State of the Art of the Theory of Parametric Control in Continua and its Industrial and Technological Applications

The study presents the development of physical, mathematical, and numerical models for the parametric excitation and suppression of oscillations at the interfaces separating continuous media. Through computational, physical, and natural experiments, the research reveals new phenomena and parametric effects that enhance existing and create new high-efficiency technological processes. The scientific novelty lies in advancing the theory and applications of parametric excitation and suppression of oscillations on continua boundaries, exemplified in three task classes: flat and radial spreading film flows of viscous incompressible liquids, surfaces of phase transition from liquid to solid, and heterogeneous granular media. The research considers external actions such as alternating electromagnetic, vibration, acoustic, and thermal fields. Both linear and non-linear parametric oscillations, including strongly non-linear cases, are explored, with theoretical findings substantiated by experimental data. The study discusses the general and specific features of parametrically excited oscillations and newly discovered parametric effects for technical and technological applications. It evaluates the current state, analyzes features, prospects for further development, and addresses main methodological, mathematical, and applied challenges.