Viscous pressure-driven flows and their stability in channels with vertically oscillating walls

We study viscous flows in infinitely long two-dimensional channels which are driven by vertical periodic oscillations of their rigid walls and a prescribed horizontal pressure gradient. Exact solutions of the Navier-Stokes equations exist for these flows in the sense that a coupled system of partial differential equations (involving time and the vertical coordinate alone) determines the solutions. The flow due to the vertical periodic wall oscillations is of stagnation point form (the horizontal velocity is linear in the horizontal coordinate x) and couples with the prescribed horizontal pressure gradient to produce a horizontal velocity which is x-independent. We investigate the solutions computationally for wide ranges of Reynolds numbers as the amplitude of the wall oscillations increases. It is found that the x-independent flow loses stability (Floquet theory is used to quantify this) at values of the Reynolds number below those required to drive the purely oscillatory flow into the chaotic regime. A similar bifurcation picture emerges when the pressure gradients are time oscillatory. In addition, computations show that when the wall and pressure oscillations are synchronous, their amplitudes can be used to produce a change in the sign of the net flux across the channel over one time period. The computations identify parameter regimes that achieve a desired direction and magnitude of the time-averaged net flux across the channel.