In this paper, we present a Hamiltonian and thermodynamic theory
of heat transport on various levels of description. Transport of heat is formulated
within kinetic theory of polarized phonons, kinetic theory of unpolarized phonons,
hydrodynamics of polarized phonons, and hydrodynamics of unpolarized phonons.
These various levels of description are linked by Poisson reductions, where no
linearizations are made. Consequently, we obtain a new phonon hydrodynamics that
contains convective terms dependent on vorticity of the heat flux, which are missing
in the standard theories of phonon hydrodynamics. Within the zero-order Chapman-
Enskog reduction, the resulting hydrodynamic equations are hyperbolic and Galilean
invariant, while the first Chapman-Enskog expansion gives additional viscous-like
terms. The vorticity-dependent terms violate the alignment of the heat flux with
the temperature gradient even in the stationary state, which is expressed by a Fourier-
Crocco equation. Those terms also cause that temperature plays in heat transport a
similar role as pressure in aerodynamics, which is illustrated on numerical simulations
of flow past a cylinder. In particular, we show that the vorticity-dependent terms lead
to a colder spot just behind the cylinder, and for high-enough Reynolds numbers they
lead to the von Kármán vortex street.