Weakly nonlinear coupling comes from deterministic noises due tofixed boundaries and mean stochastic motion of molecular gasesinside interior domains is investigated. The effect ofdeterministic surface wavy-roughness on molecular flows insideannuli with microfabricated solid walls was presented. Thecontinuum-limit equations are being solved using perturbationmethods with incorporated microscopic slip conditions along thesmall wavy-wall. The volume flow rate thus obtained, consideringfully-developed parallel flow cases, contains terms which arerelated to the Knudsen number (K n ) of the fluid; the ratio ofthe inner to outer radius ( $r_1/r_2$ ); roughness ratio(ϵ), phase shift (β) and wave number (k) of thesmall wavy-roughness elements. There are critical pairs(k,β) which tell us under certain K n and larger $r_1/r_2$ cases, the trend for the second order volume flow rateis increasing or decreasing.