Collision efficiency of a pollutant particle onto a long cylinder in low Reynolds number fluid flow

A method for calculating the collision efficiency of a small pollutant particle onto a solid long circular cylinder in a low Reynolds number fluid flow with inertia affects is presented. The cylinder is considered at rest in a uniform undisturbed flow at infinity, in the direction perpendicular to the cylinder axis.Assuming that the Reynolds number R based on cylinder radius b is very small but not zero ($R ll 1$), and the Reynolds number Re based on cylinder length l is of order unity, the force per unit length of the cylinder, correct to the order of R, is obtained, first for a general flow direction and then for the case of flow perpendicular to the cylinder axis. This is done by using the Naiver-Stokes equations in long slender bodies theory and applying matched asymptotic expansions in terms of the ratio $kappa$ of radius to body length. Flow field around the cylinder is calculated and the equation of particle motion is developed by applying Newton's second law of motion. The initial particle velocity far from the cylinder is calculated analytically and the particle trajectory course is solved numerically as an initial value problem by using Richardson Extrapolation and the Bulirsch-Stoer method.The collision Efficiency E is obtained by trial and error and is plotted against the dimensionless particle parameter p for different values of R (from 10$sp{-6}$ to 1). The numerical calculations show that the curves have a tendency to move to the right and become like a straight-line as R gets very small. The points at which E is less than 0.005 are also predicted