Abstract
First, the special-relativistic Theory of Irreversible Processes for a multi-component fluid is formulated. It is based on (i) the balance equations of the particle number and the energy-momentum for the total system (i. e., the mixture of the components) as well as the sub-systems (i. e., the components) and (ii) the dissipation inequality and the Gibbs equation for the mixture. In order to allow for reactions between the single components, in contrast to the total system, the sub-systems are assumed to be open, which means that their particle number and energy-momentum are not constrained by conservation laws. Without making any assumptions on the thermodynamic behavior of the interacting components, one arrives at a thermodynamic description of the mixture showing now heat conduction and viscosity. In particular, this makes it possible to calculate the entropy production and, thus, to identify thermodynamic currents and forces. In a second part, the post-Newtonian limit of this theory is calculated to show that for the mixture there result relations known from classical Extended Thermodynamics that partly are corrected by entrainment terms. The mathematical origin and physical consequences of these terms are discussed.
Addendum
After finishing our work, we became aware of two papers by W. Muschik which are related to our topic. First (W. Muschik, Entropy 21 (2019), 1034.), wherein a non-symmetric energy-momentum tensor is assumed in the framework of srTIP and second (W. Muschik, Entropy 20 (2018), 740.), where in the classical theory the possibility of different temperatures of the components of the mixture is considered.
Appendix A Appendix
A.1 The entropy identity for a multi-component fluid
From eqs. (46) and (32) we have the following relation:
On the left-hand side one writes the quantities for the mixture and on the right-hand side one does the same for the components. As already mentioned, we assume eq. (20), i. e., a perfect fluid for the components such that
Here, it is necessary to observe that the quantities for the mixture
From eqs. (24), (25) and (27) one derives
such that, with the help of eq. (20), the entropy identity simplifies to
If one uses the expression eq. (25) for the effective energy the second line vanishes identically, so that one remains with a condition for the entropy parts. However, this is also identically fulfilled by the decomposition of the entropy current eq. (32) if one uses eqs. (34) and (35). Therefore, the identity is proved.
A.2 Identification of fluxes and forces in Gibbs’ equation
One has to eliminate the time derivatives in the Gibbs equation eq. (38), i. e.,
First, eq. (38) is formulated for the K-th component and the second term can be rewritten as follows:
where in the last step one has added a zero. Altogether, it gives
and the last term vanishes because
The sum in the last equation is the total mass. One finally obtains
Therefore, the Gibbs equation can be written as
Now, one replaces the time derivative by (a) the internal energy, (b) the effective mass and (c) the concentrations.
To (a): From the balances of the energy eq. (29) and the mass eq. (10) it follows that
To (b): This can be directly read off from the balance equation for the mass eq. (10):
To (c): The time derivative of the definition of the concentration gives
which leads to
This can be also written as
Therefore, one writes
and using the balance equation of mass eq. (10) it follows that
Now, one considers the mass balance for a single component,
Adding a zero, one gets
and rearranging the terms leads to
The expression in the first bracket is the diffusion flux
and, together with eq. (103), for each component it finally follows that
and therefore also
Now, with the help of eqs. (97), (98) and (109), eq. (96) can be rewritten, where similar terms are already summarized. We have
Introducing instead of the mass productions
In order to bring the equation into the required form one has to add on the right-hand side the zero
The first term on the right-hand side is just
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