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Influence of side walls and undulated topography on viscous gravity-driven film flow
(2012)
- While a gravity-driven viscous film flow down an inclined flat plane of infinite extent can be described by an easy analytical solution, flow problems in nature, like glacier movements or the liquid film on the human eye are much more complex. Also to optimize a large number of technical applications, like coating applications or heat exchanger devices, one has to investigate and understand how different influencing factors, like topological features on the substrate or a finite width of the system, influence the flow and its stability isolated from each other. By introducing a wavy structure to the underlying topography, which could be for example a model for roughness, new effects emerge in the flow, which cannot be observed in flows over a flat incline. Eddies can separate from the main flow at the lee side of the undulation for kinematic reasons, or induced by inertial effects. In biological systems these eddies are dead water areas, which are cut off from nutrient supply, in heat exchanger applications their appearance has a strong impact on the convective heat transport within the liquid. Furthermore, the amplitude of free surface of the liquid can be amplified immensely when the liquid is in resonance with the undulation of the underlying topography. In this work we study experimentally as well as numerically the complex interaction of this resonance phenomenon with the appearing of eddy structures in the valleys of the undulation and show, that one can suppress flow separation selectively even at rather high Reynolds numbers when one exploits the resonance phenomenon specifically. Another part of this work deals with the question how the presence of side walls and the contact angle of the liquid there influences the free surface shape of the liquid, the velocity field and the globally transported volume flux. While an additional no--slip condition at the wall causes additional friction and leads thus to a lower volume flux, capillary elevation at the side walls can generate a velocity overshoot in the vicinity of the walls, depending on the film thickness and the wetting properties of the liquid, which counteracts the additional friction coming from the walls. An extensive theoretical parameter study, which is supplemented with experimental data, provides criteria for the first onset of the velocity overshoot and gives answer to the question when the counteracting influences on the global volume flux just cancel each other. An experimental study of the free surface shape of a draining flow shows that this configuration cannot be described by a series of quasi-steady states, even when a dynamic contact angle is taken into consideration, although the flow changes only very slowly in time. Additional time dependent numerical simulations of the draining flow reveal an indentation of the free surface in the vicinity of the side wall, which could promote film rupture in technical thin film applications. Furthermore, side wall effects play an important role for the physical stability of the flow. Waves develop at the free surface of a gravity--driven flow and grow while they are traveling downstream, when a critical volume flux is exceeded. It is shown by experimental variation of the contact angle, the film thickness and the side wall distance, that the presence of side walls generates different effects which have competing influences on the stability of the flow. Capillary elevation leads to a pretensioning of the free surface, which tends to stabilize the free surface, just as the additional no-slip condition at the wall does. The emerging of a velocity overshoot in the capillary elevation on the other hand leads to a destabilization of the flow. In the system studied here the stabilizing influence of the side walls dominates over the destabilizing influence which is of comparatively short range, which means that this flow configuration is more stable than the corresponding flow of infinite extent. However, the results suggest that the destabilizing influences should dominate over the stabilizing influences in similar flow configurations when the film would become even thinner. While free surface film flows typically form long waves at first, we find for this flow configuration, that the type of instability changes from a long-wave type in the middle of the channel to a short-wave type instability, as it is well known for boundary layer flows, as the side wall distance is reduced.
