SPECTRUM OF DENSITY-FLUCTUATIONS IN A PARTICLE FLUID SYSTEM .1. MONODISPERSE SPHERES

A method for calculating the density autocorrelation [rho'(x)rho'(x r)] for a homogeneous particle-fluid system in both physical and Fouri er transform space has been developed. The density autocorrelation was related to two quantities, the Overlap function which is defined as t he volume of intersection of two spheres as a function of the separati on distance and the radial distribution function (RDF) of the particle s. In dimensionless co-ordinates, the parameter that characterizes the density autocorrelation is the volume fraction of particles, alpha(1) , or equivalently the dimensionless mean separation distance (normaliz ed by the particle diameter), lambda = (3) root(2 alpha(2)/alpha(1)). For an isotropic randomly distributed system of particles, the density autocorrelation was observed to oscillate with the correlation distan ce r, with a wavelength that was proportional to lambda. The Fourier t ransform of the autocorrelation likewise oscillated with the wavenumbe r k, however the effect of changes in the particle volume fraction was limited to the first peak only. Subsequent peaks were more closely as sociated with the Overlap function. The results for the density autoco rrelation were extended to a particle-fluid system which experienced a n asymptotically large pressure gradient. This initially produced a un iform relative motion between the two fields. In this limit, other hig her-order moments such as the Reynolds stress can be related to the de nsity autocorrelation in a straightforward manner. Moreover the spectr al shapes of all moments collapse onto the density autocorrelation spe ctrum in this limit. It was pointed out that the uniform relative moti on will eventually become unstable because of hydrodynamic forces on t he particles induced by the relative motion. This effect was estimated by introducing a mildly attractive force into the RDF. The results de monstrated that the induced hydrodynamic force promoted a shift in the density spectrum toward small k (large scale) indicating an alternati ve mechanism for growth in the integral length scale.