On the asymptotic analysis of pulsating planar Poiseuille flow instabilities

The linear stability analysis of a pulsating flow between parallel plates at finite amplitudes and different frequencies demonstrates the presence of instability with a different nature to that initially reported in earlier studies at Wo<25. The time-dependent stability problem is formulated using a Floquet expansion of the perturbation and solved in a matrix approach. At higher pulsation frequencies (i.e. Wo>30), pulsation amplitudes and sufficiently large Reynolds numbers, the time-dependent baseflow displays multiple inflexion points throughout the entire duration of the pulsation. A scaling analysis of the temporal growth rate reveals that the instability is of inviscid type and scales with Re-1. Freezing the time-dependent baseflow for a particular set of parameters, the growth rate is simultaneously controlled by viscosity solely acting on the perturbation, and the pulsating frequency coupling the harmonics. In the asymptotic limit (Re ->+infinity), we demonstrate that the temporal problem simplifies to a set of steady stability analyses dependent on the phase of the pulsation. In particular, we show that in this limit, Rayleigh and Fj & oslash;rtoft's criteria for instability become valid again, that sub-harmonic periodic orbits can be used to estimate the inviscid growth rate and match with the asymptotic values predicted from the viscous stability analysis.