In this paper, the linear stability property of oscillatory flow in a circular pipe under a periodic pressure gradient is examined. Our results of neutral curve show the existence of a much more remarkable finger-like protrusions structure. In this case, we may speak about two neutral curves: one inside which corresponds to a rather late critical threshold and the other one which makes this threshold weaker but closer to experiment observations. The underlying mechanism is explained both from mathematical point of view and from physical point of view.
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