Mechanism of particle collision in the one-dimensional dynamics of gas-particle mixtures

A theory is developed for the one-dimensional flow of a gas, containing solid particles of two different sizes, in which the effect of particle collisions is accounted for as well as the interaction between the particles and the gas. It is assumed that the particles behave as smooth elastic spheres, that they follow the Stokes drag law and exchange heat with the gas at a Nusselt number of unity. It is shown that there exists a range of parameters which provides that (i) the viscous flow fields about each particle do not interfere during collision, and (ii) the random velocities imparted by one collision are damped before either particle suffers another collision. Using the assumption of small particle slip, the one-dimensional flow problem is solved explicitly up to first order terms in the small slip. It is found, of course, that the tendency of collisions is to cause the two particle-slip speeds to have more nearly the same value than they would in the absence of interparticle collision. It appears that, although the physical assumptions restrict the magnitude of the interparticle forces, the model does provide the proper limit for very strong particle interaction and can conceivably be applied in this range also without gross error