Gravity-driven thin liquid films : rivulets and flow dynamics

TLDR: In this article, the authors focus on the formation of rivulets at the advancing front of a spreading liquid and the inner flow structures formed when a continuous fluid film flows over a substrate on whose surface topographical features are present, and formulate the governing equations for each problem in two distinct forms: one using the long-wave approximation theory and the other the full Navier-Stokes and continuity equations.

Abstract: The work presented in this thesis focuses on the classical problem of gravity-driven thin films flowing over rigid substrate. Two problems are considered, the formation of rivulets at the advancing front of a spreading liquid and the inner flow structures formed when a continuous fluid film flows over a substrate on whose surface topographical features are present. The governing equations for each problem are thus formulated in two distinct forms: one using the long-wave approximation theory and the other the full Navier-Stokes and continuity equations. Accordingly, two state-of-the-art computational methodologies are developed and utilised to extract tractable numerical solutions from the two equation sets. The first problem of rivulet formation, explored using an error-controlled adaptive multigrid method to solve the lubrication equations, builds on the seminal work of Huppert (1982). By constructing a systematic and thorough data set for both fully and partially wetting liquids, a new expression for the wavelength of the rivulet pattern is obtained incorporating the wetting properties of the film. Long-time solutions uncover the transient dynamics that are associated with rivulet formation such as the merging of neighbouring fingers. The study is extended to consider film flow on the outer and inner surfaces of a cylinder; curvature effects becoming prevalent as the radius of the cylinder decreases. The cylinder’s circumference counter-acts curvature effects in that at a critical value, the evolution of the contact line is restricted to a single rivulet. The impact of surface heterogeneities (topographic and chemical), as well as the presence of surface tension gradients, on rivulet evolution is also explored. Distinctly different in focus, the induction of the transport of liquid from separated re-circulating regions in the valleys of substrate topography is investigated. Results from this preliminary work demonstrate how pulsed surface waves passing over the topography to break the symmetry and excite the separatrix, forming lobes which transport liquid across the boundary between the bulk and eddy flow. Using particle tracking calculations to visualise this phenomena reveals the dependence of the transport enhancement on the size of the free-surface disturbance created.