2. Governing Equations

The following process is detailed in the following sections

Integral balance equations for primary physical quantities. First, integral balance equations for closed systems are written as a manifestation of principles of classical mechanics for closed systems, namely mass conservation, second principle of mehcanics, and first principle of thermodynamics. Starting from integral balance equations for closed systems (material systems, Lagrange description), Reynolds transport theorem is used to derive integral balance equations for open systems, either stationary in space (control volume, Eulerian description) or with arbitrary motion (arbitrary description).

Differential balance equations for primary physical quantities. Starting from integral balance equations, under the assumption of sufficient regularity of the physical quantities, divergence theorem and arbitrariety of the domain is used to derive differential (local) balance equations of primary physical quantities.

Differential balance equations for derived physical quantities. Starting from differential equations of primary physical quantities, differential balance equations are derived for other physical quantities, as an example kinetic energy, internal energy and entropy.

Integral balance equations for derived physical quantities. Starting from differential balance equations, and exploiting divergence theorem (in the “opposite direction” w.r.t. what has been done before, to get differential from integral equations), integral balance equations are derived for derived quantities.