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Author: Bothe, Dieter × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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Now showing 1 - 5 of 5
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    Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Bothe, Dieter; Druet, Pierre-Étienne
    We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick-Onsager or Maxwell- Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolic-hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier- Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity.
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    Multicomponent incompressible fluids -- An asymptotic study
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Bothe, Dieter; Dreyer, Wolfgang; Druet, Pierre-Étienne
    This 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.
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    Well-posedness analysis of multicomponent incompressible flow models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bothe, Dieter; Druet, Pierre-Étienne
    In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities of the species stays constant. In this type of models, non solenoidal effects affect the velocity field in the Navier--Stokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.
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    Continuum thermodynamics of chemically reacting fluid mixtures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bothe, Dieter; Dreyer, Wolfgang
    We 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.
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    On the structure of continuum thermodynamical diffusion fluxes -- A novel closure scheme and its relation to the Maxwell--Stefan and the Fick--Onsager approach
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bothe, Dieter; Druet, Pierre-Étienne
    This paper revisits the modeling of multicomponent diffusion within the framework of thermodynamics of irreversible processes. We briefly review the two well-known main approaches, leading to the generalized Fick--Onsager multicomponent diffusion fluxes or to the generalized Maxwell--Stefan equations. The latter approach has the advantage that the resulting fluxes are consistent with non-negativity of the partial mass densities for non-singular and non-degenerate Maxwell--Stefan diffusivities. On the other hand, this approach requires computationally expensive matrix inversions since the fluxes are only implicitly given. We propose and discuss a novel and more direct closure which avoids the inversion of the Maxwell--Stefan equations. It is shown that all three closures are actually equivalent under the natural requirement of positivity for the concentrations, thus revealing the general structure of continuum thermodynamical diffusion fluxes.