (September 2022, ICCFD 11 PROCEEDINGS)

   Brehm, Christop; Pandya, Shishir (Ed.)

   Paraffin wax is a prominent solid fuel for hybrid rockets. The atomization process of the paraffin wax fuel into he hybrid rocket combustion involves the droplets pinching off from the fuel surface. Therefore, droplet formation and pinch-off dynam- ics is analyzed using a one-dimensional axisymmetric approximation to understand droplet size distribution and pinch-off time. A mixed finite element formulation is used to solve the numerical problem. The computational algorithm uses adaptive mesh refinement to capture singularity and runs self-consistently to calculate droplet elongation. The code is verified using the Method of Manufactured Solution (MMS) and validated against laboratory experiments. Moreover, paraffin wax simulations are explored for varying inlet radius and it is found that the droplet size increases very slightly with the increasing inlet radius. Also, the pinch-off time increases up to a point where it starts to decrease as we increase the inlet radius. This behavior leads to a conjecture for the theoretical maximum radius that the droplet approaches as the inlet radius increases, which is a motivation for the future work. 

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2. Multiclass Reinforced Active Learning for Droplet Pinch-Off Behaviors Identification in Inkjet Printing

   https://doi.org/10.1115/1.4057002  

   Li, Zebin; Segura, Luis Javier; Li, Yifu; Zhou, Chi; Sun, Hongyue (July 2023, Journal of Manufacturing Science and Engineering)

   Abstract Inkjet printing (IJP) is one of the promising additive manufacturing techniques that yield many innovations in electronic and biomedical products. In IJP, the products are fabricated by depositing droplets on substrates, and the quality of the products is highly affected by the droplet pinch-off behaviors. Therefore, identifying pinch-off behaviors of droplets is critical. However, annotating the pinch-off behaviors is burdensome since a large amount of images of pinch-off behaviors can be collected. Active learning (AL) is a machine learning technique which extracts human knowledge by iteratively acquiring human annotation and updating the classification model for the pinch-off behaviors identification. Consequently, a good classification performance can be achieved with limited labels. However, during the query process, the most informative instances (i.e., images) are varying and most query strategies in AL cannot handle these dynamics since they are handcrafted. Thus, this paper proposes a multiclass reinforced active learning (MCRAL) framework in which a query strategy is trained by reinforcement learning (RL). We designed a unique intrinsic reward signal to improve the classification model performance. Moreover, how to extract the features from images for pinch-off behavior identification is not trivial. Thus, we used a graph convolutional network for droplet image feature extraction. The results show that MCRAL excels AL and can reduce human efforts in pinch-off behavior identification. We further demonstrated that, by linking the process parameters to the predicted droplet pinch-off behaviors, the droplet pinch-off behavior can be adjusted based on MCRAL. 

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3. Droplet detachment and pinch-off of bidisperse particulate suspensions

   https://doi.org/10.1039/D1SM00593F  

   Thiévenaz, Virgile; Rajesh, Sreeram; Sauret, Alban (January 2021, Soft Matter)

   null (Ed.)

   When a droplet is generated, the ligament connecting the drop to the nozzle thins down and eventually pinches off. Adding solid particles to the liquid phase leads to a more complex dynamic, notably by increasing the shear viscosity. Moreover, it introduces an additional length scale to the system, the diameter of the particles, which eventually becomes comparable to the diameter of the ligament. In this paper, we experimentally investigate the thinning and pinch-off of drops of suspensions with two different sizes of particles. We characterize the thinning for different particle size ratios and different proportions of small particles. Long before the pinch-off, the thinning rate is that of an equivalent liquid whose viscosity is that of the suspension. Later, when the ligament thickness approaches the size of the large particles, the thinning accelerates and leads to an early pinch-off. We explain how the bidisperse particle size distribution lowers the viscosity by making the packing more efficient, which speeds up the thinning. This result can be used to predict the dynamics of droplet formation with bidisperse suspensions. 

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4. Droplet formation simulation using mixed finite elements

   https://doi.org/10.1063/5.0089752  

   Nathawani, Darsh K.; Knepley, Matthew G. (June 2022, Physics of Fluids)

   Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate the size of a droplet but fails to approximate the shape of the droplet. This is due to a highly nonlinear flow description near the point where the first pinch-off happens. A one-dimensional axisymmetric mathematical model was first developed by Eggers and Dupont [“Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] using asymptotic analysis. This asymptotic approach to the Navier–Stokes equations leads to a universal scaling explaining the self-similar nature of the solution. Numerical models for the one-dimensional model were developed using the finite difference [Eggers and Dupont, “Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and finite element method [Ambravaneswaran et al., “Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops,” Phys. Fluids 14, 2606–2621 (2002)]. The focus of this study is to provide a robust computational model for one-dimensional axisymmetric droplet formation using the Portable, Extensible Toolkit for Scientific Computation. The code is verified using the Method of Manufactured Solutions and validated using previous experimental studies done by Zhang and Basaran [“An experimental study of dynamics of drop formation,” Phys. Fluids 7, 1184–1203 (1995)]. The present model is used for simulating pendant drops of water, glycerol, and paraffin wax, with an aspiration of extending the application to simulate more complex pinch-off phenomena. 

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5. Prediction of the Atomization Process in Respimat® Soft MistTM Inhalers Using a Volume of Fluid-to-Discrete Phase Model

   https://doi.org/10.3390/bioengineering12030264  

   Sperry, Ted; Feng, Yu (March 2025, Bioengineering)

   This study investigates the atomization process in Respimat® Soft MistTM Inhalers (SMIs) using a validated Volume of Fluid (VOF)-to-Discrete Phase Model (DPM) to simulate the transition from colliding liquid jets to aerosolized droplets. Key parameters, including colliding jet inlet velocity, surface tension, and liquid viscosity, were systematically varied to analyze their impact on the atomization, i.e., aerosolized droplet size distributions. The VOF-to-DPM simulation results indicate that higher jet inlet velocities enhance ligament fragmentation, producing finer and more uniform droplets while reducing total atomized droplet mass. The relationship between surface tension and atomization performance in colliding jet atomization is not monotonic. Reducing surface tension plays a complex dual role in the atomization process. On the one hand, lower surface tension enhances the likelihood of liquid jet breakup into a liquid sheet, leading to the formation of smaller ligaments under the same airflow conditions and shear forces. This increases the probability of generating more secondary droplets. On the other hand, reduced surface tension also destabilizes the liquid surface shape, decreasing the formation of fine, high-sphericity droplets in regimes where surface tension is a dominant force. Viscosity also influences atomization through complex mechanisms, i.e., lower viscosity reduces resistance to ligament breakup but promotes droplet interactions and coalescence, while higher viscosity suppresses ligament fragmentation, generating larger droplets and reducing atomization efficiency. The validated VOF-to-DPM framework provides critical insights for enhancing the performance and efficiency of inhalation therapies. Future work will incorporate nozzle geometry, jet impingement angles, and surfactant effects to better understand and optimize the atomization process in SMIs, focusing on achieving preferred droplet size distributions and emitted doses for enhanced drug delivery efficiency in human respiratory systems. 

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