The present work is devoted to the solution of the steady-state diffusion equation for radon or thoron decay products within a fluid which moves with a parabolic velocity distribution through a circular tube. Taking into consideration axial diffusion, formation and attachment of the decay products to aerosol particles, the concentration distribution of the radioactive decay products was calculated. The transmission rate parameters for both free and attached decay products are given as a function of Peclet number and attachment rate. These parameters, as calculated from the present solution, show considerable dependence on the Peclet number. The obtained theoretical results can be extended to determine experimentally the diffusion coefficient of radioactive gases decay products and attachment rates of small ions to aerosol particles via the two filter method.