Euler-Poincare formulation and elliptic instability for nth-gradient fluids

The energy of an $n^{th}-$gradient fluid depends on its Eulerian velocity gradients of order $n$. A variational principle is introduced for the dynamics of $n^{th}-$gradient fluids and their properties are reviewed in the context of Noether’s theorem. The stability properties of Craik-Criminale solutions for first and second gradient fluids are examined.