  This paper surveys results found by the authors in the previous papers (see
for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for
spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130,
288-292 (2018)) concerning calculation and classification of symmetry Lie
algebra and differential invariants for a variety of fundamental systems in
fluid mechanics, including the Euler and Navier-Stokes equations on a plane,
three-dimensional space, a sphere, and a spherical layer. In each case, the
symmetry Lie algebra is found, followed by a list of generating differential
invariants and invariant derivations for the group action. The main extension
is a detailed analysis of thermodynamic states, symmetries, and differential
invariants. This analysis is based on consideration of Riemannian structure
naturally associated with Lagrangian manifolds that represent thermodynamic
states. This approach radically changes the description of the thermodynamic
part of the symmetry algebra as well as the field of differential invariants.
