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

Abstract

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.

Surface-tension-driven dewetting of Newtonian and power-law fluids

“…Analysis of fluids with shear-dependent viscosity has previously been carried out in [5][6][7][8], for example, in the contexts of droplet spreading and film drainage. Other applications of thin-film modelling of power-law fluids include rivulet flows [9].…”

Local strong solutions to a quasilinear degenerate fourth-order thin-film equation

“…In the work [6] Ansini and Giacomelli establish the existence of global non-negative weak solutions to (1.3) for p > 2 and p−1 2 < n < 2p − 1. In [5] the same authors verify the existence of travelling-wave solutions and study a class of quasi-self-similar solutions to (1.1). Moreover, in [33] the authors establish the existence of weak solutions to a non-Newtonian Stokes equation with a viscosity that depends on the fluid's shear rate and its pressure at the same time.…”

Non-linear higher-order boundary value problems describing thin viscous flows near edges

“…In the case k = 0, Laugesen and Pugh [27][28][29][30] have studied in detail the existence and the stability properties of solutions to (8) which are either positive periodic or non-negative with equal contact-angle; for these solutions one necessarily has F = 0, that is q = 0 in terms of Problem (I). For appropriate choices of the functions involved and of the boundary conditions, well-posedness and/or properties of solutions to (8) have been considered in the contexts of wetting, coating and Tanner's law [2,3,5,6,11,16,18,19,25,36,37], dewetting [8,21,22], blowup [7,35,41] and shock formation [9,10,13] (see also the references therein and [20,32] for related PDE approaches). In all these cases F (that is q in terms of Problem (I)) is not an unknown of the problem (whereas the solution's domain often is) and the boundary condition are different, too.…”