The growth, saturation, and scaling behaviour of one- and two-dimensional disturbances in fluidized beds

It is well-known that fluidized beds are usually unstable to small perturbations and that this leads to the primary bifurcation of vertically travelling plane wavetrains. These one-dimensional periodic waves have been shown recently to be unstable to two-dimensional perturbations of large transverse wavelength in gas-fluidized beds. Here, this result is generalized to include liquid-fluidized beds and to compare typical beds fluidized with either air or water. It is shown that the instability mechanism remains the same but there are big differences in the ratio of the primary and secondary growth rates in the two cases. The tendency is that the secondary growth rates, scaled with the amplitude of a fully developed plane wave, are of similar magnitude for both gas- and liquid-fluidized beds, while the primary growth rate is much larger in the gas-fluidized bed. This means that the secondary instability is accordingly stronger than the primary instability in the liquid-fluidized bed, and consequently sets in at a much smaller amplitude of the primary wave. However, since the waves in the liquid-fluidized bed develop on a larger time and length scale, the primary perturbations need longer time and thereby travel farther until they reach the critical amplitude. Which patterns are more amenable to being visually recognized depends on the magnitude of the initially imposed disturbance and the dimensions of the apparatus. This difference in scale plays a key role in bringing about the differences between gas- and liquid-fluidized beds; it is produced mainly by the different values of the Froude number.