Optimal Control of Thin Liquid Films and Transverse Mode Effects

Abstract

We consider the control of a three-dimensional thin liquid film on a flat substrate, inclined at a non-zero angle to the horizontal. Controls are applied via same-fluid blowing and suction through the substrate surface. We consider both overlying and hanging films, where the liquid lies above or below the substrate, respectively. We study the weakly nonlinear evolution of the system, which is governed by a forced Kuramoto-Sivashinsky equation in two space dimensions. The uncontrolled problem exhibits three ranges of dynamics depending on the incline of the substrate: stable flat film solution, bounded chaotic dynamics, or unbounded exponential growth of unstable transverse modes. We proceed with the assumption that we may actuate at every point on the substrate. The main focus is the optimal control problem, which we first study in the special case that the forcing may only vary in the spanwise direction. The structure of the Kuramoto-Sivashinsky equation allows the explicit construction of optimal controls in this case using the classical theory of linear quadratic regulators. Such controls are employed to prevent the exponential growth of transverse waves in the case of a hanging film, revealing complex dynamics for the streamwise and mixed modes. We then consider the optimal control problem in full generality, and prove the existence of an optimal control. For numerical simulations, we employ an iterative gradient descent algorithm. In the final section, we consider the effects of transverse mode forcing on the chaotic dynamics present in the streamwise and mixed modes for the case of a vertical film flow. Coupling through nonlinearity allows us to reduce the average energy in solutions without directly forcing the linearly unstable dominant modes. arXiv:1805.08236v1 [physics.flu-dyn] 21 May 2018 * k2 − q k2 /γ.

Point-actuated feedback control of multidimensional interfaces

“…The BDFs were also utilised in Tomlin et al (2017) for a non-local problem. These schemes have been employed for both the 1D and 2D optimal control problems for the KSE in Gomes et al (2017) and Tomlin et al (2019), respectively. We predominantly utilised the fourth-order BDF scheme, and performed tests with the other schemes for validation.…”

“…in the context of thin film flows on inclined flat substrates -see Tomlin et al (2019) for example. The schematic in Figure 1 shows the set-up for an overlying thin film flow with same-fluid blowing and suction controls at the substrate surface.…”

“…Additionally, there are the usual instabilities in the streamwise and mixed modes present for 0 κ < 1. Optimal controls involving the transverse modes alone were applied to (1.2) in (Tomlin et al, 2019), revealing windows of chaotic and travelling wave attractors for the streamwise and mixed modes. The success of the controls will be measured by two objectives; the first being the successful suppression of transverse growth resulting in bounded solutions, and the second being the stronger property of exponential stability of the flat film solution.…”

“…However, it may be the case that such a simplification overlooks important instabilities or mechanisms which are only observed from the full three-dimensional (3D) formulation of the original multiphase problem, e.g. Rayleigh-Taylor instabilities or electrostatically induced instabilities in liquid films (Tomlin et al, 2017(Tomlin et al, , 2019.…”

Pancake making and surface coating: Optimal control of a gravity-driven liquid film

“…Applications of optimal control in thin layer flows are reviewed in [13]. Relevant contributions include that of Grigoriev where the author finds optimal heating strategies to actively suppress evaporative instabilities in thin liquid films [14], that of Sellier and Panda where the optimal substrate shape to control the free surface of a liquid film is inferred [15], or that of Papageorgiou et al where the authors find the optimal source/sink distribution to control and stabilize falling liquid films [16,17]. Because thin liquid films are commonly described by the long-wave approximation which leads to second-order or fourth-order parabolic partial differential equations (PDEs), depending on whether the effect of surface tension are prevalent or not, it is natural to apply the standard framework of optimal control of PDEs as described in [18,19].…”