Transverse instability of plane wavetrains in gas-fluidized beds

Abstract

Using a two-fluid model of gas-fluidized beds, it is shown that periodic plane voidage waves travelling against gravity are unstable to perturbations with large transverse wavelength. This secondary instability sets in at arbitrarily small amplitudes of the plane wave and correspondingly small transverse wavenumbers of the two-dimensional perturbation. More precisely, if the bed is wide enough to accommodate sufficiently long horizontal waves, then the plane wave becomes unstable as soon as its amplitude has grown to the order of the square of the transverse wavenumber. The instability can be stationary or oscillatory in nature and has its origin in the interaction between the plane wave and four least-stable modes with small transverse wavenumber. Two of them represent a pair of bubble-like ‘mixed modes‘; the other two are initially, i. e. at the onset of the primary wave, pure transverse modes, one consisting only of an initially pure vertical velocity perturbation of the state of uniform fluidization. Depending on a relation between the eigenvalues of the least-stable modes at the primary bifurcation point, either one of these can be the dominant mode, which becomes (most) unstable along the growing vertically travelling plane wave. While the transverse modes gain longitudinal structure during this process, the mixed modes obtain a vertical component of the vertically averaged velocity as well, so that it appears that the secondary instability described here is a variant of Batchelor & Nitsche's (1991) ‘overturning’ instability found recently for unbounded stratified fluids, see also Batchelor (1993).