Local interaction between vorticity and shear in a perfect incompressible fluid

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Abstract
We show from a simple model related to the Euler equations that the flow of an incompressible and inviscid fluid diverges in a finite time. For this we look at the local interaction between vorticity and shear by neglecting the gradients of these two quantities in their equations of motion. A non linear system of 8 first order differential equations is obtained whose asymptotic behaviour can be easily obtained. The two largest eigenvalues of the shear tensor diverge to + ∞ and the smallest one to — ∞. The vorticity vector also diverges and lies along the eigenvector of the shear tensor which corresponds to the (positive) intermediate eigenvalue, thus giving a positive sign to the energy spreading function of the von Kármán-Howarth equation. At the same time the rotation of the shear principal axis stops
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