Heterogeneous structures of a particle-fluid system at constant fluid flow velocity and constant pressure drop are simulated by the so-called pseudo-particle modeling (PPM). The mass specific energy consumption N-st for suspending and transporting the solids is obtained by a weighted average method simultaneously. The stability criterion, N-st = min, proposed in the energy-minimization multi-scale (EMMS) model is demonstrated numerically, which reflects the spatial and temporal compromise of the movement tendencies of the fluid and solid phases, and leads to the formation of heterogeneous structures. Further work shows that when the total mass specific energy consumption for the solids (N-T) is variable, N-st/N-T = min should be used as a more general expression of the original stability criterion. Analysis of simulation results also shows dilute-to-dense phase drag responsible for the difference between N-st and N-T. (c) 2005 Elsevier Ltd. All rights reserved.