ON THE ORIGIN OF WAVE PATTERNS IN FLUIDIZED-BEDS

In this report, a model designed for the description of the flow of two miscible phases in a fluidized bed is discussed. Apart from basic problems of modelling accurately such multi-phase flows, little analytical progress had been achieved in the investigation of a certain standard model based on the theory of interacting continua. It turns out, however, that the model under consideration can be investigated with the help of bifurcation theory. In particular, the methods of the theory of bifurcation with symmetry can be applied owing to the symmetries of the system. In general, a stationary homogeneous state exists in fluidized beds which can become unstable when the physical parameters of the system are varied. Then pattern formation takes place, e.g. in the form of one- and/or two-dimensional waves, bubbles, or convection patterns; also turbulent behaviour has been observed. In order to understand the occurrence of wave patterns and other phenomena as an inherent feature of the system, a finite, but periodically continued two-dimensional bed is investigated. While this suppresses certain boundary effects, it gives us thorough insight into the principal behaviour of this complicated system. In particular, it allows us not only to perform easily a linear stability analysis of the basic state of uniform fluidization, but also to conclude that bifurcation of travelling waves occurs when this state becomes unstable. Well-known patterns like vertical and oblique travelling waves (OTW) of the form u(x, y, t) = u(x - wt +/- ky), k greater-than-or-equal-to 0, are discovered. Owing to symmetry, the existence of standing travelling waves (STW) of the form u(x, y, t) = u(x - wt, y) is also expected, but regrettably no mathematically rigorous proof of this last conjecture is presently available. Bubble formation can also be approached via the instability of a vertical travelling plane wave train to transverse perturbations. Then a secondary stationary bifurcation to another kind of standing travelling waves takes place. This scenario is also in agreement with experimental observations. In addition, the occurrence of bifurcations of higher order, which lead to more and more complex wave patterns and are to be found on the route to turbulence, can be deduced.