Infection of a Mixture of Small Particles Into a Flow, Treated as a Perturbation Problem

Small identical solid spherical particles are injected from a wall into a laminar flow of an incompressible viscous fluid. The concentration of particles is assumed to be low. The goal of this paper is to study the action of the particles on the velocity profile of the fluid. The equations used for the fluid-particles mixture reduce the effect of each particle to a Stokes force acting on its center. A perturbation method is applied to solve these equations: the non-dimensional number S=t_p/te, where to is a characteristic time for a particle and t_e a characteristic time of the flow, is required to be small. It is shown that this condition is compatible with the other assumptions, but that the equations for the mixture are valid only within a limited region close to the wall. The results prove that the velocity profile of the fluid is affected by the particles if the concentration of the particles at the wall is of order 1/Re S at least, where Re is the fluid flow Reynolds number. For the case where this condition is satisfied, velocity profiles are computed and written in closed form. Some typical profiles are shown for an uniform injection of particles.