Symmetric solutions for the oscillatory channel flow with arbitrary suction

This paper considers a porous channel in which a suction-driven flow is modulated by arbitrary levels of fluid extraction acting uniformly along its porous boundaries. When small longitudinal oscillations are enabled, a rotational wave motion is established that this study attempts to analyse. For an elongated channel, two asymptotic methods are used. The first technique is based on a two-variable multiple-scale expansion that takes into account the thin boundary layer near the wall. While retaining generality of expression, the multiple-scale procedure is carried out until a closed-form solution for the velocity field is obtained for an arbitrary mean-flow function. An alternative approach based on WKB exponentials is also employed. The WKB expansion is then pursued to arbitrary order. These asymptotic formulations are shown to agree with one another and with numeric simulations of the problem for three specific mean-flow functions.