Constitutive relations for the energy transfer in nonsaturated continuous mixtures (original) (raw)
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The continuum theory of mixtures, specially developed to model multiphase phenomena, in which the phases are treated as overlapping continuous constituents, requires additional source terms, absent in a contimmm mechanics approach, in order to couple the transport processes among the constituents. A physical interpretation for the internal heat source, present in the energy balance equations for a solid-fluid mixture, is proposed by relating the heat transfer processes between both constituents in two distinct approaches: continuum theory of mixtures and continuum mechanics. This leads to an analogy between the local temperature difference and the usual heat transfer coefficients.
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Bulk kinematic properties of mixtures such as velocity are known to be the density weighed averages of the constituent velocities. No such paradigm exists for the heat flux of mixtures when the constituents have different temperatures. Using standard principles such as frame indifference, we address this topic by developing linear constitutive equations for the constituent heat fluxes, the interaction force between constituents, and the stresses for a mixture of two fluids. Although these equations contain 18 phenomenological coefficients, we are able to use the Clausius-Duhem inequality to obtain inequalities involving the principal and cross flux coefficients. The theory is applied to some special cases and shown to reduce to standard results when the constituents have the same temperature.
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The subject of this paper is the steady-state heat transfer process in a rigid mixture with N continuous constituents, each of them representing a given continuous body. A continuous mixture consists of a convenient representation for bodies composed by several different materials or phases, in which the actual interfaces do not allow an adequate Classical Continuum Mechanics approach, once that the boundary conditions make the mathematical description of the problem unfeasible (as for instance in reinforced concrete, polymer strengthened concrete, and porous media). The phenomenon is mathematically described by a set of N partial differential equations coupled by temperature-dependent terms that play the role of internal energy sources. These internal energy sources arise because, at each spatial point, there are different temperatures, each one associated with one constituent of the mixture. The coupling among the partial differential equations arises from the thermal interchange ...
A thermomechanical theory of solid-fluid mixtures
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144 S. KRISHNASWAMY and R. С. BATRA An advantage of Rivlin's approach is that entropy is a constructed quantity for which a prescription is provided. The constitutive restrictions (see (4.4) through (4.6) and (4.11)) are identical to those obtained by others who assume entropy to be a primitive variable and postulate that the Clausius-Duhem inequality be satisfied in every thermomechanical process.
A local model for the energy transfer in a saturated static mixture
Meccanica, 1997
In the present work the transient energy transfer in a nonsaturated porous medium is studied, using a mixture theory viewpoint. The porous matrix is assumed homogeneous, rigid and isotropic, while the fluid is a Newtonian incompressible one and both are assumed static. Since the homogeneous matrix is not saturated, gradients of concentration are present. The porous medium and the fluid (a liquid) will be regarded as continuous constituents of a mixture that will have also a third constituent, an inert gas, assumed with zero mass density and thermal conductivity. The problem is described by a set of two partial differential equations which represent the energy balances for the fluid and the solid constituents. Isovalues for these two constituents are plotted, considering representative time instants and selected values for the energy equations coefficients and for the saturation. Sommario. Nel presente lavoro si studia il trasferimento di energia transiente in un mezzo poroso, utilizzando il punto di vista della teoria delle miscele. Si ipotizza che la matrice porosa sia omogenea, rigida ed isotropa, che il fluido sia newtoniano ed incompressibile e che entrambi siano in condizioni statiche. Poichè la matrice omogeneà e non satura, sono presenti gradienti di concentrazione. Il mezzo poroso ed il fluido vengono considerati come costituenti continui di una miscela dotata di un terzo costituente, un gas inerte, che si ipotizza abbia densità di massa e conduttività termica nulle. Il problema viene descritto tramite un insieme di due equazioni differenziali alle derivate parziali che rappresentano il bilancio di energia per i costituenti solido e fluido. Per questi due costituenti sono disegnate curve di egual valore, considerando istanti rappresentativi e selezionando alcuni valori sia per i coefficient delle equazioni dell' energia che per la saturazione.
International Journal of Engineering Science, 2011
A thermoelastic-plastic body consisting of two phases, a solid and a fluid, each comprising two constituents is considered where one constituent in one phase is allowed to exchange mass with another constituent (of the same substance) in the other phase. A large strain setting is adopted and the formulation applies to general anisotropy and the existence of residual stresses. Generalized forms of Fourier's, Fick's and Darcy's laws are derived and also the stresses on the constituent, phase and mixture level are established; in addition, the evolution law for general plasticity is given. Finally, and in particular, a general evolution law for the rate of deformation tensor related to mass exchange is proposed and this leads to general absorption and desorption evolution laws for mass exchange between two constituents (of the same substance), one belonging to the solid phase and the other to the fluid phase. Equilibrium curves for absorption and desorption also emerge from the theory.
Total stress tensors and heat fluxes of single flow through a porous viscoelastic medium
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Using the second law of thermodynamics, we examine the macroscopic equations for mass, momentum, energy and entropy for a biphasic system whose interface has thermodynamic properties. This system is made up of a mesoscopic particle and a fluid, including mass exchange and different phase temperatures. By exploiting the entropy inequality in terms of Coleman and Noll's method we obtain nonequilibrium and equilibrium results. We show how the solid phase stress tensor depends on the solid phase pressure, the Terzaghi stress, the hydration stress and the stress contributed by the interface properties, which is similar to the Terzaghi stress. We determine the heat fluxes. We further linearize the non-equilibrium parts of their constitutive forms in terms of heat conduction, fluid viscosity and viscoelasticity about the equilibrium. Finally we obtain expressions of the total stress and the total heat flux for a particle.
Heat and momentum transport in a multicomponent mixture far from equilibrium
Physica A: Statistical Mechanics and its Applications, 2001
Explicit expressions for the heat and momentum fluxes are given for a low-density multicomponent mixture in a steady state with temperature and velocity gradients. The results are obtained from a formally exact solution of the Gross-Krook model [Phys. Rev. 102, 593 (1956)] of the Boltzmann equation for a multicomponent mixture. The transport coefficients (shear viscosity, viscometric functions, thermal conductivity and a cross coefficient measuring the heat flux orthogonal to the thermal gradient) are nonlinear functions of the velocity and temperature gradients and the parameters of the mixture (particle masses, concentrations, and force constants). The description applies for conditions arbitrarily far from equilibrium and is not restricted to any range of mass ratios, molar fractions and/or size ratios. The results show that, in general, the presence of the shear flow produces an inhibition in the transport of momentum and energy with respect to that of the Navier-Stokes regime. In the particular case of particles mechanically equivalent and in the tracer limit, previous results are recovered.
Continuum Mechanics and Thermodynamics, 2002
In this first part of the work on the thermodynamics of microstructured flowing media, a complete set of balance equations and jump conditions for single and mixed continua is presented. These equations are appropriate for a wide range of applications, from the creep of polycrystals to the flow of granular media and chemically reacting mixtures of liquid crystals. Thanks to the use of the framework of mixtures with continuous diversity, both isotropic and anisotropic media can be considered at the same footing, and a straightforward comparison of the present results with those found in kinetic, statistical and continuum theories is allowed. Among other conclusions, it is shown that most of the previous theories are either oversimplified or conceptually deficient, due to different reasons. Four different levels of description of the balance equations and jump conditions are addressed, from orientation-dependent relations for the constituents to mixture relations. In particular, the adoption of an orientation-dependent description of microstructured mixtures reveals the occurrence of net orientational diffusive fluxes of mass, which arise in such mixtures from a combination of inertial effects with rotatory diffusion. According to the definitions employed here, these orientational mass fluxes are neither conductive nor convective and their effects disappear after consideration of all microstructural orientations.