Abstract
Assuming linear relations between the thermodynamic fluxes and the thermodynamic forces derived in Chap. 7, in this chapter we apply Onsager’s reciprocal relations to determine the most general symmetry conditions that are satisfied by the phenomenological coefficients (Sects. 8.1 and 8.2). Then, in Sect. 8.3, these conditions are applied to particular cases, obtaining the constitutive relations of pure fluids, multicomponent mixtures, metals and non-isotropic media. Finally, in Sect. 8.4, we see how these same symmetry conditions can be applied to find the cross correlation properties of the fluctuating thermodynamic fluxes.
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Notes
- 1.
Examples are Fourier’s law of heat transport, Fick’s law of mass transport and Newton’s law of momentum transport.
- 2.
In certain cases, this fact is trivial. For example, the phenomenological coefficient relating a vectorial flux and a second-order tensorial force must be a third-order tensor, and there is no isotropic third-order tensor. For details, see [5].
- 3.
Note that ∑ m ϵ ijm ϵ mkl =δ ik δ jl −δ il δ jk and ∑ jk ϵ ijk ϵ jkl =2δ il .
- 4.
Here we have used Gibbs’ notation: \(\mathbf{A}\mathbf{:}\mathbf{B} = A_{ij} B_{ij} \).
- 5.
Clearly, this is always true in liquid systems.
- 6.
A simple example of thermo-diffusion is when the hot rod of an electric heater is surrounded by tobacco smoke: as the small particles of air nearest the hot rod are heated, they create a fast flow away from the rod, down the temperature gradient, thereby carrying with them the slower-moving particles of the tobacco smoke.
- 7.
Care should be paid to implement this experiment, as the final stationary state is reached after a time L 2/D, which may be very long.
- 8.
Sometimes, it is preferred to use the thermal diffusion factor, k T =ϕ (1) ϕ (2) TD′/D, with values laying between 0.01 and 1.
- 9.
For details, see [4].
- 10.
Note that, consequently, E′=−∇ϕ (e)=E−∇μ (e)/z (e) is an effective electric field.
- 11.
See the beautiful description in [8].
- 12.
Here we assume that the variables are all of the x-type, i.e. they are invariant to time reversal transformation. Generalization to y-type variables is straightforward.
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Mauri, R. (2013). Constitutive Relations. In: Non-Equilibrium Thermodynamics in Multiphase Flows. Soft and Biological Matter. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5461-4_8
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