  An internal energy function of the mass density, the volumetric entropy and
their gradients at n-order generates the representation of multi-gradient
fluids. Thanks to Hamilton's principle, we obtain a thermodynamical form of the
equation of motion which generalizes the case of perfect compressible fluids.
First integrals of flows are extended cases of perfect compressible fluids. The
equation of motion and the equation of energy are written for dissipative
cases, and are compatible with the second law of thermodynamics.
