Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Doi
Abstract
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