On the stability of the gravity-driven viscous channel flow

Abstract

Gravity-driven viscous film flow is a classic problem in fluid mechanics. It represents a simplified model for a variety of technical and natural systems. In the simplest example, the plane film flow over an inclined plane of infinite extent, both the stationary solution - known as Nusselt flow - and the primary instability can be determined by simple means from the Navier-Stokes equations. The Nusselt flow with its parabolic velocity profile offers a good estimate of flow behavior in real systems, i.e. it is possible to predict both the stationary flow and its instability. The latter is often of great interest, particularly for technical systems, since surface waves caused by the instability are usually undesirable. A prominent example are coating processes that require a film of constant thickness. Research on film flows has surpassed these simple model in recent decades for two main reasons: 1) Simplified models are not sophisticated enough to represent real systems. The surfaces in technical and natural systems are often not perfectly smooth but rough, either accidentally or intentionally. In addition, a real flow cannot be infinitely extended. It must be limited, for example, by side walls. To describe these complex multidimensional flows, refined models and further experimental investigations are necessary. 2) Since particularly for many technical systems the formation of surface waves should be suppressed, control of the primary instability is crucial. This can be achieved by varying the substrate, but also by appropriate use of sidewall effects. This thesis deals mainly with the influence of side walls on film flows over flat surfaces, i.e. the difference between channel flow and plane film flow. For this purpose, the stationary flow, the primary instability and the shape of the resulting waves were investigated with different experimental methods and compared with the numerically obtained results for the plane film flow. In a first step, the equations needed to calculate the primary instability of the plane film flow - namingly the Orr-Sommerfeld equation and the corresponding boundary conditions - are derived. These equations can only be solved analytically in the limit of very long waves, but with numerical methods the whole parameter space is accessible. Furthermore, it must be ensured that all measurements of steady-state flow and stability are not influenced by inflow effects of the channel. For this purpose, the inflow area was characterized by the measurement of the film thickness and surface velocity along the channel. The measurement positions for all further measurements were chosen in such a way that all inflow effects have decayed to this point. It will be shown that side walls have a stabilizing effect on film flow and the stabilization increases significantly with decreasing channel width. This can be explained by the influence of side walls on the stationary flow: In close vicinity of the sidewalls the flow is slowed down. In this disturbed area, surface waves are…