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^{1}Retired from Ecole Polytechnique and Total, France

**Keywords:** EoS, partial derivatives**property:** speed of sound**material:** -

We present a general version of the method of iterates previously introduced in the thermodynamics of simple systems [1].

Iterates (It) are, in present context, indexed scalar quantities that depend only on d, the degree of freedom of a thermodynamic system. There are 1,2,… k-iterates, the number of indices is 1+k(d-1). k-Its allow to compute kth order partial derivatives (PD) but the converse is not true. Iteration allows to pass from k-It to k+1-It (from which their name). Equations of state are 0-It, and, when known, higher order iterates may be explicitly computed. Iterates are arranged in a hierarchy with linear and quadratic structure relations between them at each level. We detail such algebraic systems in the case of d=1,2 and 3, thus covering the cases of quasi-static processes, single component and binary mixtures. Critical point or phase change loci are treated simply by adding relations between iterates. By solving these algebraic systems we get dictionaries of iterates that are useful for practical/theoretical purposes, i.e. for computing PD or finding PDE, as done in [1].

The method also applies to fluid dynamics by extending the 4D thermodynamic space (TS) with three additional dimensions v,x,t, (v the fluid velocity, z the space variable, t the time), we get a 7D fluid dynamic space (FD) with coordinates S,ρ,T,p,v,z,t. The Euler equations for the unidirectional motion of a unviscid fluid are rewritten in terms of 1- iterates. By solving the so called Cauchy problem for the complete system, we recover that this system is hyperbolic, it has two real characteristics that describe simple waves, propagating at a finite velocity. The new result is that both characteristics project onto a single isentrope in TS thus proving that the governing equation in [1] is parabolic just because it is a vertical projection of the hyperbolic Euler equations. Furthermore, this seamless integration of thermodynamics and fluid dynamics provides a physical interpretation of Riemann Invariants.

Binary mixtures are represented by (d=3) submanifolds S laying in a 6D TS, with coordinates, e.g. S,ρ,x_{1},T,p, μ_{1} ( x_{1} is the molar fraction of species 1, μ_{1} its chemical potential). We write down their algebraic systems. Dictionaries of 1- and 2-iterates are then established in various coordinate systems including the Leung and Griffiths one. A GE equation in density arises in the same way as for the single component case [1] and is again parabolic.

Thus, the method of iterates appear as promizing tool to solve thermodynamic formulation problems.

- Valentin P., Balian R., 18th ECTP conference, 31/8-4/9 2008, Pau

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