Liquid drop impact on granular beds: the influence of drop inertia and grain size

This paper explores crater formation resulting from the impact of a liquid drop on a densely packed granular bed composed of lightweight polystyrene beads. Several regimes based on the drop impact velocity v and diameter D, and the grain diameter d_g are identified. These regimes are discussed in terms of several dimensionless numbers, including a Froude number Fr, which compares the droplet's kinetic energy to its potential energy at impact, the Weber number We, which compares the inertial to capillary forces, and the size ratio d_g/D. At low We, Fr, and d_g/D, the dimensionless crater diameter D_max/D follows a power-law scaling as We^1/4, consistent with previous studies on droplet impacts on granular surfaces, where the crater size reflects the maximum droplet spreading observed on a solid surface. This situation is thus analysed using a so-called signature approach. In this situation, the crater size is also shown to quantitatively depend on d_g/D. When We exceeds a critical value We_c(d_g/D), the scaling deviates from We^1/4 and the crater size depends mainly on d_g/D. This transition is discussed in connection with the onset of droplet splashing. For larger d_g/D, a different power-law scaling emerges with an exponent smaller than 1/4, regardless of the value of Fr or We, and the splash transition no longer occurs under these conditions. This is consistent with other studies, highlighting the significant amount of energy transfer in crater formation, therefore referred to as the energetic approach. Overall, the final crater size is found to depend strongly on d_g/D among the droplet impact characteristics. To unify part of these observations, the role of local dissipation due to grain contact friction during crater formation is incorporated. This leads to the definition of a new dimensionless number , which combines the effects of grain-to-drop size ratio d_g/D and droplet inertia (via Fr). This parameter enables the collapse of D_max/D data onto a single curve for the range of parameters investigated in this study.