Droplet Hamilton Dynamics 1: the rolling angle of a water droplet on an inclined surface

The rolling angle calculation of a water droplet on an inclined surface is not only an everyday phenomenon but also forms an essential part of many industrial processes. Previous researchers always used Newtonian mechanics (the vector mechanics) to set up the differential equation by analyzing the forces affecting on the droplet on an inclined surface and get the rolling angle. Here, by constructing the single-valued mapping from the droplet mass center motion parameters to the moving droplet body, we use the Hamilton's principle (the energy method) and the Karush-Kuhn-Tucher (KKT) theory to set up Euler equation sets of mass center motions. In initial conditions of “sphere shape-zero velocity”, we solve Euler equation sets and gain the droplet mass center motion equations. By judging whether the singularities (velocity zero points) exist or not in the phase space of mass center motions, we judge the droplet “is pinned” or “continuously rolls” on the inclined surface. Finally, we calculate out the critical tilt angle of the inclined surface, which just makes the droplet “continuously roll”, as the droplet rolling angle. Our results from the energy method can also improve analyzing the kinetic behaviors of droplets in multi-physical fields, such as phase-change droplets, triboelectrically induced electrostatic droplets and polar droplets in fixed electrostatic fields.