Entropy inequality

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Here as usual [A]=A+-A. andO is the discontinuity speed. The entropy inequality yields... 

Solution (2.5) (see Fig. 4) describes the nucleation of a phase 2 (w = w ) which is accompanied by a generation of shock wave precursors in phase 1 (w = w ) and is satisfactory only until the first shock wave reaches the boundary of the segment [0,1]. The entropy inequality (2.4) is automatically satisfied for the precursors (moving with the speedDi) and is satisfied for the phase boundaries (moving with the speed if the area... 

Abstract In a thermodynamic framework which exploits the entropy inequality to obtain constitutive equations, it is common practice to assume charge neutrality and enforce this restriction using Lagrange multipliers. In this paper we show that the Lagrange multiplier used to enforce charge neutrality does not correspond to any known physical parameter, raising the question of whether charge neutrality can really be enforced. 

As a reference to something more familiar, consider the case of a fluid where incompressibility is enforced via a Lagrange multiplier. For a Stokesian fluid, it is assumed that the constitutive variables (stress, energy, heat flux) are a function of density, p, temperature, T, rate of deformation tensor, d, and possibly other variables (such as the gradients of density and temperature). Exploiting the entropy inequality in this framework produces the following constitutive restriction for the Cauchy stress tensor [10]... 

Hybrid mixture theory is a hybridization of classical volume averaging of field equations (conservation of mass, momenta, energy) and classical theory of mixtures [8] whose theory of constitution results from the exploitation of the entropy inequality in the sense of Coleman and Noll [9], In [4] the microscale field equations for each species of each phase, modified appropriately to include charges, polarization, and an electric field, are averaged to the macroscale, defined to be the scale where the phases are indistinguishable. Thus at the macroscale the porous media is viewed a mixture, with each thermodynamic property for each constituent of each phase defined at each point in space. 

Next we present the results of exploiting the entropy inequality which involve the Lagrange multiplier A, and/or the charge of a constituent, zaA In each case, the first form is the result from enforcing charge neutrality ... 

In connection with the dissipation mechanism of the entropy inequality, the relations for the heat flux vector... 

Additionally, the thermo-elastic behaviour will be described. Although, not all constitutive relations can be identified right now, the theoretical treatment of the entropy inequality is finished for the three phase model. 

In the above mentioned field equations the number of unknown quantities does not correspond to the number of equations, thus we have to conclude the problem with the constitutive equations for the partial stress tensors T , the interaction forces p", the partial internal energies ea and the partial heat flows q . From the evaluation of the entropy inequality of the saturated porous body, see de Boer [4], we obtain for the solid phase and the mobile phases with Index j3 = L, G the constitutive relations for T and p ... 

In addition to the momentum balance equation (6), one generally needs an equation that expresses conservation of mass, but no other balance laws are required for so-called purely mechanical theories, in which temperature plays no role (as mentioned, balance of angular momentum has already been included in the definition of stress). If thermal effects are included, one also needs an equation for the balance of energy (that expresses the first law of thermodynamics energy is conserved) and an entropy inequality (that follows from the second law of thermodynamics the entropy of a closed system cannot decrease). The entropy inequality is, strictly speaking, not a balance law but rather imposes restrictions on the material models. 

The entropy inequality principle must hold for all locations of the system. The closure law formulation is restricted by the second law of thermodynamics. 

When the temperature is not constant, the bulk heat transfer equation complements the system and involves Equations 5.240, 5.241, and 5.276. The heat transfer equation is a special case of the energy balance equation. It should be noted that more than 20 various forms of the overall differential energy balance for multicomponent systems are available in the literature." " The corresponding boundary condition can be obtained as an interfacial energy balance." - Based on the derivation of the buUc and interfaciaT entropy inequalities (using the Onsager theory), various constitutive equations for the thermodynamic mass, heat, and stress fluxes have been obtained. 

By applying the requirement according to which also total growth of entropy vanishes, the entropy inequality for the mixture reads... 

Balance of Energy and Entropy Inequality in Reacting Mixture ... 

The construction of entropy and absolute temperature (even in nonequiUbrium) fulfilling entropy inequality is done by Silhavy s method in terms of the primitives work, heat and empirical temperature (for the latter, see AppendixA.1 cf. Zemansky cited in [17, p. 53]). Moreover, the existence of energy satisfying the energy balance will be also proved. 

Existence of entropy and the entropy inequality for each system in the universe there exists a state function S, called entropy, such that for every process in the system... 

Thus, we obtain entropy inequality (1.21) for entropies defined relative to the same... 

Thus, the entropy inequality (1.21) was proved for any process p from state to af with entropies 5, and 5/ respectively, defined relative to the same (equilibrium stable) reference state ao (states a,, a/ and processes p between them may be arbitrary). 

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