959 research outputs found
Continuum thermodynamics of chemically reacting fluid mixtures
We consider viscous, 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, we derive a closed
system of partial mass and partial momentum balances plus a mixture balance of
internal energy. This is achieved by careful exploitation of the entropy
principle and requires appropriate definitions of absolute temperature and
chemical potentials, based on an adequate definition of thermal energy
excluding diffusive contributions. The resulting interaction forces split into
a thermo-mechanical and a chemical part, where the former turns out to be
symmetric in case of binary interactions. For chemically reacting systems and
as a new result, the chemical interaction force is a contribution being
non-symmetric outside of chemical equilibrium. 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. Moreover,
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 become
a strict consequence. With a classification of the factors in the binary
products of the entropy production according to their parity--instead of the
classical partition into 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 Maxwell-Stefan
equations follow in the case without chemistry, thereby neglecting wave
phenomena in the diffusive motion. This results in a reduced model with only
mass being balanced individually. In the reactive case ..
A compressible mixture model with phase transition
We introduce a new thermodynamically consistent diffuse interface model of Allen--Cahn/Navier--Stokes type for multi-component flows with phase transitions and chemical reactions.
For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques.
We consider two scaling regimes, i.e.~a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized
Allen-Cahn/Euler system for mixtures with chemical
reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young--Laplace and a Stefan type law
On jump conditions at phase boundaries for ordered and disordered phases
This is a study on jump conditions across the interface between two adjacent phases. The interface behaves as a free boundary, and in sharp interface models jump conditions are used to determine the values of thermodynamic fields at the free boundaries. In this study the jump conditions are derived from balance equations for singular surfaces that do not have singular lines, i.e. triple junctions are not considered here. At first we present the most general form of jump conditions to give a general framework, from where we consider various special cases with a focus on the influence of mechanical fields on the interfacial processes. The special cases include the Hoffmann/Cahn capillarity vector theory and jump conditions for interfaces where order/disorder transitions are involved. Furthermore we discuss interfacial chemical reaction laws, and in particular the creation and annihilation of vacancies at a liquid/solid interface
A study of the coarsening in tin/lead solders
This paper presents a model, which is capable to simulate the coarsening process observed during thermo-mechanical treatment of binary tin-lead solders. Fourier transforms and spectral theory are used for the numerical treatment of the thermo-elastic as well as of the diffusion problem encountered during phase separation in these alloys. More specifically, the analysis is based exclusively on continuum theory, first, relies on the numerical computation of the local stresses and strains in a representative volume element (RVE). Second, this information is used in an extended diffusion equation to predict the local concentrations of the constituents of the solder. Besides the classical driving forces for phase separation, as introduced by Fick and Cahn-Hilliard, this equation contains an additional term which links the mechanical to the thermodynamical problem. It connects internal and external stresses, strains, temperature, as well as concentrations and allows for a comprehensive study of the coarsening and aging process
Modeling diffusional coarsening in eutectic tin/lead solders: A quantitative approach
This paper presents a quantitative simulation of the phase separation and coarsening phenomenon in eutectic tin/lead (SnPb) solders. The computer modeling is based on continuum theory and field phase models which were evaluated using the most recently available data for the free energy of the tin/lead system, diffusional and mobility coefficients, elastic constants as well as surface tensions of both phases. The model presented allows to study the influence as well as the interaction between classical diffusion of the Fickean type, surface energies according to Cahn and Hilliard, as well as stresses and strains on phase separation and coarsening. An attempt is made to compare the temporal development of a eutectic SnPb microstructure at different temperature levels and subjected to different stress levels as predicted by the model to actual experiments
Kinetic flux-vector splitting schemes for the hyperbolic heat conduction
A kinetic solver is developed for the initial and boundary value problems (IBVP) of the symmetric hyperbolic moment system. This nonlinear system of equations is related to the heat conduction in solids at low temperatures. The system consists of a conservation equation for the energy density e and a balance equation for the heat flux 혘푖, where 푒 and 혘푖 are the four basic fields of the theory. We use kinetic flux vector splitting (KFVS) scheme to solve these equations in one and two space dimensions. The flux vectors of the equations are splitted on the basis of the local equilibrium distribution of phonons. The resulting computational procedure is efficient and straightforward to implement. The second order accuracy of the scheme is acheived by using MUSCL-type reconstruction and min-mod nonlinear limitters. The solutions exhibit second order accuracy, and satisfactory resolution of gradients with no spurious oscillations. The secheme is extended to the two-dimensional case in a usual dimensionally split manner. In order to prescribe the boundary data we need the knowledge of the 푒 and 혘푖. However, in experiments only one of the quantities can be controlled at the boundary. This problem is removed by using a continuity condition. It turned out that after some short time energy and heat flux are related to each other according to Rankine Hugoniot jump relations. To illustrate the performance of the KFVS scheme, we perform several one- and two-dimensional test computations. For the comparison of our results we use high order central schemes. The present study demonstrates that this kinetic method is effective in handling such problems
Micro-macro transitions by interpolation, smoothing, averaging and scaling of particle trajectories
We consider a Newtonian system of many diatomicmolecules each of which consisting of two atoms of equal mass whichare separated by a fixed distance. The barycenters are allowed to movealong some fixed straight line. Moreover each molecule has an additionalrotational degree of freedom. The atoms of neighbouring moleculesinteract to each other by a generic pair potential. By means of thisexample we propose a new method for deriving macroscopic models from microscopic ones. The method is based on the definition of macroscopicobservables and the derivation of corresponding balance laws by interpolation smoothing/averaging and subsequent scaling of particle trajectories
Cold, thermal and oscillator closure of the atomic chain
We consider a simple microscopic model for a solid body and study the problematic nature of micro/macro transitions. The microscopic model describes the solid body by a many particle system that develops according to NEWTONs equations of motion. We discuss various initial value problems that lead to the propagation of waves. The initial value problems are solved directly from the microscopic equations of motion. Additionally these equations serve to establish macroscopic field equations. The macroscopic field equations consist of conservation laws, which follow rigorously from the microscopic equations, and of closure relations which are completely determined by the distributions of the microscopic motion. In particular we consider three kinds of closure relations which correspond to three different kinds of equilibrium. It turns out that closure relations cannot be given appropriately without relating them to the initial conditions, and that closure relations might change during the temporal development of the initial data, because the body undergoes several transitions between different states of local equilibrium. In those examples that we have considered, the macroscopic variables mass density and temperature do not constitute an unique kind of microscopic motion
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Continuum thermodynamics of chemically reacting fluid mixtures
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
On the approximation of periodic traveling waves for the nonlinear atomic chain
We study a scheme from \cite{FV99}, which allows to approximate periodic traveling waves in the nonlinear atomic chain with nearest neighbour interactions. We prove a compactness result for this scheme, and derive some generalizations. Moreover, we discuss the thermodynamic properties of traveling waves
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