Numerical and Analytical Study of the Impact of Droplets on Substrates of Various Topologies

Abstract

Droplet impact, a common occurrence in everyday life such as rain hitting surfaces or ink spreading, has captured considerable interest across various scientific and technological domains like medicine, aerospace, and materials science. When a liquid droplet meets a solid surface, its behavior is shaped by a complex interplay of physical forces, including interfacial tension, gravity, and viscous effects, influencing its motion and spreading. This research primarily focuses on numerically and analytically studying the impact of droplets on substrates with various topologies. The present computational study utilizes finite volume-based axisymmetric simulations, employing the volume of fluid (VOF) method to anticipate intricate hydrodynamic phenomena. To conduct these simulations, the conservation equations for mass, momentum, and volume fraction are solved using the ANSYS Fluent 18.1 solver. Initially, the droplet surface undergoes continuous deformation upon impacting the thin cylindrical target, progressing through several critical stages: free fall, impact, cap formation, encapsulation, uncovering, and detachment. The computational study considers a range of cylinder-to-droplet diameter ratios (𝐷𝑐 𝐷𝑜⁄ ) from 0.13 to 0.4 to observe various deformation patterns of the droplet. The influence of parameters such as contact angle (𝜃), (𝐷𝑐 𝐷𝑜⁄ ), Weber number (𝑊𝑒), Ohnesorge number (𝑂ℎ), and Bond number (𝐵𝑜), on the maximum deformation factor is analyzed based on numerical results. The findings indicate that the maximum deformation factor increases with rising 𝑊𝑒 and decreasing contact angle. An analytical model is developed to explain the maximum deformation factor, showing excellent agreement with numerical findings. Additionally, a correlation is established to predict maximal deformation factors in terms of 𝜃, 𝐷𝑐 𝐷𝑜⁄ , 𝑊𝑒, and 𝑂ℎ, demonstrating strong accuracy within ±1% of the computational data. Following that, numerical studies have investigated the impingement and spreading dynamics of a water droplet around a small right-angled circular cone suspended in the air. An increase in the Weber number (𝑊𝑒) leads to a shorter interaction duration between the droplet and the substrate, particularly for specific values of 𝜃, 𝑂ℎ, and (𝐷𝑐 𝐷𝑜⁄ ). Additionally, the interaction time significantly decreases with an increase in the Ohnesorge number (𝑂ℎ), while 𝜃, 𝑊𝑒, and 𝐷𝑐 𝐷𝑜⁄ kept constant. The droplet dynamics of each stage were clearly observed using pressure contour and velocity vector. Moreover, to characterize the morphological and hydrodynamic behavior of water droplets impingement onto the hemispherical substrate, it has been studied computationally. The effects of various parameters are hemisphere-to-droplet diameter ratio (𝐷ℎ ⁄ 𝐷𝑜), contact angle (𝜃), Bond number (Bo), Ohnesorge number (𝑂ℎ), and release height (ℎ 𝐷𝑜⁄ ) on deformation factor (𝜉) of the droplet is delineated thoroughly. The droplet fails to detach from the target at higher Oh and greater 𝐷ℎ ⁄ 𝐷𝑜. Based on this, a scatter regime plot has been represented to distinguish between two different hydrodynamic behavior of droplets. Furthermore, the impingement mechanism of a liquid droplet on a solid torus surface is explored through numerical simulations and analytical methods. Key findings reveal that the central film ruptures early when the ratio of torus diameter to droplet diameter (𝐷𝑡 𝐷𝑜⁄ ) is lower, attributed to the development of a relatively thin film. Simultaneously, tiny droplets pinch off at 𝐷𝑡 𝐷𝑜⁄ = 0.83, while larger detached drops are observed at lower 𝐷𝑡 𝐷𝑜⁄ = 0.16 due to increased drainage through the hole. Moreover, the first pinch-off occurs more rapidly with the continuous increase of the Weber number (𝑊𝑒) for a specific 𝐷𝑡 𝐷𝑜⁄ and 𝜃 value, along with a scattered regime map aiding in distinguishing droplet configurations during impinge. A numerical investigation into the binary head- on collision of vertically aligned drops of equal size on a cylindrical substrate is presented. Various dimensionless parameters, including Weber number (𝑊𝑒), contact angle (𝜃), Ohnesorge number (𝑊𝑒), Bond number (𝐵𝑜), and diameter ratio (𝐷𝑐 𝐷𝑜⁄ ), are employed to characterize the coalescence and subsequent impingement on the cylindrical substrate. A higher value of 𝛽𝑓,𝑚𝑎𝑥is attained at greater Weber numbers (𝑊𝑒), holding 𝜃, 𝐷𝑐 𝐷𝑜⁄ , and 𝑂ℎ constant. When the diameter ratio (𝐷𝑐 𝐷𝑜⁄ ) is lower, and 𝑊𝑒 is higher, the ring drop separates from the merged parent drop. Additionally, a theoretical model has been formulated to ascertain the maximum deformation factor. Again, the mechanism of collision and drainage of liquid mass around the spherical substrate suspended within the hollow cylinder using Gerris open-source code by employing VOF methodology. The pattern of the interfacial morphology of droplet collision and drainage mechanism is presented using numerical contours. It has been observed that quantify the drainage of liquid volume passes through the passage, which is denoted as (𝑄∗ = 𝑄 𝑄𝑜⁄ ) is increasing pattern of 𝑄 𝑄𝑜⁄ with continuous progress of time stamp for all cases of 𝐷𝑠 ⁄ 𝐷𝑜 for a fixed value of 𝑊𝑒. Finally, numerical simulations have demonstrated the impingement mechanism of a hollow droplet on a solid cylindrical surface. The key findings show that the maximum deformation on the cylindrical target increases with a higher Weber number (We), a larger target-to-droplet diameter ratio (𝐷𝑐 𝐷𝑜⁄ ), and a decreasing contact angle. Additionally, as the Ohnesorge number (Oh) decreases, both the spreading diameter and the height of the counter-jet formed after the hollow droplet impact increase.