Diffusion in Mixtures of Reacting Thermoelastic Solids (original) (raw)

Diffusion of chemically reacting fluids through nonlinear elastic solids: mixture model and stabilized methods

Mathematics and Mechanics of Solids, 2014

This paper presents a stabilized mixed finite element method for advection-diffusion-reaction phenomena that involve an anisotropic viscous fluid diffusing and chemically reacting with an anisotropic elastic solid. The reactive fluid–solid mixture theory of Hall and Rajagopal (Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body within the context of mixture theory. Math Mech Solid 2012; 17: 131–164) is employed wherein energy and entropy production relations are captured via an equation describing the Lagrange multiplier that results from imposing the constraint of maximum rate of entropy production. The primary partial differential equations are thus reduced to the balance of mass and balance of linear momentum equations for the fluid and the solid, together with an equation for the Lagrange multiplier. Present implementation considers a simplification of the full system of governing equations in the context of isothermal problems, although anisot...

A continuum theory of chemically reacting mixtures of fluids and solids

International Journal of Engineering Science, 1981

In this work, a continuum theory of a chemically reacting mixture composed of solids and fluids is presented. Following a different line of localization, a set of balance equations applicable to every component in the mixture is obtained. These are: conservation of mass, balance of momenta, conservation of energy and the principle of entropy which is also given for every species in the mixture. The localization is so applied that the individual energy equations are form invariant under time dependent motions of the spatial frame of reference and every state variables appearing explicitly in these equations can be considered as objective quantities. By use of the concept of reactive mass density for solid continuum and modifying the axiom of equipresence, originally developed for single bodies, a set of thermodynamically admissible constitutive relations is derived for a reacting mixture composed of a solid and a non-Newtonian fluid. Finally, utilizing the linearized field and constitutive equations, the propagation of simple harmonic waves in such a medium is studied and several particular cases are discussed.

Modelling of thermal diffusion and thermodynamic restrictions

International Journal of Engineering Science, 2014

Upon a review of the balance equations for a mixture, diffusion fluxes are shown to satisfy an evolution equation which, with slight differences, describes diffusion relative to a constituent or relative to the barycentric reference. As a consequence, the particular case of binary mixtures is considered with the view of establishing a connection with the current models. The fact that the diffusion flux is governed by a balance (evolution) equation is quite unusual in the literature since customary models are based on constitutive equations (Fick's law, Soret and Dufour models). This in turn motivates the analysis for a more appropriate thermodynamic setting. The second law for the whole mixture, along with some constitutive assumptions in classical (nonequilibrium) thermodynamics, is shown to provide the constitutive equations which are currently applied in the literature. Instead, the model of diffusion flux through a balance equation proves to be fully consistent with rational thermodynamics and the assumption that the functions, relative to a single constituent, depend on variables pertaining to that constituent whereas interaction terms are allowed to depend also on quantities pertaining to the other constituents.

Constitutive Relations of Thermal and Mass Diffusion

Journal of Non-Equilibrium Thermodynamics

Non-equilibrium thermodynamics provides a general framework for the description of mass and thermal diffusion, thereby including also cross-thermal and material diffusion effects, which are generally modeled through the Onsager coupling terms within the constitutive equations relating heat and mass flux to the gradients of temperature and chemical potential. These so-called Soret and Dufour coefficients are not uniquely defined, though, as they can be derived by adopting one of the several constitutive relations satisfying the principles of non-equilibrium thermodynamics. Therefore, mass diffusion induced by a temperature gradient and heat conduction induced by a composition gradient can be implicitly, and unexpectedly, predicted even in the absence of coupling terms. This study presents a critical analysis of different formulations of the constitutive relations, with special focus on regular binary mixtures. It is shown that, among the different formulations presented, the one whic...

Thermodynamic parameters of mixtures with allowance for phase transition components under shock-wave loading

Epitoanyag - Journal of Silicate Based and Composite Materials, 2017

the shock-wave synthesis and compaction using powder mixtures are the one of perspective directions of new materials creation. the results of numerical experiments on modeling of shock wave loading of mixtures with allowance for phase transition components in their composition are presented. the significant change in volume in the region of phase transition components included in the mixtures allows us to expand the range of variation of thermodynamic parameters of the mixtures under shock wave loading. the calculation model is based on the assumption that all components of mixture under shock-wave loading are in thermodynamic equilibrium (model tec). the model tec allows us to describe the region of the polymorphic phase transition, considering the material in the region of phase transition as a mixture of low-pressure phase and high-pressure phase. the good agreement of these model calculations with the data of different authors defined on the basis of experiments is obtained. thermodynamic parameters of the nitrides mixture, solid and porous mixtures with quartz as component were reliably described. this model is useful for determining the compositions and volume fractions of the components of the mixture to obtain the specified parameters of solid and porous materials under shock-wave loading. keywords: High pressure engineering, integrated circuits, mixtures, Phase transitions, Porous materials, shock waves, thermodynamic equilibria Sergey A. Kinelovskii Doctor of sciences, full Professor, chief scientist in LiH sB rAs. Author and co-author of more than 100 papers that includes Wos and scopus publications. Konstantin K. Maevskii scientist in LiH sB rAs. scientific degree candidate of Physico-mathematical sciences 2011. topic of PhD work "the model of dynamic loading of heterogeneous porous materials".

Thermodynamics of diffusing mixtures

1979

We present a formal description of a set of field equations for the thermodynamics of mixtures, proceeding from an integral formulation of the balance and imbalance laws to local equations. Our treatment is general in that we include volume-change as a kinematic descriptor and special in that we exclude multipolar effects. We apply the theory to derive restrictions on constitutive equations for a mixture of two fluids and a mixture of a solid and a fluid.

On the asymptotic spatial behaviour in the theory of mixtures of thermoelastic solids

International Journal of Solids and Structures, 2008

This paper is concerned with the study of asymptotic spatial behaviour of solutions in a mixture consisting of two thermoelastic solids. A second-order differential inequality for an adequate volumetric measure and the maximum principle for solutions of the one-dimensional heat equation are used to establish a spatial decay estimate of solutions in an unbounded body occupied by the mixture. For a fixed time, the result in question proves that the mechanical and thermal effects are controlled by an exponential decay estimate in terms of the square of the distance from the support of the external given data. The decay constant depends only on the thermal constitutive coefficients of the mixture.

Qualitative results for a mixture of Green-Lindsay thermoelastic solids

2018

We study qualitative properties of the solutions of a PDE system modeling thermomechanical deformations for mixtures of thermoelastic solids when the theory of Green and Lindsay for the heat conduction is considered. Three dissipation mechanisms are introduced in the system: thermal dissipation, viscosity effects on one constituent of the mixture and damping in the relative velocity of the two displacements of both constituents. We prove the existence and uniqueness of the solutions and their stability over the time. We use the semigroup arguments to establish our results.

On the thermodynamics of mixtures with several temperatures

International Journal of Engineering Science, 1970

Abstract This work concerns the formulation of a thermomechanical theory of a mixture where each constituent has its own temperature field. The theory also contains the effects of nonlinear elasticity, nonlinear heat conduction, nonlinear viscosity and diffusion. A linearized version of the general theory is also presented.