Viscous/viscoelastic Rayleigh–Taylor instability accounting for the number of fingers in a droplet impacting a plane surface

In this study, we develop unified and analytical frameworks to examine the effect of viscosity, elasticity, and viscoelasticity on the Rayleigh–Taylor instability (RTI), which underlies finger formation during prompt splashing as a droplet impacts a flat metal surface. We complement our theoretical developments with experimental validations designed to match our theoretical predictions. A new dimensionless number, R=Re/We3/4, is introduced to characterize the evolution of the finger patterns. Three distinctive regimes are identified based on our analysis: when R≲1, the number of fingers scales with Re2/3; for 1≲R≲10, the finger count is influenced by both Re and We, a regime not extensively studied previously; and for R≳10, the count becomes insensitive to Re. We also discern a transient deceleration effect, represented by g=16V02/D, which prompts perturbation development due to RTI. It is noted that the constant 16 is dependent on fluid and surface physical properties. Though our theoretical predictions closely align with experimental observations, it is noteworthy that in experimental settings, g exhibits significant temporal variability. Further, our study extends to include viscoelastic effects, facilitating comparisons with recent advancements in managing finger formation in splashing scenarios. Additional experiments targeting medium R values further corroborate our theoretical model. This comprehensive analysis not only reaffirms but also enhances the understanding of splashing dynamics by integrating complex material behaviors and characteristics, thus offering a substantive benchmark for future research in the field.