Optimal control of thin liquid films and transverse mode effects - Tomlin,R; Gomes,SN; Pavliotis,G; Papageorgiou,D (SIAM Journal on Applied Dynamical Systems, 2019)
We consider the control of a three-dimensional thin liquid film on a flat substrate, inclined at a nonzero angle to the horizontal. Controls are applied via same-fluid blowing and suction through the substrate surface. The film may be either overlying or hanging, where the liquid lies above or below the substrate, respectively. We study the weakly nonlinear evolution of the fluid interface, 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 location 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. Next, we consider the optimal control problem in full generality and prove the existence of an optimal control. For numerical simulations, an iterative gradient descent algorithm is employed. Finally, 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 dominant linearly unstable modes.