Stability of periodic travelling waves in trickle beds

In trickle bed reactors, a gas and a liquid flow together through the interstices of a random array of solid packing. A pulsing flow pattern is obtained at high gas and liquid fluxes. A macroscopic volume averaged model can be used to mathematically represent the hydrodynamics of two-phase flow in packed beds. In one dimension, the onset and characteristics of pulsing flow can be modelled as periodic travelling waves. In this paper, the stability of the one-dimensional travelling waves is computed in the context of one-dimensional and two-dimensional perturbations of the full model. The 1-D waves are imposed as solutions of a pseudo spectral discretization of the 2-D equations. Floquet theory is applied to analyze the linear stability of these time-periodic solutions. The results show marked differences between the stability of fully developed waves and the corresponding uniform flow solutions when subjected to the same perturbations. The analysis also shows that two-dimensional flow patterns are likely to evolve from the early stages of pulse growth, and not from breakup of fully-developed 1-D waves.