Shear-thinning liquid films: Macroscopic and asymptotic behaviour by quasi self-similar solutions

We consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity in the complete wetting regime. In the case of constant viscosity, the no-slip condition leads to a force singularity at advancing contact lines. It is well known nowadays that the introduction of appropriate slip conditions removes this paradox and alters only logarithmically the macroscopic behaviour of solutions at intermediate timescales. Here, we investigate a different approach, which consists in keeping the no-slip condition and assuming instead a shear-thinning rheology. This relaxation leads, in lubrication approximation, to fourth order degenerate parabolic equations of quasilinear type. By analysing a class of quasi-self-similar solutions to these equations in the limit of Newtonian rheology, we obtain a scaling law in time for macroscopic quantities (such as macroscopic profile, effective contact-angle) which is only logarithmically affected by the shear-thinning parameters. As opposed to positive slippage models, the scaling law is uniform for large times as far as the macroscopic support is well defined, and thus could also describe the asymptotic behaviour of a large class of solutions for fixed shear-thinning rheology