A unifying approach to account for droplet impact forces

Droplet impact occurs in diverse applications such as pesticide treatment and inkjet printing. The maximum impact force of the droplet, a crucial factor in these applications, is dictated by the interplay of inertial, capillary, and viscous forces. Through direct numerical simulations, we demonstrate that the viscous dissipation within the droplet dictates the maximum impact force. For low Ohnesorge numbers (dimensionless viscosity of the liquid drop), the maximum impact force is a function of the impact Weber number (dimensionless inertia of the impacting drop) only, as shown by Zhang et al., Phys. Rev. Lett. 129, 104501 (2022). However, at high Ohnesorge numbers, the maximum impact force becomes independent of the Weber number, as viscous dissipation consumes the entire initial kinetic energy of the impacting droplet. A force balance explains why the impact force scales with the square root of the Ohnesorge number in that regime. To systematically account for the scaling behaviors in the various regimes, we extend the energy dissipation splitting approach of the Grossmann-Lohse theory for wall-bounded turbulent flows [Grossmann & Lohse, J. Fluid Mech. 407, 27-56 (2000)] to the droplet impact problem. With this, we can elucidate the parameter dependences of the impact force of liquid droplets falling on superhydrophobic surfaces.