Evaluation of the exact coagulation kernel under simultaneous Brownian motion and gravitational settling

The analytic representation of the exact coagulation kernel for combined Brownian motion and gravitational settling entails an infinite sum of ratios of modified Bessel functions. Heretofore, the evaluation of this sum has been limited to relatively small arguments due to computational difficulties, hindering the numerical solution of the general dynamic equation in computational aerosol transport simulations. An approximation and an asymptotic extension have been available, but their accuracy for large arguments has not been established, and they have been shown to lead to numerical diffusion and oscillation. Using multiple-precision arithmetic, exact values for the kernel are presented in this paper for large arguments, and it is shown that both the asymptotic and approximating formulae provide a poor fit in the range of practical interest. Based on the results, the asymptotic formula is rectified and it is shown that the approximation is no longer needed. Practical formulae are given to facilitate rapid numerical evaluation of the exact kernel for all arguments.