A thermodynamic analysis of rigid heat conductors

A thermodynamic approach to rigid heat conductors is proposed: it introduces the heat flux vector as independent variable while its temporal evolution is governed by a first order differential equation. The form of the second law is that proposed by Müller wherein the entropy flux and the entropy source are not given a priori but determined through constitutive equations. Restrictions on the constitutive equations are placed by the second law. Some properties, valid in the vicinity of equilibrium are established. In particular, it is shown that the present theory leads to a hyperbolic heat conduction equation, allowing for the propagation of heat as a thermal wave with a finite velocity. The concept of thermodynamic forces and fluxes is also introduced. The latter are seen to derive from a potential function plus an additional term. Finally, it is established under which conditions symmetry relations are satisfied.     

Bi-velocity hydrodynamics: Single-component fluids

Acceptance of the Navier-Stokes-Fourier (NSF) equations as the fundamental equations of single-component continuum fluid mechanics for liquids and gases is noted to be inseparably linked to Euler’s implicit, but unproved, hypothesis that but a single-velocity field is required to characterize the four physically different, context-specific, velocities appearing in the mass, momentum, and energy equations. To test Euler’s hypothesis, velocity is added to the usual list of quantities requiring constitutive formulation - namely the heat flux q and viscous stress T - in order to effect closure of the mass, momentum, and energy equations. Establishment of this enlarged set of constitutive relations is effected by using conventional linear irreversible thermodynamics (LIT) principles governing the behavior of simple fluid continua, importantly including satisfaction of Onsager reciprocity as a fundamental continuum requirement. The resulting analysis shows that, in general, two velocities rather than one are required and, concomitantly, that additional driving forces must be added to each of the standard constitutive equations for the Fourier’s-law heat flux q = −k∇T and the Newton’s-law viscous stress (wherein the “mass velocity” v_m is the context-specific velocity appearing in the continuity equation ∂ρ/∂t + ∇ · (ρv_m) = 0). For the particular case of dilute gaseous continua explicit expressions are established for the phenomenological coefficients appearing in these additional constitutive contributions. Determination of these coefficients is effected using data derived from the Chapman-Enskog-Burnett constitutive expressions for q and T, the latter obtained by solving the Boltzmann equation at small Knudsen numbers, including so-called rarefied-gas contributions. These coefficients are found to be nonzero, confirming the conclusion, inter alia, that two velocities are constitutively required to quantify hydrodynamic behavior for gases and, by inference, for liquids too. Collectively, these velocity, heat flux, and stress constitutive findings collectively negate the current belief that the NSF equations fully describe the physics of viscous fluid continua. Rather, they do so only in limiting cases where the additional constitutive terms than we have found necessary for completeness are asymptotically small.     

On the characterization of fluxes in nonlinear irreversible thermodynamics

The linear Onsager theory of irreversible thermodynamics is extended to include nonlinear phenomenological relations by means of Onsager fluxes. Such fluxes satisfy a full system of reciprocity relations, vanish in thermodynamic equilibrium, and give a non-negative production of entropy. A complete characterization of Onsager fluxes is obtained in terms of non-negative scalar valued functions which vanish in thermodynamic equilibrium. These same functions are also shown to characterize all C² fluxes which satisfy the second law of thermodynamics. Each system of Onsager fluxes is shown to derive from a dissipation function which attains its absolute minimum in thermodynamic equilibrium. The reaction rates given by reaction kinetics are shown to be Onsager fluxes and their dissipation functions are explicitly calculated.     

A thermodynamical formulation for chemically active multiphase turbulent flows

A general thermodynamic theory for chemically active multiphase solid–fluid mixtures in turbulent state of motion is formulated. The global equations of balance for each phase are ensemble averaged and the local conservation laws for the mean motions are derived. As a classical treatment, the averaged form of the Clausius–Duhem inequality is used and the thermodynamics of the chemically active mixtures in turbulent state of motion is studied. Particular attention is given to the species concentration of the miscible fluid constituents and chemical reaction effects, in addition to the transport of the phasic fluctuation energies between phases. Based on the averaged entropy inequality, constitutive equations for the stresses, energy, heat and mass fluxes of various species are developed. Explicit governing equations of motion, along with the equation of the dissipation rate of the turbulent kinetic energy are also derived and discussed. A particular emphasis is on the thermodynamically consistent formulation of different solid–fluid interaction terms in these equations.     

Modeling the non-linear viscoelastic response of high temperature polyimides

A constitutive model is developed to predict the viscoelastic response of polyimide resins that are used in high temperature applications. This model is based on a thermodynamic framework that uses the notion that the ‘natural configuration’ of a body evolves as the body undergoes a process and the evolution is determined by maximizing the rate of entropy production in general and the rate of dissipation within purely mechanical considerations. We constitutively prescribe forms for the specific Helmholtz potential and the rate of dissipation (which is the product of density, temperature and the rate of entropy production), and the model is derived by maximizing the rate of dissipation with the constraint of incompressibility, and the reduced energy dissipation equation is also regarded as a constraint in that it is required to be met in every process that the body undergoes. The efficacy of the model is ascertained by comparing the predictions of the model with the experimental data for PMR-15 and HFPE-II-52 polyimide resins.     

A consideration of the thermodynamics of continuous media using the entropy flux assumption of muller and the inclusion of more constitutive variables

In this paper we attempt to study the effect of the entropy flux of Muller and of the inclusion of other constitutive variables on the new heat equation, which is capable of predicting a finite velocity of propagation of the heat disturbance, and the form of the stress tensor. We define a new free energy function. The heat equation obtained is shown to be a generalisation of some obtained so far. It turns out that the inclusion of the constitutive variables T, the material time derivative of the temperature and the velocity gradient may be necessary and the adoption of Muller's new entropy flux in thermoelastic materials explored.     

含损伤复合材料的湿热弹性响应

依据不可逆热力学理论, 未引入任何附加假设, 建立了湿热弹性各向异性损伤复合材料的一般理论。应用建立损伤本构方程的本构泛函展开法, 推导出湿热弹性损伤材料全部本构方程的一般形式, 其中包括比自由能密度表达式、 应力-应变关系、 熵密度方程、 损伤应变能释放率表达式、 吸湿对偶力表达式、 湿-热-固-损伤耦合的热传导方程和损伤演化方程。研究表明, 在本构方程中含有若干损伤效应函数, 表征损伤对材料宏观力学性能与湿、 热性能的影响, 其具体形式可由细观力学解确定, 从而使连续损伤力学与细观损伤力学有机结合在一起。最后, 从细观力学与实验观测两个角度, 举例说明损伤效应函数与系数张量的确定方法, 为分析变温变湿环境下复合材料的损伤问题提供重要的理论依据。      

An Extended Equation of State Modeling Method II. Mixtures

This work is the extension of previous work dedicated to pure fluids. The same method is extended to the representation of thermodynamic properties of a mixture through a fundamental equation of state in terms of the Helmholtz energy. The proposed technique exploits the extended corresponding-states concept of distorting the independent variables of a dedicated equation of state for a reference fluid using suitable scale factor functions to adapt the equation to experimental data of a target system. An existing equation of state for the target mixture is used instead of an equation for the reference fluid, completely avoiding the need for a reference fluid. In particular, a Soave–Redlich–Kwong cubic equation with van der Waals mixing rules is chosen. The scale factors, which are functions of temperature, density, and mole fraction of the target mixture, are expressed in the form of a multilayer feedforward neural network, whose coefficients are regressed by minimizing a suitable objective function involving different kinds of mixture thermodynamic data. As a preliminary test, the model is applied to five binary and two ternary haloalkane mixtures, using data generated from existing dedicated equations of state for the selected mixtures. The results show that the method is robust and straightforward for the effective development of a mixture- specific equation of state directly from experimental data.     

Asymptotic analysis of extinction behaviour in fast nonlinear diffusion

A number of initial-boundary-value problems for the equation of fast diffusion are analysed (at varying levels of detail and completeness), i.e., $$\frac{\partial u}{\partial t} = \nabla \cdot (u^{-n} \nabla u)$$ with n > 0, in dimension N > 2 and with zero-Dirichlet boundary data, namely (i) the Cauchy problem (no boundary), mainly summarising existing results, (ii) the interior problem for a simply connected bounded domain (in large part revisiting earlier results), (iii) the problem exterior to a simply connected bounded domain and (iv) the half-space problem (for which we include N =2). The critical (borderline) case ${n = n_{s} \equiv 4/(N+2)}$ , which arises in Yamabe flow, is the subject of particular focus, in part because it provides considerable insight into both the subcritical case, 0 < n < n _s , and the supercritical one, n _s < n < 1. The results are based on formal-asymptotic analysis and suggest a range of conjectures that could be the subject of rigorous studies. The role of distinct types of similarity solutions is highlighted.     

Constitutive modeling of hot deformation behavior of H62 brass

The hot deformation behaviors of H62 brass are investigated by isothermal compression tests on a Gleeble 1500 thermal-mechanics simulator in the temperature range of 650-800 °C and the strain rate range of 0.01-10 s⁻¹. Most of the stress-strain curves exhibit a single peak stress, indicating a typical dynamic recrystallization (DRX) behavior of the alloy. Further microstructural observation confirms the occurrence of DRX behavior and the β → α phase transformation of H62 brass under the deformation conditions. A new constitutive equation coupling flow stress with strain, strain rate and deformation temperature is developed on the basis of the Arrhenius-type equation, in which the Zener-Hollomon parameter is modified by considering the compensation of the strain rate. In the constitutive equation, the material constants α, n, Q and A are found to be functions of the strain. The flow stress predicted by the constitutive equation shows good agreement with the experimental stress, which validates the efficiency of the constitutive equation in describing the deformation behavior of the material.     

Unsteady mixed-convection boundary layer flow along a symmetric wedge with variable surface temperature

Laminar two-dimensional unsteady mixed-convection boundary-layer flow of a viscous incompressible fluid past a sharp wedge has been studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations is reduced to local non-similarity boundary layer equations, which are solved analytically for small time. Perturbation solutions are also obtained for small and large dimensionless time, τ. Solutions of the governing equations for all time are obtained employing the implicit finite difference method. Here we have focused our attention on the evolution of skin-friction coefficient (C_f) and local Nusselt number (Nu) (heat transfer rate), fluid velocity and fluid temperature with the effects of different governing parameters such as different time, τ, the exponent, m (=0.2, 0.4, 0.6, 0.8, 1.0), mixed convection parameter, λ (= 0.0, 0.5, 1.0) for fluids having Prandtl number, Pr = 0.1, 0.7, and 7.0.     

A BEM sensitivity and shape identification analysis for acoustic scattering in fluid–solid problems

In this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented. Also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems. The formulation developed utilizes the boundary integral equation of the Helmholtz equation for the external problem and the Cauchy–Navier equation for the internal elastic problem. The sensitivities are obtained by the implicit differentiation technique. Examples are presented to demonstrate the accuracy of the proposed formulations. © 1998 John Wiley & Sons, Ltd.     

On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media

Macroscopic balance equations for components, momentum and energy are established for a multiphase flow with diffusion, chemical reactions, heat transfer and exchanges of components between phases in a porous medium. These equations are established separately for each fluid phase, for the solid part of the medium, and for interfaces, by starting from the corresponding equations valid at the pore level and taking their mean value around each point. Then macroscopic entropy balance equations are derived. The entropy source density shows clearly the generalized fluxes and forces which appear in the problem, and suggests how to choose phenomenological equations. A simple example illustrating the method is given in the last paragraph, for a single phase flow with heat transfer in a porous medium. One obtains a generalized form of Darcy's equation. Rigorous conditions along the interfaces and contact lines in a multiphasic medium are given in Appendix.     

Entropy of systems with internal variables

Two entropy functions are currently in use: the thermostatic entropy, defined by Carathéodory's theory, and the thermodynamic entropy, defined by the theory of irreversible processes. Both entropy concepts are confined to systems without internal variables, and both can be shown to be equal by substituting the respective balance of internal energy to which they are related by the integrating factor, 1/. When irreversible internal phenomena are present, represented by internal coordinates and their conjugate affinities, they become part of the entropy production, but not of the energy balance, and the two entropies are no longer equal. It has been shown in the literature that by multiplication by a second integrating factor, an extended entropy function for systems with internal variables can be derived. It is the purpose of this paper to present a method for the determination of this integrating factor. Under certain conditions, the latter may be unity; such is shown to be the case with the Gibbs equation for gas mixtures.Paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

Modelling of Diffuse Interfaces with Temperature Gradients

The work is devoted to capillary phenomena in miscible liquids under the assumption that they have a constant and the same density. The model consists of the heat equation, diffusion equation, and the Navier-Stokes equations with the Korteweg stress. We study several configurations corresponding to the microgravity experiments planned for the International Space Station. The basic conclusion of the numerical simulations is that transient capillary phenomena in miscible liquids exist and can produce convective flows sufficiently strong to be observed experimentally. In particular, there exists a miscible analogue to the Marangoni convection where the temperature gradient is applied along the transition zone between two fluids. Convection also appears if, instead of the temperature gradient, the case where the width of the transition zone varies in space is considered. Finally, similar to the immiscible case, miscible drops move in a temperature gradient.     

Boundary element analysis of planar drop deformation in confined flow. Part II. Viscoelastic fluids

A boundary element formulation is presented for the general two-dimensional simulation of confined two-phase incompressible flow of viscoelastic fluids. A boundary-only formulation is implemented for fluids obeying the Oldroyd-B constitutive equation. Similarly to the formulation in Part I for Newtonian fluids [Khayat et al. Engng Anal Boundary Elements 1997;19:279], the method requires the solution of two simultaneous integral equations on the interface between the two fluids and the confining solid boundary. Although the problem is formulated for any confining geometry, the method is illustrated for a deforming drop as it is driven by the ambient flow inside a convergent channel. The accuracy and convergence of the method are assessed by comparison with the analytical solution for two-phase Taylor–Couette flow, leading to excellent agreement. Further assessment is made by varying the time increment and mesh size of the discretized boundary for a drop deforming in a convergent channel. The influence of fluid elasticity is examined when one or both fluids are viscoelastic.     

Electroviscous effects in purely pressure driven flow and stationary plane analysis in electroosmotic flow of power-law fluids in a slit microchannel

This paper is a comprehensive work on flow of power-law fluids in a slit microchannel. The first part of the paper deals with study of electrokinetic effects in steady, fully developed, laminar pressure driven flow of power-law fluids. The second part of the work provides analysis of stationary plane that is formed during electroosmotic flow (EOF) of power-law fluids inside a closed slit microchannel. In the entire analysis, the flow of power-law fluid is characterized by the modified Navier-Stokes equation incorporating the electric body force term. The electric double layer (EDL) potential is described by the Poisson-Boltzmann distribution under Debye Hückel approximation. In pressure driven flow, analytical expressions for velocity profiles of various power-law fluids are obtained for n = 1, 1/2, 1/3. Numerical simulation is carried out for all values of n to find induced streaming potential without any approximations for entire range of flow behavior indices. The analytical solutions are compared with numerical results. The effects of flow behavior index, zeta potential and channel dimension on velocity distribution, streaming potential, apparent viscosity, volumetric flow rate and friction coefficient are discussed. In the analysis of EOF in closed slit microchannel, the positions of stationary planes for various flow behavior indices at different EDL thicknesses are found both analytically and numerically. It is found that the electroosmotic counter pressure developed inside the cell is strongly dependent on zeta potential and weakly dependent on EDL thickness.