Exploring Droplet Deformation and Jet Formation within a Liquid Medium: A Numerical Investigation

We conducted a numerical investigation centered on the distortion of droplets immersed in a liquid medium and the generation of jets utilizing the Volume of Fluid (VOF) technique. When a droplet accelerates, it results in the creation of a jet resembling a mushroom in shape. Our observations revealed that the velocity profile of the jet closely adheres to the similarity solution for a symmetrical free jet, which can be expressed as u / um = (1 + η²)^(-2). In this equation, η = σ`r/x is the similarity variable, where σ` is a constant, and um represents the axial velocity along the central line (u(x,y=0)). By investigating the capillary time scale and energy balance, we ascertained that the jet velocity (Vj) can be determined as V (1 - Oh)^(-1/2), where Oh = μ / (ρ R σ)^1/2. Here, R and V denote the droplet's radius and velocity, while ρ, µ, and σ represent the liquid's density, viscosity, and interfacial tension coefficient, respectively. When the Ohnsorge number is low (Oh ≈ 0.01), the jet velocity is approximately equal to the droplet velocity. As the Weber number (We = ρ V^2 R /σ) of the droplet increases, the jet velocity escalates, and the jet advances, piercing the droplet in a toroidal form. Furthermore, we formulated an analytical projection concerning the critical Weber number required for the jet to entirely traverse the droplet. This estimate was established through an energy analysis akin to Edgerton's experiment involving a bullet passing through an apple. Our discoveries indicated that the critical Weber number for jet formation is approximately 23, which closely aligns with our numerical findings. In this situation, the jet velocity closely mirrors the initial droplet velocity. Additionally, we observed that the maximum height of the jet (Hjmax ≈ 0.056 We) and the maximum jet radius (Rjmax ≈ - 0.036 We) exhibit a linear correlation with the Weber number. This linear correlation implies that the jet becomes slimmer and elongated as the droplet velocity increases. We categorized the formation of droplet shapes into three groups: short jet, long jet, and stretching break-up. We found that stretching break-up occurs when the droplet reaches a maximum height three times its diameter.