3 research outputs found
Shear-thinning liquid films: Macroscopic and asymptotic behaviour by quasi self-similar solutions
No full textWe 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
Doubly nonlinear thin-film equations in one space dimension
No full textWe consider a free-boundary problem for a class of fourth-order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative. The problem is relevant in the description of the surface-tension driven spreading of a non-Newtonian liquid over a solid surface in the "complete wetting" regime. Relying solely on global and local energy estimates and on Bernis' inequalities, we prove existence of solutions to this problem, and obtain sharp upper bounds for the propagation of their support. A necessary condition for the occurrence of waiting-time phenomena is also derived
