Quasi-self-similarity for wetting drops.

Abstract

We develop and experimentally test a quasi-self-similar solution for the spreading of viscous nonvolatile droplets over a dry and horizontal solid substrate, under the condition of complete wetting (spreading parameter $S>0$) with both gravity and Laplace pressure as driving forces. The problem does not admit a self-similar solution because two dimensional characteristic parameters, namely, the slipping length $\ensuremath{\lambda}$ and the capillary distance $a$, cannot be ruled out. Therefore, we approximate the solution by the members of a family of self-similar solutions, each one corresponding to different values of the ratios $\frac{{x}_{f}}{a}$ and $\frac{{h}_{0}}{\ensuremath{\lambda}}$, where ${x}_{f}$ and ${h}_{0}$ are the instantaneous drop extension and central thickness, respectively. This treatment of the problem also produces as explicit formula (which must be integrated) to predict the drop radius. The excellent agreement with our own and other authors' experimental data suggests that the approach can be considered as an interesting tool for solving problems where strict self-similarity fails.