Theoretical Analysis of Some Thin-Film Flows over Complex Surfaces

Abstract

In this thesis, three different aspects of thin-film flow over complex surfaces are investigated. First, locally-unidirectional rivulet flow on a slippery surface is considered. We study rivulets with prescribed flux and either fixed semi-width or fixed contact angle. In both cases we determined the effect of varying the slip length on the rivulet. We found that in the limit of strong slip, for a rivulet of a perfectly wetting fluid and a rivulet with constant width, the velocity becomes large and plug-like, and the rivulet becomes shallow, while for a rivulet with positive constant contact angle, the velocity becomes large and plug-like, and the rivulet becomes narrow and shallow. Second, rivulet flow over and through a permeable membrane is considered. We study rivulets with prescribed

flux and either fixed semi-width or fixed contact angle. We found that whereas there is a physically realisable pendant rivulet solution only if the semi-width does not exceed a critical value, there are physically realisable sessile and vertical rivulet solutions for all values of the semi-width; moreover, a sessile rivulet with fixed semi-width has a finite maximum possible length which is attained in the limit of a wide rivulet. Lastly, patterns formed in a two-dimensional thin film with a Derjaguin-type disjoining pressure on a planar substrate with periodic wettability stripes is considered. Using Liapunov-Schmidt reduction, we study the local bifurcation structure of the problem for spatially homogeneous disjoining pressure and how the structure depends on the average film thickness. Using methods of local bifurcation theory and the continuation software package AUTO, we perform a continuation analysis of the steady state solutions and establish the existence of both nucleation and metastable regimes. The dependence of the steady state solutions on the wettability contrast are investigated for two forms of spatially non-homogeneous disjoining pressure.