Influence of particle dynamics on the instability for pattern formation in shallow pulsed beds

A granular layer can form standing-wave patterns, such as squares, stripes, and hexagons, when it is fluidized with a pulsed gas flow. These patterns resemble the well-known patterns formed in vertically vibrated granular layers, but are governed by different dimensionless numbers. Recent research [de Martin et al., Phys. Rev. Fluids 3, 034303 (2018)] reveals that the onset to pattern formation in shallow pulsed beds can be understood in terms of the dimensionless number Gamma(p) = u(a )/ u(t)(phi) over bar, where u(a), is the amplitude of the gas velocity, u(t) is the terminal velocity of the particles, and (phi) over bar is the average solids volume fraction. In contrast, pattern formation in vertically vibrated granular layers in vacuo is governed by the dimensionless number Gamma(v) = 4 pi(2) f(2) d/g, where f and d are the frequency and displacement of the vibrated plate, respectively, and g is the gravitational acceleration. In addition, the threshold for pattern formation in pulsed beds exhibits a strong dependence with the frequency of the excitation that is not observed in the threshold for pattern formation in vibrated systems. This work explores the origin of these differences by simulating the dynamics of a one-dimensional pulsed array of particles. Simulations reproduce well the experimental stability curves, and reveal that the criterion for instability in shallow pulsed and vibrated systems is actually the same; the layer flight time must be equal to 1/f. In pulsed beds, this criterion is determined by the traveling time of the kinematic wave that forms in each flow pulse. These results provide a theoretical basis to the recent experimental observations and highlights commonalities between the mechanisms behind pattern formation in thin vibrated granular layers and shallow pulsed fluidized beds.