Macroscopic Streamline Integral Relations for Two-Phase Flows

A spout-fluidized bed is an example of a situation where a nonuniform fluid flow through a bed of particles causes particle circulation. Several integral relations are derived from a steady, two-dimensional two-phase flow model. The vorticity of the particle motion enclosed by a particle streamline is shown to be equal to the fluid vorticity enclosed by that streamline. The net flux of fluid vorticity through a particle streamline is shown to be equal to zero. The pressure drop along a fluid streamline is related to the net drag force along that streamline. The net flux of particle vorticity through a fluid streamline is given in terms of the pressure drop. Implications of these relations to the mechanics of particle-fluid flows are discussed. Relations giving the particle vorticity in terms of an integral of the fluid vorticity, and vice versa, are presented. A possible numerical scheme for calculation of the flow fields is discussed.