Abstract:

Based on the Galerkin method, the approximate analytical solution to the transient velocity field is obtained for the high-viscosity fluid in a square cavity driven by the vibration of upper and bottom lids. Then, a dynamic equation describing the mixing process is proposed and the configuration variation of the tracer with time is obtained via the front tracking of passive tracer that numerically integrated by the fourth-order Runge-Kutta scheme. It is found that there exists periodic chaos in the square cavity because the flow field is sensitive to the initial position of particles, that when the micro-elements of fluid with different initial orientations are advected from the same initial position, the interface tension approaches to an asymptotical distribution mode and displays an exponential growth with the time, while the length ratio keeps constant, and that not only the interface tension of the tracers advected from different initial positions but also the geometrical configurations of tracer interfaces display a self-similarity.

Key words: periodicity, chaotic mixing, interface, self-similarity, Runge-Kutta scheme