This study investigates the formation of transient self-similar jets that develop at the rear side of an inertially driven liquid droplet. Similar jetting behavior is observed in various fluid dynamic systems, including collapsing standing waves, bursting bubbles, and cavity collapse following droplet impacts. We consider a droplet of diameter d, droplet density ρd, and dynamic viscosity μd, which is imparted with an initial velocity V0. The corresponding Weber number, We=ρdV02d/σ, within the range 140We708, and the Ohnesorge number, Oh=μ/ρlσd, remains below 0.1. Time is nondimensionalized using the inertial timescale T=d/V0. Within the interval 1.4t/T2.8, we find that the jet dynamics are largely independent of capillary and viscous effects, indicating that inertia dominates the flow evolution in this regime. To describe this behavior, we develop a self-similar framework based on negligible vorticity and inviscid potential flow. Within this framework, we identify distinct scaling laws characterized by the conditions that the far-field flow rate per unit length is approximately constant and the jet Weber number (Wej=ρlVjb2rj/σ)∝(t/T)−1/2, where ρl is the surrounding liquid velocity, Vjb is the jet base velocity, and rj is the local jet radius. Incorporating this behavior into our formulation yields a self-similar exponent (ε)=1/2, contrasting with the classical inertio-capillary exponent of 2/3. Our results confirm the existence of a transient self-similar inertial regime and offer new insights into the physical mechanisms driving jet formation in inertially dominated flows.
Senapati et al. (Thu,) studied this question.