Viscous flow through unsteady symmetric channels

The high-Reynolds-number (K) flow through a symmetric channel, with walls whose shape is time dependent, is studied. The distortion of the walls is of non-dimensional height $O(K^{-\frac{1}{3}}$ ) and length O(1), this particular size of perturbation being chosen such that (for the first regime of unsteadiness studied) the effects of the unsteadiness, viscous diffusion and advection all interact nonlinearly in the region of the fluid near the walls. For this first regime of unsteadiness the problem is solved numerically. This leads on to analytic descriptions for progressively faster time variations of wall shape, and in fact the entire range of unsteadiness is covered for this particular size of distortion.