Isothermal Case: Two-Dimensional Flow

Abstract

We analyze the linear and nonlinear stage of the instability of a falling liquid film by using the average models developed in Chap. 6. Their linear stability characteristics, e.g. their description of spatially growing disturbances in relation to the convective nature of the instability, are shown to be in good agreement with the Orr–Sommerfeld eigenvalue problem (Chap. 3). By using the average models, the mechanism of the primary instability, already discussed in Chap. 3, is then re-investigated within the framework of the wave hierarchy analysis proposed by Whitham. We emphasize the similarities between roll waves in open channels and solitary waves in film flows at large Reynolds numbers. In particular, two-equation models of film flows have a structure similar to the Saint-Venant equations for shallow-water flows. In both cases, the mechanism of the primary instability can be understood in terms of a wave hierarchy as the competition between kinematic and dynamic waves. We scrutinize the influence of dispersive effects associated with the stream-wise second-order viscous terms, a phenomenon we refer to as “viscous dispersion,” onto the kinematic waves: viscous damping of high-frequency waves reduces the kinematic wave speed which in turn reduces the gap in speed between kinematic and dynamic waves. As far as the nonlinear stage of the dynamics of a falling liquid film is concerned, it is dominated by a competition between the primary instability of the Nusselt flat film flow and the secondary instabilities of the traveling waves with saturated amplitudes. This competition is characterized by a variety of nonlinear processes (e.g., spatial and temporal modulations, phase locking) which are still not fully understood. Applying a periodic forcing at the inlet may regularize the flow, leading further downstream to regular periodic wave-trains whose properties can be obtained using elements from dynamical systems theory. We construct bifurcation diagrams of permanent-form traveling waves including solitary waves. Particular attention is given to the role of stream-wise viscous effects on the properties, such as shape, speed and solution branches of the traveling waves. Taking into account these effects is crucial for a proper description of the dynamics of wavy film flows.