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 ( β ₘ ) are derived: β ₘ ~ (We/ Oh)1/6 for low-viscosity regimes ( Oh 0.1 ) and β ₘ ~ 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, β ₘ ~ (A + We/ Oh)1/6 , where A is a fitting parameter accounting for finite spreading ( β ₘ ≈ 1 ) at negligible impact velocities. This semi-empirical law provides an effective description of β ₘ for a broad experimental range of 10⁻³ Oh 10⁰ and 10¹ We 10³ .