Linear stability and numerical analysis of vertical dense particulate flows in hydraulic conveying

In this paper, we investigate the concentration instabilities that arise in vertical dense particulate flows during hydraulic conveying. The widely used two-fluid model for dense particulate flows is employed, and existing closure relations for drag correlation, stress tensors, etc., found in the literature are utilized. The ratio between the pipe diameter and particle size should be greater than 60 in this model. Then, an investigation is conducted to analyze the linear stability of the two-fluid model in a homogeneous and bounded pipe to determine the growth rates, frequency, and propagation velocity of small disturbances. Results show that the vertical dense particulate flows in hydraulic conveying exhibit instability across a broad spectrum of controlling parameters. We also obtain a fully nonlinear transient numerical solution for the two-fluid system using the finite difference method. The numerical solution of the model shows a high level of concordance with the linear stability analysis conducted, accurately capturing the corresponding growth rates in the region of wave formation. In addition, it has been found that small disturbances transform into saturation waves with finite amplitudes when the nonlinear effect becomes dominant. The saturation waves are asymmetrical due to the imbalance of inertia, wall shear stress, particle pressure, and particle viscosity mechanisms. The instable mode corresponds to plug flow or slug flow.