Weakly nonlinear coupling comes from deterministic noises due to fixed boundaries and mean stochastic motion of molecular gases inside interior domains is investigated. The effect of deterministic surface wavy-roughness on molecular flows inside annuli with microfabricated solid walls was presented. The continuum-limit equations are being solved using perturbation methods with incorporated microscopic slip conditions along the small wavy-wall. The volume flow rate thus obtained, considering fully-developed parallel flow cases, contains terms which are related to the Knudsen number (Kn) of the fluid; the ratio of the inner to outer radius ($r_1/r_2$); roughness ratio (ϵ), phase shift (β) and wave number (k) of the small wavy-roughness elements. There are critical pairs (k,β) which tell us under certain Kn and larger $r_1/r_2$ cases, the trend for the second order volume flow rate is increasing or decreasing.