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The dynamics along the particle trajectories for the 3D axisymmetric Eulerequations are considered. It is shown that if the inflow is rapidly increasing(pushy) in time, the corresponding laminar profile of the incompressible Eulerflow is not (in some sense) stable provided that the swirling component is notzero. It is also shown that if the vorticity on the axis is not zero (with someextra assumptions), then there is no steady flow. We can rephrase theseinstability to an instantaneous blow-up. In the proof, Frenet-Serret formulasand orthonormal moving frame are essentially used.

Mathematical analysis of pulsatile flow, vortex breakdown and instantaneous blow-up for the axisymmetric Euler equations