On Steady Flow through a Channel Consisting of an Uneven Wall and a Plane Wall : Part 1. Case of No Relative Motion in Two Walls

A steady viscous flow through a two-dimensional, infinite channel consisting of an uneven wall and a plane wall is theoretically investigated under a given pressure gradient. The profile of the uneven wall is assumed to be periodic in the direction of the mean flow. A systematic expansion procedure is developed for the case where the ratio of the mean channel width to the period of the uneven wall, k, is small. An approximate solution is obtained up to the order of k2 . As a result, the flow rate is explicitly found as a function of the profile of the uneven wall, k and the Reynolds number. The unevenness of the wall always decreases the flow rate in proportion to k2 . The inertia effect on the flow rate first arises from the second order of k. The flow rate and the stream line are concretely obtained for the case of a sinusoidal wall. The approximate solutions show in good agreement with the solutions obtained by a numerical calculation.