T. Krasnopolskaya

We propose a new model and study the properties of a piezoceramic transducer interacting with an excitation device of limited power-supply. A special attention is given to the examination of the origin and stages of development of deterministic chaos in this system. It is shown that a great variety of effects typical for problems of chaotic dynamics is inherent in the system. The presence of several types of chaotic attractors is established, and moreover the existence of hyper-chaos is revealed. Various scenarios of transition from regular regimes to chaotic ones are explored. The phase portraits, Poincare surfaces of section and maps of some chaotic attractors are investigated. For some of the chaotic attractors, the spectral densities and distributions of invariant measures are obtained and explored.

The paper presents a new methodology for investigating and evaluating the basic properties of distributive laminar mixing in creeping flows. Our analysis is based upon conservation of some topological properties (e.g. connectedness and orientation) of the Lagrangian interface line under continuous transformations induced by an Eulerian velocity field. The principal advantage of our approach for line tracking is that despite complicated stretching and folding, the area of the blob, enclosed by the contour, is preserved. This gives possibility to develop three criteria for estimating the quality of mixing. The objective of this article is to expound the methodology of investigation of distributive laminar mixing of highly viscous materials by considering a typical example of two-dimensional Stokes flow in an annular wedge cavity.

Our methodology is based on the following steps: (1) determination of an analytical solution for the velocity field in the cavity; (2) observations on the deformation of the interface contour line of the stirring blob; (3) finding and classification of periodic points; (4) construction of statistical quantity measures for estimation of the quality of mixing at any given moment of time.