New solutions for flow in a channel with porous walls and/or non-rigid walls

Abstract

We consider a class of exact solutions for flow in a channel. The resulting flows are three-dimensional and involve some known two-dimensional channel flows driven by suction/injection at the channel walls and accelerating/decelerating channel walls. After reviewing previous work we first consider a two-dimensional hybrid problem involving flow in a channel whose walls are porous with negative acceleration; a two-dimensional simple exact solution of the Navier–Stokes equations is known in this case. We also give some details of the temporal and spatial stability of this flow and we then generalise the steady solution to a class of exact three-dimensional solutions which involves unsteady and steady flows. The new component of the flow, which gives rise to an exact solution of the Navier–Stokes equations, is found to decay for increasing time if R, the Reynolds number, is less than a critical value, R 0, and grows without limit if R> R 0. A steady solution exists if R= R 0. This critical value of the Reynolds number is much less than the value of the Reynolds number at which the two-dimensional simple exact solution becomes unstable. Using the process of continuation the locus of the critical value of the Reynolds number, at which the flow is three-dimensionally growing, is determined for the channel flow driven by suction at the channel walls only and by accelerating walls only.