A Unified Transformation Framework for Studying Various Situations of Vertical/Oblique Drop Impact on Horizontal/Inclined Stationary/Moving Flat Surfaces

Document Type : Regular Article


Department of Aerospace Engineering, Sharif University of Technology, Tehran, 11155-9161, Iran



There are various situations of drop impact on solid surfaces widely occurred in natural phenomenon or used in different industrial applications. However, comparing and classifying these drop impact situations is not easy due to different states of the parameters affecting drop impact dynamics. In this article, a unified transformation framework is proposed to study various situations of vertical/oblique drop impact on horizontal/inclined stationary/moving flat surfaces with/without a crossflow. This simple framework consists of a coordinate with normal and tangential axes on a horizontal stationary surface. For each drop impact situation, the drop velocity, gravitational acceleration, possible induced flow due to the moving surface, and possible crossflow are transformed into the framework. Comparing the transformed versions of considered drop impact situations facilitates identification of their physical similarities/differences and determines which situations (and under what conditions) lead to identical results and can be used interchangeably. Although common situations of drop impact on moving surfaces (having tangential component of surface velocity) lead to asymmetric drop spreading, the possibility of symmetric drop spreading on moving surfaces is demonstrated and analyzed using the proposed transformation framework. This interesting possibility means that for related production lines or experimental setups, where symmetrical drop spreading is required, the surface does not need to be stationary. In such applications/setups, the use of moving surfaces (rather than stationary surfaces) can considerably accelerate the symmetric drop impact process. Our simulation results of several of the considered drop impact situations well confirm the facilities/predictions of the proposed transformation framework.


Main Subjects

Aboud, D. G., & Kietzig, A. M. (2015). Splashing threshold of oblique droplet impacts on surfaces of various wettability. Langmuir, 31(36), 10100-10111. https://doi.org/10.1021/acs.langmuir.5b02447##
Aboud, D. G., & Kietzig, A. M. (2018). On the oblique impact dynamics of drops on superhydrophobic surfaces. Part I: sliding length and maximum spreading diameter. Langmuir, 34(34), 9879-9888. https://doi.org/10.1021/acs.langmuir.8b02034##
Almohammadi, H., & Amirfazli, A. (2017a). Understanding the drop impact on moving hydrophilic and hydrophobic surfaces. Soft Matter, 13(10), 2040-2053. https://doi.org/10.1039/C6SM02514E##
Almohammadi, H., & Amirfazli, A. (2017b). Asymmetric Spreading of a Drop upon Impact onto a Surface. Langmuir, 33(23), 5957-5964. https://doi.org/10.1021/acs.langmuir.7b00704##
Antonini, C., Villa, F., & Marengo, M. (2014). Oblique impacts of water drops onto hydrophobic and superhydrophobic surfaces: outcomes, timing, and rebound maps. Experiments in Fluids, 55(4), 1-9. https://doi.org/10.1007/s00348-014-1713-9##
Azadi, E., & Taeibi Rahni, M. (2023). A three-dimensional mass-conserved multiphase lattice Boltzmann flux solver for incompressible flows with large density and viscosity ratios. Accepted to be Published by Advances in Applied Mathematics and Mechanics.##
Benther, J. D., Pelaez-Restrepo, J. D., Stanley, C., & Rosengarten, G. (2021). Heat transfer during multiple droplet impingement and spray cooling: Review and prospects for enhanced surfaces. International Journal of Heat and Mass Transfer, 178, 121587. https://doi.org/10.1016/j.ijheatmasstransfer.2021.121587##
Bird, J. C., Tsai, S. S., & Stone, H. A. (2009). Inclined to splash: triggering and inhibiting a splash with tangential velocity. New Journal of Physics, 11(6), 063017. https://doi.org/10.1088/1367-2630/11/6/063017##
Buksh, S., Almohammadi, H., Marengo, M., & Amirfazli, A. (2019). Spreading of low-viscous liquids on a stationary and a moving surface. Experiments in Fluids, 60(4), 1-12. https://doi.org/10.1007/s00348-019-2715-4##
Buksh, S., Marengo, M., & Amirfazli, A. (2020). Impacting of droplets on moving surface and inclined surfaces. Atomization and Sprays, 30(8). https://doi.org/10.1615/AtomizSpr.2020033015##
Carrolo, G., Ribeiro, D., Barata, J. M., & Silva, A. R. (2019). Aerodynamic breakup of a single droplet due to a crossflowed airstream. AIAA Scitech 2019 Forum, 0628. https://doi.org/10.2514/6.2019-0628##
Chen, R. H., & Wang, H. W. (2005). Effects of tangential speed on low-normal-speed liquid drop impact on a non-wettable solid surface. Experiments in Fluids, 39(4), 754-760. https://doi.org/10.1007/s00348-005-0008-6##
Cimpeanu, R., & Papageorgiou, D. T. (2018). Three-dimensional high speed drop impact onto solid surfaces at arbitrary angles. International Journal of Multiphase Flow, 107, 192-207. https://doi.org/10.1016/j.ijmultiphaseflow.2018.06.011##
Cui, J., Chen, X., Wang, F., Gong, X., & Yu, Z. (2009). Study of liquid droplets impact on dry inclined surface. Asia‐Pacific Journal of Chemical Engineering, 4(5), 643-648. https://doi.org/10.1002/apj.309##
Cunha, N., Ribeiro, D., Barata, J. M., & Silva, A. R. (2018). The splash deposition transition limits of a biofuel droplet wall impact with a and without crossflow. 14th International Conference on Liquid Atomization and Spray Systems 2020. ILASS Europe, Institute for Liquid Atomization and Spray Systems. ICLASS 2018-14th International Conference on Liquid Atomization and Spray Systems 2020. ILASS Europe, Institute for Liquid Atomization and Spray Systems. Retrieved from http://hdl.handle.net/10400.6/12166##
Dai, B., Liu, C., Liu, S., Wang, D., Wang, Q., Zou, T., & Zhou, X. (2023). Life cycle techno-enviro-economic assessment of dual-temperature evaporation transcritical CO2 high-temperature heat pump systems for industrial waste heat recovery. Applied Thermal Engineering, 219, 119570. https://doi.org/10.1016/j.applthermaleng.2022.119570##
Eyo, A. E., Ogbonna, N., & Ekpenyong, M. E. (2012). Comparison of the exact and approximate values of certain parameters in laminar boundary layer flow using some velocity profiles. Journal of Mathematics Research, 4(5), 17. https://doi.org/10.5539/jmr.v4n5p17##
Fakhari, A., & Bolster, D. (2017). Diffuse interface modeling of three-phase contact line dynamics on curved boundaries: A lattice Boltzmann model for large density and viscosity ratios. Journal of Computational Physics, 334, 620-638. https://doi.org/10.1016/j.jcp.2017.01.025##
Fathi, S., Dickens, P., & Fouchal, F. (2010). Regimes of droplet train impact on a moving surface in an additive manufacturing process. Journal of Materials Processing Technology, 210(3), 550-559. https://doi.org/10.1016/j.jmatprotec.2009.10.018##
Ferrao, I., Ribeiro, D., Barata, J. M., & Silva, A. R. (2019). Comparative study of droplet impact onto sloped surface versus a droplet impact onto a surface with a crossflow. AIAA Scitech 2019 Forum, 0629. https://doi.org/10.2514/6.2019-0629##
Ferrao, I., Vasconcelos, D., Ribeiro, D., Silva, A., & Barata, J. (2020). A study of droplet deformation: The effect of crossflow velocity on jet fuel and biofuel droplets impinging onto a dry smooth surface. Fuel, 279, 118321. https://doi.org/10.1016/j.fuel.2020.118321##
García-Geijo, P., Riboux, G., & Gordillo, J. M. (2020). Inclined impact of drops. Journal of Fluid Mechanics, 897, A12-46. https://doi.org/10.1017/jfm.2020.373##
Gordillo, J. M., Riboux, G., & Quintero, E. S. (2019). A theory on the spreading of impacting droplets. Journal of Fluid Mechanics, 866, 298-315. https://doi.org/10.1017/jfm.2019.117##
Hao, J., & Green, S. I. (2017). Splash threshold of a droplet impacting a moving substrate. Physics of Fluids, 29(1), 012103. https://doi.org/10.1063/1.4972976##
Hao, J., Lu, J., Lee, L., Wu, Z., Hu, G., & Floryan, J. M. (2019). Droplet splashing on an inclined surface. Physical Review Letters, 122(5), 054501. https://doi.org/10.1103/PhysRevLett.122.054501##
Jiang, M., & Zhou, B. (2020). Droplet behaviors on inclined surfaces with dynamic contact angle. International Journal of Hydrogen Energy, 45(54), 29848-29860. https://doi.org/10.1016/j.ijhydene.2019.07.173##
Josserand, C., & Thoroddsen, S. T. (2016). Drop impact on a solid surface. Annual Review of Fluid Mechanics, 48(1), 365-391. https://doi.org/10.1146/annurev-fluid-122414-034401##
LeClear, S., LeClear, J., Park, K. C., & Choi, W. (2016). Drop impact on inclined superhydrophobic surfaces. Journal of Colloid and Interface Science, 461, 114-121. https://doi.org/10.1016/j.jcis.2015.09.026##
Li, H. (2013). Drop impact on dry surfaces with phase change. [Doctoral Thesis, Mechanical Engineering, Technische Universität Darmstadt: Darmstadt]. Retrieved from https://tuprints.ulb.tu-darmstadt.de/id/eprint/3550##
Li, Y., Niu, X. D., Wang, Y., Khan, A., & Li, Q. Z. (2019). An interfacial lattice Boltzmann flux solver for simulation of multiphase flows at large density ratio. International Journal of Multiphase Flow, 116, 100-112. https://doi.org/10.1016/j.ijmultiphaseflow.2019.04.006##
Liang, H., Liu, H., Chai, Z., & Shi, B. (2019). Lattice Boltzmann method for contact-line motion of binary fluids with high density ratio. Physical Review E, 99(6), 063306. https://doi.org/10.1103/PhysRevE.99.063306##
Lunkad, S. F., Buwa, V. V., & Nigam, K. P. (2007). Numerical simulations of drop impact and spreading on horizontal and inclined surfaces. Chemical Engineering Science, 62(24), 7214-7224. https://doi.org/10.1016/j.ces.2007.07.036##
Marengo, M., Antonini, C., Roisman, I. V., & Tropea, C. (2011). Drop collisions with simple and complex surfaces. Current Opinion in Colloid & Interface Science, 16(4), 292-302. https://doi.org/10.1016/j.cocis.2011.06.009##
Mohammad Karim, A. (2023). Physics of droplet impact on various substrates and its current advancements in interfacial science: A review. Journal of Applied Physics, 133(3). https://doi.org/10.1063/5.0130043##
Moreira, A. N., Moita, A. S., & Panao, M. R. (2010). Advances and challenges in explaining fuel spray impingement: How much of single droplet impact research is useful? Progress in Energy and Combustion Science, 36(5), 554-580. https://doi.org/10.1016/j.pecs.2010.01.002##
Niu, X. D., Li, Y., Ma, Y. R., Chen, M. F., Li, X., & Li, Q. Z. (2018). A mass-conserving multiphase lattice Boltzmann model for simulation of multiphase flows. Physics of Fluids, 30(1), 013302. https://doi.org/10.1063/1.5004724##
Pasandideh-Fard, M., Bhola, R., Chandra, S., & Mostaghimi, J. (1998). Deposition of tin droplets on a steel plate: simulations and experiments. International Journal of Heat and Mass Transfer, 41(19), 2929-2945. https://doi.org/10.1016/S0017-9310(98)00023-4##
Pasandideh-Fard, M., Qiao, Y. M., Chandra, S., & Mostaghimi, J. (1996). Capillary effects during droplet impact on a solid surface. Physics of Fluids, 8(3), 650-659. https://doi.org/10.1063/1.868850##
Pereira, F. L. (2019). Effect of crossflow variation on impacting droplets. [Doctoral dissertation, Universidade da Beira Interior]. Portugal.##
Raman, K. A. (2019). Normal and oblique droplet impingement dynamics on moving dry walls. Physical Review E, 99(5), 053108. https://doi.org/10.1103/PhysRevE.99.053108##
Rioboo, R., Tropea, C., & Marengo, M. (2001). Outcomes from a drop impact on solid surfaces. Atomization and Sprays, 11(2). https://doi.org/10.1615/AtomizSpr.v11.i2.40##
Sahoo, N., Khurana, G., Harikrishnan, A. R., Samanta, D., & Dhar, P. (2020). Post impact droplet hydrodynamics on inclined planes of variant wettabilities. European Journal of Mechanics-B/Fluids, 79, 27-37. https://doi.org/10.1016/j.euromechflu.2019.08.013##
Shen, C., Yu, C., & Chen, Y. (2016). Spreading dynamics of droplet on an inclined surface. Theoretical and Computational Fluid Dynamics, 30(3), 237-252. https://doi.org/10.1007/s00162-015-0377-2##
Shusheng, Z., Hao, L., Li-Zhi, Z., Saffa, R., Zafer, U., & Huaguan, Z. (2020). A lattice Boltzmann simulation of oblique impact of a single rain droplet on super-hydrophobic surface with randomly distributed rough structures. International Journal of Low-Carbon Technologies, 15(3), 443-449. https://doi.org/10.1093/ijlct/ctaa004##
Šikalo, Š., Tropea, C., & Ganić, E. N. (2005). Impact of droplets onto inclined surfaces. Journal of Colloid and Interface Science, 286(2), 661-669. https://doi.org/10.1016/j.jcis.2005.01.050##
Wang, X., Xu, B., Chen, Z., Del Col, D., Li, D., Zhang, L., & Cao, Q. (2022). Review of droplet dynamics and dropwise condensation enhancement: Theory, experiments and applications. Advances in Colloid and Interface Science, 102684. https://doi.org/10.1016/j.cis.2022.102684##
Wang, Y., Shu, C., & Yang, L. M. (2015). An improved multiphase lattice Boltzmann flux solver for three-dimensional flows with large density ratio and high Reynolds number. Journal of Computational Physics, 302, 41-58. https://doi.org/10.1016/j.jcp.2015.08.049##
Xu, L. (2007). Liquid drop splashing on smooth, rough, and textured surfaces. Physical Review E, 75(5), 056316. https://doi.org/10.1103/PhysRevE.75.056316##
Xu, L., Zhang,, W. W., & Nagel, S. R. (2005). Drop splashing on a dry smooth surface. Physical Review Letters, 94(18), 184505. https://doi.org/10.1103/PhysRevLett.94.184505##
Xu, M., Zhang, J., & Chen, R. (2022). Cooling effect of droplet impacting on heated solid surface. International Journal of Heat and Mass Transfer, 183, 122070. https://doi.org/10.1016/j.ijheatmasstransfer.2021.122070##
Yang, L., Shu, C., Chen, Z., Wang, Y., & Hou, G. (2021). A simplified lattice Boltzmann flux solver for multiphase flows with large density ratio. International Journal for Numerical Methods in Fluids, 93(6), 1895-1912. https://doi.org/10.1002/fld.4958##
Yin, C., Wang, T., Che, Z., Jia, M., & Sun, K. (2018). Oblique impact of droplets on microstructured superhydrophobic surfaces. International Journal of Heat and Mass Transfer, 123, 693-704. https://doi.org/10.1016/j.ijheatmasstransfer.2018.02.060##
Zen, T. S., Chou, F. C., & Ma, J. L. (2010). Ethanol drop impact on an inclined moving surface. International Communications in Heat and Mass Transfer, 38(8), 1025-1030. https://doi.org/10.1016/j.icheatmasstransfer.2010.05.003##
Zhan, H., Lu, C., Liu, C., Wang, Z., Lv, C., & Liu, Y. (2021). Horizontal motion of a superhydrophobic substrate affects the drop bouncing dynamics. Physical Review Letters, 126(23), 234503. https://doi.org/10.1103/PhysRevLett.126.234503##
Zhang, R., Hao, P., & He, F. (2017). Drop impact on oblique superhydrophobic surfaces with two-tier roughness. Langmuir, 33(14), 3556-3567. https://doi.org/10.1021/acs.langmuir.7b00569##
Zhao, H., Han, X., Li, J., Li, W., Huang, T., Yu, P., & Wang, L. (2022). Numerical investigation of a droplet impacting obliquely on a horizontal solid surface. Physical Review Fluids, 7(1), 013601. https://doi.org/10.1103/PhysRevFluids.7.013601##