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 ( $\beta _m$ ) are derived: $\beta _m \sim ({\textit{We}}/ {\textit{Oh}})^{1/6}$ for low-viscosity regimes ( ${\textit{Oh}} \lesssim 0.1$ ) and $\beta _m \sim \textit{Re}^{1/5}$ for high-viscosity regimes ( ${\textit{Oh}} \gt 1$ ), where $\textit{We}$ , $\textit{Re}$ and $\textit{Oh}$ are the Weber, Reynolds and Ohnesorge numbers, respectively. Both scaling laws show good agreement with the experimental data for their respective validity ranges of $\textit{Oh}$ . Furthermore, to better model experimental data at vanishing $\textit{Re}$ , we introduce a semi-empirical scaling law, $\beta _m \sim (A + {\textit{We}}/ {\textit{Oh}})^{1/6}$ , where $A$ is a fitting parameter accounting for finite spreading ( $\beta _m \approx 1$ ) at negligible impact velocities. This semi-empirical law provides an effective description of $\beta _m$ for a broad experimental range of $10^{-3} \leqslant {\textit{Oh}} \leqslant 10^0$ and $10^1 \leqslant {\textit{We}} \leqslant 10^3$ .