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Generalized Hydrodynamics and Transport Processes

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Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics

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Abstract

The generalized hydrodynamic equations derived from various kinetic equations for nonequilibrium ensembles in the previous chapters are capable of describing transport processes and hydrodynamic phenomena in fluids.

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Notes

  1. 1.

    If the fluid is chemically reactive, there appears a dissipation term representing reaction rates in (9.2). See Chap. 2 for the phenomenological treatment of chemical reacting fluids.

  2. 2.

    It, in fact, should be 13 plus \(3\left( r-1\right) \) moments if the independent diffusion fluxes are included. We use this pure fluid terminology for want of a more appropriate terminology, even if the system is a mixture and diffusions are present.

  3. 3.

    The nonlinear factor \(q_{n}^{-1}\) can be replaced by the equivalent quantity \(q_{L}=\sinh ^{-1}\Gamma /\Gamma \), where \(\Gamma ^{2}\) is a quadratic form of the thermodynamic gradients in (9.58)–(9.61). The details about this replacement procedure is discussed below; see the steps leading to (9.85)–(9.88) of this chapter.

  4. 4.

    The compressibility \(\kappa _{\widehat{\Psi }}^{*}\) is defined for the system evolving on the constant calortropy surface \(\widehat{\Psi }\) because the generalized hydrodynamic equations used here are subject to the calortropy differential as shown in the kinetic theory chapter of this work. This compressibility tends to the usual adiabatic compressibility \(\kappa _{S}^{*}\) as the system tends to equilibrium in the limit of \(N_{\delta }\rightarrow 0\). Equation (9.111) can be shown as follows: Since the calortropy density is described by an exact differential form in the thermodynamic space in the thermodynamic manifold for the present irreversible system, we find [30]

    $$\begin{aligned} \gamma _{0}&=\left( \frac{\partial \widehat{\Psi }}{\partial T}\right) _{p}\!/\!\left( \frac{\partial \widehat{\Psi }}{\partial T}\right) _{v} =\frac{\partial \!\left( \widehat{\Psi },p\right) }{\partial \!\left( T,p\right) }\frac{\partial \!\left( T,v\right) }{\partial \!\left( \widehat{\Psi },v\right) }\\&=\frac{\partial \!\left( v,p\right) }{\partial \!\left( T,p\right) } \frac{\partial \!\left( \widehat{\Psi },p\right) }{\partial \!\left( \widehat{\Psi },v\right) }=\left( \frac{\partial v}{\partial p}\right) _{T}\!/\!\left( \frac{\partial v}{\partial p}\right) _{\widehat{\Psi }}. \end{aligned}$$

    This proves (9.111).

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  1. Department of Chemistry, McGill University, Montreal, QC, Canada

    Byung Chan Eu

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Eu, B.C. (2016). Generalized Hydrodynamics and Transport Processes. In: Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-41147-7_9

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