Dynamics of fluidized suspensions of spheres of finite size

We propose a one-dimensional theory of fluidized suspensions in which the fluids and solids momentum equations are decoupled by using a new mean drag law for the particles. Our mean drag law differs from the standard drag laws frequently used in that the drag is assumed to depend on the area fraction rather than the number density. For a monodisperse suspension of spheres of radius R, the area fraction and the number density are related by a simple geometrical construction that takes into account the area of intersection of the spheres with a plane perpendicular to the flow. For the linearized theory uniformly fluidized suspension is unstable but not Hadamard unstable. However, there is a distinguished set of marginally stable modes belonging to a countable set of blocked wave numbers α: α = 4.493 R, 7.7253 R, 10.904 R,... The nonlinear theory contains bounded solutions when a certain dimensionless "growth rate" parameter is below a critical value. The power spectrum of these bounded solutions is broad banded in both space and time, and is very low for the wave numbers that are marginally stable in the linear theory. These results agree with our experiments, as well as witht he previous experimental results from diffraction studies.