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- ItemMulticomponent incompressible fluids -- An asymptotic study(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Bothe, Dieter; Dreyer, Wolfgang; Druet, Pierre-ÉtienneThis paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems: (i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the 2nd law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the non-appropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi-- or Gamma--convergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDE-system relying on the equations of balance for partial masses, momentum and the internal energy.
- ItemContinuum thermodynamics of chemically reacting fluid mixtures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bothe, Dieter; Dreyer, WolfgangWe consider viscous and heat conducting mixtures of molecularly miscible chemical species forming a fluid in which the constituents can undergo chemical reactions. Assuming a common temperature for all components, a first main aim is the derivation of a closed system of partial mass and partial momentum balances plus a common balance of internal energy. This is achieved by careful exploitation of the entropy principle which, in particular, requires appropriate definitions of absolute temperature and chemical potentials based on an adequate definition of thermal energy that excludes diffusive contributions. The latter is crucial in order to obtain a closure framework for the interaction forces between the different species. The interaction forces split into a thermo-mechanical and a chemical part, where the former turns out to be symmetric if binary interactions are assumed. In the non-reactive case, this leads to a system of Navier-Stokes type sub-systems, coupled by interspecies friction forces. For chemically reacting systems and as a new result, the chemical interaction force is identified as a contribution which is non-symmetric, unless chemical equilibrium holds. The theory also provides a rigorous derivation of the so-called generalized thermodynamic driving forces, avoiding the use of approximate solutions to the Boltzmann equations which is common in the engineering literature. Moreover, starting with a continuum thermodynamic field theory right away, local versions of fundamental relations known from thermodynamics of homogeneous systems, like the Gibbs-Duhem equation, are derived. Furthermore, using an appropriately extended version of the entropy principle and introducing cross-effects already before closure as entropy invariant couplings between principal dissipative mechanisms, the Onsager symmetry relations are a strict consequence. With a classification of the factors forming the binary products in the entropy production according to their parity instead of the classical distinction between so-called fluxes and driving forces, the apparent anti-symmetry of certain couplings is thereby also revealed. If the diffusion velocities are small compared to the speed of sound, the well-known Maxwell-Stefan equations together with the so-called generalized thermodynamic driving forces follow in the special case without chemical reactions, thereby neglecting wave phenomena in the diffusive motion. This results in a reduced model having only the constituents mass balances individually. In the reactive case, this approximation via a scale separation argument is no longer possible. Instead, we first employ the partial mass and mixture internal energy balances, common to both model classes, to identify all constitutive quantities. Combined with the concept of entropy invariant model reduction, leaving the entropy production unchanged under the reduction from partial momentum balances to a single common mixture momentum balance, the chemical interactions yield an additional contribution to the transport coefficients, leading to an extension of the Maxwell-Stefan equations to chemically active mixtures. Within the considered model class for reactive fluid mixtures the new results are achieved for arbitrary free energy functions.