Flow in a slowly-tapering channel with oscillating walls

The flow of a fluid in a channel with walls inclined at an angle to each other is investigated at arbitrary Reynolds number. The flow is driven by an oscillatory motion of the wall incorporating a time-periodic displacement perpendicular to the channel centreline. The gap between the walls varies linearly with distance along the channel and is a prescribed periodic function of time. An approximate solution is constructed assuming that the angle of inclination of the walls is small. At leading order the flow corresponds to that in a channel with parallel, vertically oscillating walls examined by Hall and Papageorgiou \cite{HP}.
A careful study of the governing partial differential system for the first order approximation controlling the tapering flow due to the wall
inclination is conducted. It is found that as the Reynolds number is increased from zero the tapering flow loses symmetry and undergoes exponential growth in time. The loss of symmetry occurs at a lower Reynolds number
than the symmetry-breaking for the parallel-wall flow.
A window of asymmetric, time-periodic solutions is found at higher Reynolds number, and these are reached via a quasiperiodic transient from a given set of initial conditions.
Beyond this window stability is again lost to exponentially growing solutions as the Reynolds number is increased.