On the maximum spreading of viscous droplets impacting flat solid surfaces

We experimentally and theoretically examine the maximum spreading of viscous droplets impacting ultra-smooth solid surfaces, where viscosity plays a dominant role in governing droplet spreading. For low-viscosity droplets, viscous dissipation occurs mainly in a thin boundary layer near the liquid-solid interface, whereas for high-viscosity droplets, dissipation is expected to extend throughout the droplet bulk. Incorporating these dissipation mechanisms with energy conservation principles, two distinct theoretical scaling laws for the maximum spreading factor (βm) are derived: βm ∼(We/Oh)1/6 for low-viscosity regimes (Oh≲ 0.1) and βm ∼Re1/5 for high-viscosity regimes (Oh>1), where We, Re and Oh are the Weber, Reynolds and Ohnesorge numbers, respectively. Both scaling laws show good agreement with the experimental data for their respective validity ranges of Oh. Furthermore, to better model experimental data at vanishing Re, we introduce a semi-empirical scaling law, βm ∼(A +We/Oh)1/6, where A is a fitting parameter accounting for finite spreading (βm ≈1) at negligible impact velocities. This semi-empirical law provides an effective description of βm for a broad experimental range of 10-3 ≤ Oh ≤ 100 and 101 ≤ We ≤ 103.