On the quasi-static relaxation of a drop in a combined model of dissipation

The spontaneous quasi-static relaxation of liquid drops on solid surfaces in the partial wetting regime is studied. We base our study on the combined approach suggested in de Ruijter et al. (Langmuir 1999, 15, 2209), which uses the standard mechanical description of dissipative system dynamics. Two types of dissipation are considered: a hydrodynamic dissipation in the core of the droplet and a dissipation term proportional to the contact line length. In the case of the spherical cap approximation of the drop shape, we derive asymptotic solutions of the differential equations describing the base radius and the contact angle relaxation. The asymptotic solutions are derived by a perturbation technique for small initial deviations of the base radius from the final equilibrium value. We find that the time relaxation of the base radius and the cosine of the contact angle are given by a linear combination of exponential functions. We compare the asymptotic solutions with numerically obtained solutions, and we find very good agreement for small initial deviations. For larger initial deviations, the numerical solutions are well approximated by exponential decay functions of first or second order.