The instability of oscillatory plane Poiseuille flow

Abstract

The instability of oscillatory plane Poiseuille flow, in which the pressure gradient is time-periodically modulated, is investigated by a perturbation technique. The Floquet exponents (i.e. the complex growth rates of the disturbances to the oscillatory flow) are computed by series expansions, in powers of the oscillatory to steady flow velocity amplitude ratio, about the values of the growth rates of the disturbances of the steady flow. It is shown that the oscillatory flow is more stable than the steady flow for values of Reynolds number and disturbance wave number in the vicinity of the steady flow critical point and for values of frequencies of imposed oscillation greater than about one tenth of the frequency of the steady flow neutral disturbance. At very high and low values of imposed oscillation frequency, the unsteady flow is slightly less stable than the steady flow. These results hold for the values of the velocity amplitude ratio at least up to 0·25.