  We consider the governing equations for the motion of compressible fluid on
an evolving surface from both energetic and thermodynamic points of view. We
employ our energetic variational approaches to derive the momentum equation of
our compressible fluid systems on the evolving surface. Applying the first law
of thermodynamics and the Gibbs equation, we investigate the internal energy,
enthalpy, entropy, and free energy of the fluid on the evolving surface. We
also study conservative forms and conservation laws of our compressible fluid
systems on the evolving surface. Moreover, we derive the generalized heat and
diffusion systems on an evolving surface from an energetic point of view. This
paper gives a mathematical validity of the surface stress tensor determined by
the Boussinesq-Scriven law. Using a flow map on an evolving surface and
applying the Riemannian metric induced by the flow map are key ideas to analyze
fluid flow on the evolving surface.
