Axisymmetric problem of smoothing the surface of a viscous liquid by surface tension forces

Vitalii Myntiuk, Olga Shypul, Oleh Tryfonov, Yevgen Tsegelnyk

Abstract


This study investigates an analytical solution to the problem of the surface levelling of viscous liquids under the influence of surface tension forces, focusing on the smoothing of plastic surfaces subjected to thermal energy treatment. This study aims to extend Orchard’s formula to axisymmetric surface irregularities and develop an analytical model for predicting levelling time, thereby ensuring efficient process control in thermal treatment applications. The tasks included deriving an analytical solution for axisymmetric levelling, validating it against numerical simulations in LS-DYNA, and incorporating the viscosity variation across the liquid layer. The methods involved analytical formulation and numerical simulation of surface evolution considering different initial surface geometries and viscosity distributions. Validation against numerical results demonstrated high accuracy for moderate and thick liquid layers ( ) and initial surface amplitudes up to 40% of the characteristic radius. Following validation, the model was applied to estimate levelling times for various surface configurations while maintaining simplicity while improving the predictive capabilities. Results showed that the extended formula effectively describes surface smoothing dynamics, including the cases with thickness-dependent viscosity, providing explicit expressions for levelling time. These findings enable precise control of heat input during thermal energy treatment, thereby optimizing the surface quality. In conclusion, the proposed analytical solutions offer a practical tool for surface levelling analysis, expanding the applicability of Orchard’s approach to more complex geometries and viscosity variations. In future work, we will focus on experimental validation and refinements to enhance the accuracy in industrial applications.

Keywords


Surface Levelling; Orchard's Formula; Axisymmetric Irregularities; Periodic Problem Formulation

Full Text:

PDF

References


Kurdi, A., & Chang, L. Recent advances in high performance polymers – tribological aspects. Lubricants, 2018, vol. 7, iss. 1, article no. 2. DOI: 10.3390/lubricants7010002.

Pedroso, A. F., Sebbe, N. P., Silva, F. J., Campilho, R. D., Sales-Contini, R. C., Costa, R. D., ... & Nogueira, F. R. A concise review on materials for injection moulds and their conventional and non-conventional machining processes. Machines, 2024, vol. 12, iss. 4, article no. 255. DOI: 10.3390/machines12040255.

Vambol, O., Kondratiev, A., Purhina, S., & Shevtsova, M. Determining the parameters for a 3D-printing process using the fused deposition modeling in order to manufacture an article with the required structural parameters. Eastern-European Journal of Enterprise Technologies, 2021, vol. 2, no. 1, pp. 70–80. DOI: 10.15587/1729-4061.2021.227075.

Wickramasinghe, S., Do, T., & Tran, P. FDM-based 3D printing of polymer and associated composite: A review on mechanical properties, defects and treatments. Polymers, 2020, vol. 12, iss. 7, article no. 1529. DOI: 10.3390/polym12071529.

Plankovskyy, S., Popov, V., Shypul, O., Tsegelnyk, Y., Tryfonov, O., & Brega, D. Advanced thermal energy method for finishing precision parts. In Advanced Machining and Finishing; Gupta, K., Pramanik, A., Eds.; Elsevier: Amsterdam, Netherlands, 2021, pp. 527–575. DOI: 10.1016/B978-0-12-817452-4.00014-2.

Adel, M., Abdelaal, O., Gad, A., Nasr, A. B., & Khalil, A. Polishing of fused deposition modeling products by hot air jet: evaluation of surface roughness. Journal of Materials Processing Technology, 2018, vol. 251, pp. 73–82. DOI: 10.1016/j.jmatprotec.2017.07.019.

Chai, Y., Li, R. W., Perriman, D. M., Chen, S., Qin, Q. H., & Smith, P. N. Laser polishing of thermoplastics fabricated using fused deposition modelling. The International Journal of Advanced Manufacturing Technology, 2018, vol. 96, no. 9, pp. 4295–4302. DOI: 10.1007/s00170-018-1901-5.

Plankovskyy, S., Shypul, O., Tsegelnyk, Y., Brega, D., Tryfonov, O., & Malashenko, V. Basic principles for thermoplastic parts finishing with impulse thermal energy method. In Handbook of Research on Advancements in the Processing, Characterization, and Application of Lightweight Materials; Kumar, K., Babu, B., Davim, J., Eds. IGI Global: Hershey PA, USA, 2022, pp. 49–87. DOI: 10.4018/978-1-7998-7864-3.ch003.

Orchard, S. E. On surface levelling in viscous liquids and gels. Applied Scientific Research, Section A, 1963, vol. 11, pp. 451–464. DOI: 10.1007/BF03184629.

Wapler, D. Das rheologische Verhalten der Anstrichstoffe bei der Verarbeitung (2a) – Untersuchungen über das Verlaufen. Farbe und Lack, 1975, vol. 81, no. 8, pp. 717–723.

Overdiep, W. S. The levelling of paints. Progress in Organic Coatings, 1986, vol. 14, iss. 2, pp. 159–175. DOI: 10.1016/0033-0655(86)80010-3.

Weidner, D. E. Leveling of a model paint film with a yield stress. Journal of Coatings Technology and Research, 2020, vol. 17, no. 4, pp. 851–863. DOI: 10.1007/s11998-019-00260-z.

Seeler, F., Hager, C., Tiedje, O., & Schneider, M. Simulations and experimental investigation of paint film leveling. Journal of Coatings Technology and Research, 2017, vol. 14, no. 4, pp. 767–781. DOI: 10.1007/s11998-017-9934-5

Klarskov, M., Jakobsen, J., & Saarnak, A. Verification of the Orchard leveling analysis. Rheology, 1992, vol. 92, pp. 30–39.

Xie, Y., Zhao, T., & Liu, K. Theoretical modeling of key parameters of wet film in roll coating process of coating panel. AIP Advances, 2023, vol. 13, no. 12, article no. 125224. DOI: 10.1063/5.0185600.

Schiltz, A. Analysis of surface tension in terms of force gradient per unit area. arXiv Preprint, 2024, arXiv no. 2406.16448. DOI: 10.48550/arXiv.2406.16448.

Takahashi, Y., Tanaka, G., Chang, F., Kato, F., & Iwata, H. Numerical analysis of sagging based on rheological properties of a paint film and proposal for a novel index to evaluate the amount of sag. IEEE Transactions on Automation Science and Engineering, 2023, vol. 21, iss. 3, pp. 2237–2250. DOI: 10.1109/TASE.2023.3337280.

Smetankina, N. V., Postnyi, O. V., Merkulova, A. I., & Merkulov, D. O. Modeling of non-stationary temperature fields in multilayer shells with film heat sources. 2020 IEEE KhPI Week on Advanced Technology (KhPIWeek), Kharkiv, Ukraine, IEEE, 2020, pp. 242–246. DOI: 10.1109/KhPIWeek51551.2020.9250139.

Shypul, O., & Myntiuk, V. Transient thermoelastic analysis of a cylinder having a varied coefficient of thermal expansion. Periodica Polytechnica Mechanical Engineering, 2020, vol. 64, no. 4, pp. 273–278. DOI: 10.3311/PPme.14733.

Boschetto, A., Bottini, L., & Veniali, F. Finishing of fused deposition modeling parts by CNC machining. Robotics and Computer-Integrated Manufacturing, 2016, vol. 41, pp. 92–101. DOI: 10.1016/j.rcim.2016.03.004.

Jin, Y., Wan, Y., Zhang, B., & Liu, Z. Modeling of the chemical finishing process for polylactic acid parts in fused deposition modeling and investigation of its tensile properties. Journal of Materials Processing Technology, 2017, vol. 240, pp. 233–239. DOI: 10.1016/j.jmatprotec.2016.10.003.

Fritz, A., Sekol, L., Koroskenyi, J., Walch, B., Minear, J., Fernandez, V., & Liu, L. Experimental analysis of thermal energy deburring process by design of experiment. Proceedings of the ASME 2012 International Mechanical Engineering Congress and Exposition, ASME, 2012, vol. 3, pp. 2035–2041. DOI: 10.1115/IMECE2012-88411.

Pan, X., Wu, C. T., & Hu, W. Incompressible Smoothed Particle Galerkin (ISPG) method for an efficient simulation of surface tension and wall adhesion effects in the 3D reflow soldering process. 16th International LS-DYNA Users Conference, ANSYS, 2020, pp. 1–17.

Zhang, X., Cao, C., Gui, N., Huang, X., Yang, X., Tu, J., Jiang, S., & Zhao, Q. A particle-scale model of surface tension for two-phase flow: model description and validation. Energies, 2022, vol. 15, iss. 19, article no. 7132. DOI: 10.3390/en15197132.




DOI: https://doi.org/10.32620/reks.2025.1.08

Refbacks

  • There are currently no refbacks.