The effect of side walls on the stability of falling films

Although the stability of infinitely wide films has been extensively studied in the literature, there exists a very limited number of experimental studies considers the influence of side walls on the stability of falling films. These studies reports a stabilization effect at certain range of wavenumbers but have no theoretical explanation to offer. In this work, we build a comprehensive theoretical and experimental framework in order to study the effect of the side walls for the range of parameters where that interesting stabilizing behavior occurs. We combine a temporal biglobal stability analysis with experiments measuring the spatial growth rate of the sinusoidal waves flowing downstream an inclined channel. A very good agreement was found when comparing the theoretical and the experimental results. A strong lateral confinement of the channel stabilizes the flow. In the wavenumber-Reynolds number space, the instability region experiences a fragmentation due to the selective damping of moderate wavenumbers. For this range of parameters, the three dimensional confined problem shows several prominent stability modes which are classified into two categories, the well known hydrodynamic instability mode (H-mode), and a new stability mode that we call wall-mode (W-mode). The two modes are stabilized with a different degree, and at sufficiently small channel widths, the W-mode only is observed. The spatial structure of the eigenmodes experiences a significant restructuring at wavenumbers smaller than the most damped wavenumber. Our results also suggest that the mode selection is based on the preservation of the even spatial symmetry. A new behavior was observed numerically at moderate spanwise confinement where the flow is destabilized at relatively high Reynolds numbers. For this limited range of parameters, a new type of unstable mode appears, that was named wall-destabilizing mode (WD-mode) to differentiate from the W-mode.