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.