PROPERTIES OF A BIDISPERSE PARTICLE GAS SUSPENSION .2. VISCOUS RELAXATION-TIME SMALL COMPARED WITH COLLISION TIME

The properties of a dilute bidisperse particle-gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit tau(v) much less than tau(c), where tau(c) is the time between successive collisions of a particle, and tau(v) is the viscous relaxation time. In this limit, the particles relax close to their terminal velocity between successive collisions, and we use a perturbation analysis in the small parameter epsilon, which is proportional to tau(v)/tau(c), about a base state in which all the particles settle at their terminal velocities. The mean velocities of the two species are O(epsilon) different from their terminal velocities, and the mean-square velocities are O(epsilon) smaller than the square of the terminal velocity. The distribution functions for the two species, which incorporate the first effects of collisions between particles settling at their terminal velocities, are derived. The velocity distribution is highly anisotropic in this limit, and the mean-square velocity in the vertical direction is twice that in the horizontal plane. The distribution function for each species is singular at its terminal velocity, and the distributions are non-zero in a finite region in velocity space between the two terminal velocities.