How to identify

Tom

Technically the classification method for higher order problems follows exactly the derivation used for second order problems; i.e. look for lines along which the pde degenerates to an ode. You need to look at one of the books

Methods om mathematical physics (vol 2) by Courant and Hilbert,

Partial differential equations by Garabedian,

there's also a description in

Linear and nonlear waves by Whitham.

The Steady (2d or 3d) fluid equations are elliptic

The unsteady Euler equations are hyperbolic

The unsteady Navier-Stokes equations are parabolic