Systems of thermodynamically coordinated laws of conservation invariant under rotations

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

“…Such connection between thermodynamics and well-posedness of differential equations of continuum mechanics was first recognized by Godunov in [3] (see also Refs. [4,5,7,24,25,26,27]). The notation that we have used in the equations above and that we shall continue to use it in the rest of this paper is the following:…”

“…We denote the Lagrangian relative velocity by the symbolŵ = (ŵ 1 ,ŵ 2 ,ŵ 3 ) T = F T w. It turns out that the relative velocity w is not a conserved quantity in the Lagrangian framework. That is the reason why, in this section, we considerŵ as a state variables instead of w. Conversely, in the Eulerian frame, we will use w as the state variable (see (25)). …”

“…Finally, it is important to note (see also [27,25]) that the Godunov class of conservation laws, and in particular equations (18), has a complementary structure implied by the summation rule (8). Namely, in order to satisfy (8) (or (20)), the fields whose time evolution is governed by conservation laws with non-zero fluxes are grouped into pairs in which the second field in the pair has the physical interpretation closely related to the rate of the first field.…”

“…[25] (some details are shown in Appendices A, B and C). We use the notation introduced in (15), i.e.…”

“…(55)) in the Eulerian framework is the subject of the series of papers [24,25,26,27,7]. We shall discuss these issues below in this section.…”