title: On the stability of the gravity-driven viscous channel flow
Creators: name: Kögel, Armin
Publisher: name: University of Bayreuth
Year: 2020
Subjects: subject: fluid mechanics
subject: film flow
subject: flow stability
subject: side walls
subject: 530
subject: 620

description: 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 damped and the flow is thus stabilized. The narrower the channel, the greater the percentage of the fluid disturbed by the side walls and thus the stronger the stabilizing effect. If the side walls are not flat, but periodically corrugated in the direction of the flow, the range of the disturbance increases significantly. This is also reflected in a correspondingly stronger stabilization. However, the influence of corrugations disappears if its periodicity is very large or very small, as in both cases the corrugated wall transitions to a flat wall. An unexpected influence of sidewalls on the stability of film flows can be seen in the fragmentation of the neutral curve. It will be shown that in channel flows, waves of a certain wavelength range are damped considerably, which may lead to a splitting of the unstable area of the stability map into two separate unstable isles. The wavelength of this selective damping depends almost exclusively on the channel width. The wider the channel, the longer the waves that are damped. A fragmentation of the neutral curve has so far only been observed in systems with undulated substrates, but not in systems with a flat substrate and flat sidewalls. Therefore the question arises how the channel width - i.e. a characteristic length transverse to the flow direction - can have such a significant influence on the damping wavelength - a characteristic length in flow direction. An attempt on explanation for this question can be provided by the shape of the surface waves that are artificially generated to measure stability: Friction on the sidewalls causes the wave crest to be curved. The curvature of the wave generally depends strongly on the wavelength, but also on the channel width. At small wavelengths (compared to the channel width) the shape of the wave is dominated by the interaction of subsequent waves, whereas the channel width has no influence. However, if the wavelengths are very large, the influence of adjacent waves vanishes and the curvature depends exclusively on the channel width. In the transition region between the two regimes, i.e. at wavelengths similar to the channel width, an unexpected but distinct maximum in curvature was measured. This maximum of curvature is approximately at the wavelengths where selective damping occurs. However, the wavelength of the maximum curvature is always slightly smaller than the damping wavelength. Based on these observations, it can only be speculated that there is a relationship between the curvature and the damping of a wave in a channel flow. However, the exact interaction between the shape of a wave and the stability of the underlying stationary flow as well as the physical background of both phenomena are highly complex and require further investigations. The damping effects presented in this thesis could be suitable for technical applications that require an undisturbed film. A stabilization of the flow can be achieved by adding side walls. This effect is further enhanced by the use of corrugated walls. Moreover, it is possible to achieve selective damping for a certain wavelength range up to arbitrarily high Reynolds numbers - at least within the measurement range of the experiments presented in this thesis. Finally, the damping wavelength can be tuned over a wide range by appropriate use of the side walls.