Stabilising falling liquid film flows using feedback control

Abstract

Falling liquid films become unstable due to inertial effects when the fluid layer is sufficiently thick or the slope sufficiently steep. This free surface flow of a single fluid layer has industrial applications including coating and heat transfer, which benefit from smooth and wavy interfaces, respectively. Here we discuss how the dynamics of the system are altered by feedback controls based on observations of the interface height, and supplied to the system via the perpendicular injection and suction of fluid through the wall. In this study, we model the system using both Benney and weightedresidual models that account for the fluid injection through the wall. We find that feedback using injection and suction is a remarkably effective control mechanism: the controls can be used to drive the system towards arbitrary steady states and travelling waves, and the qualitative effects are independent of the details of the flow modelling. Furthermore, we show that the system can still be successfully controlled when the feedback is applied via a set of localised actuators and only a small number of system observations are available, and that this is possible using both static (where the controls are based on only the most recent set of observations) and dynamic (where the controls are based on an approximation of the system which evolves over time) control schemes. This study thus provides a solid theoretical foundation for future experimental realisations of the control of falling liquid films.

Falling liquid films with blowing and suction

“…Under the assumption of small deviations from a uniform state, with small F , both the weightedresidual equations and the Benney equations reduce via a weakly nonlinear analysis to a forced version of the Kuramoto-Sivanshinsky equations, for which optimal controls have been successfully applied . Work is in progress (Thompson et al 2015) to explore the effectiveness of feedback control strategies based on the nonlinear long-wave models derived in this paper.…”

Exploiting locality and translational invariance to design effective deep reinforcement learning control of the 1-dimensional unstable falling liquid film

“…where i (integer between 1 and N, the total number of jets) is the index of the jet currently considered, which is located between x-positions l i < x < r i , and A i (t) is the strength of the corresponding jet at time t. As visible in (6), this corresponds to using a small jet following a parabolic profile, going to zero on the right and left edges of each of the forcing areas, with the centers being located at positions c i = (l i + r i )/2 and the jets having half-widths w i = (r i − l i )/2. In the following, the maximum strength of the jets, as well as their widths and locations, will be used as physical meta-parameters of the flow configuration.…”

Point-actuated feedback control of multidimensional interfaces

“…In order to apply the following theory, the assumption that η is an exact solution of (1.2) with ζ = 0 is required. Convergence to non-solutions can only be achieved transiently by such a control methodology -see the discussion in Thompson et al (2016). We consider the difference w = η − η, which evolves according to…”