Analytical solution of droplet impact on solid surfaces at short and intermediate times

Impact pressure temporal and spatial profiles caused by a droplet on a surface can assist in understanding of microscale erosion mechanisms. We derive analytical solutions of pressure profiles and impact force time-series on a surface for early and intermediate impact times, up to dimensionless time $0.27$ , using unsteady potential flow. The solutions reproduce the ‘ring pattern’ of surface pressure at early impact and the temporal evolution to a shift to, at intermediate time, the centred maximum. This is due to the emerging dominance of the steady component of the Bernoulli equation. The analytical solutions use a model of an unsteady disk in infinite liquid to induce flows similar to those within impacting drops. We identify a separation point close to the expanding edge of the impacted droplet, which interprets the Wagner condition for droplet impact and extends the validity of the analytical wet radius from time $\textit {O}(10^{-2})$ in the literature to beyond $\textit {O}(1)$ . This separation point resolves singularity issues at the wet radius and solutions address the droplet impact problem. The theoretical predictions agree with high-fidelity numerical calculations of the impact pressure at the surface centre, middle and at the radius of the ring pressure, and with the impact force for more than four time decades. The theoretical predictions remain at least qualitatively correct in dimensionless time for more than $\textit {O}(1)$ . The analytical solutions are closed and explicit functions of space, time, wet radius expanding velocity and incidence angle, and provide ready estimation of droplet impact loadings for erosion problems.