Compressible generalized Newtonian fluids

We systematically derive models that would be suitable to describe flows of compressible fluids with the material moduli depending on the symmetric part of the velocity gradient and temperature, within the context of a thermodynamic framework that has been quite successful in developing models to describe the response of bodies that produce entropy while undergoing processes.     

Constraints on equation of state for cavitating flows with thermodynamic effects

Thermodynamic effects play an important role in the cavitation dynamics of cryogenics fluids. Such flows are characterized by strong variations in fluid properties with the temperature. A compressible, multiphase, one-fluid solver was developed to study and to predict thermodynamic effects in cavitating flows. To close the system, a cavitation model is proposed to capture metastable behaviours of fluids and non isothermal thermodynamic path. The thermodynamical consistency based on entropy conditions and the evolution of the mixture speed of sound are investigated. These constraints are applied to other models. The considered working fluid is the refrigerant R-114.     

On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single‐ and multi‐temperature models

The first rational model of homogeneous mixtures of fluids was proposed by Truesdell in the context of rational thermodynamics. Afterwards, two different theories were developed: one with a single‐temperature (ST) field of the mixture and the other one with several temperatures. The two systems are from the mathematical point of view completely different and the relationship between their solutions was not clarified. In this paper, the hyperbolic multi‐temperature (MT) system of a mixture of Eulerian fluids will be explained and it will be shown that the corresponding single‐temperature differential system is a principal subsystem of the MT one. As a consequence, the subcharacteristic conditions for characteristic speeds hold and this gives an upper‐bound esteem for pulse speeds in an ST model. Global behaviour of smooth solutions for large time for both systems will also be discussed through the application of the Shizuta–Kawashima condition. Finally, as an application, the particular case of a binary mixture is considered. Copyright © 2006 John Wiley & Sons, Ltd.     

Unsaturated porous media flow with thermomechanical interaction

We propose a model for unsaturated poro‐plastic flow derived from the thermodynamic principles. For the isothermal case, the problem consists of a degenerate coupled system of two PDEs with two independent hysteresis operators describing hysteresis phenomena in both the solid and the pore fluids. Under natural hypotheses, we prove the existence of a global strong solution for this system. Copyright © 2015 John Wiley & Sons, Ltd.     

On the thermomechanics of materials that have multiple natural configurations

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasticity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.Received: February 17, 2004     

On the thermomechanics of materials that have multiple natural configurations Part I: Viscoelasticity and classical plasticity

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasicity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.     

On the thermomechanics of materials that have multiple natural configurations Part I: Viscoelasticity and classical plasticity

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasicity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.     

On the thermomechanics of materials that have multiple natural configurations Part II: Twinning and solid to solid phase transformation

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasticity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.     

Further mathematical results concerning Burgers fluids and their generalizations

In this paper, we extend the earlier work by Quintanilla and Rajagopal (Math Methods Appl Sci 29: 2133?C2147, 2006) and establish qualitative new results for a proper generalization of Burgers?? original work that stems form a general thermodynamic framework. Such fluids have been used to describe the behavior of several geological materials such as asphalt and the earth??s mantle as well as polymeric fluids. We study questions concerning stability, uniqueness and continuous dependence on initial data for the solutions of the flows of these fluids. We show that if certain conditions are not satisfied by the material moduli, the solutions could be unstable. The spatial behavior of the solutions is also analyzed.     

On stationary flow of asymmetric fluids with diffusion

We prove existence and uniqueness theorems for weak solutions of equations describing stationary isothermic motion of a mixture of two viscous incompressible fluids with asymmetric stress tensor, in a bounded subset of ?³. The model of the flow we consider here assumes that some of coefficients characterizing isotropic properties of the fluid equal zero.     

Shocks and rarefactions arise in a two-phase model with logistic growth

Multi-phase or mixture models are often used to describe the dynamics of complex fluids. In this work, we use a general transformation to reduce the two-phase system of one spatial and time variable to a system of a single variable. Then we assess the behavior of solutions for the inviscid two-phase model with logistic growth. The growth rate widely impacts the behavior of the solution, producing either shocks or rarefactions. Increasing growth increases the frequency and spread of these waves, and eliminating growth reduces solutions to continuous traveling waves. This analysis generalizes a class of asymptotic/linear results for gel swelling as well as showing the extraordinary richness in the molding framework.     

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

We provide a thermodynamic basis for the development of models that are usually referred to as ??phase-field models?? for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive ??phase-field models?? both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631?C651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier?CStokes?CFourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier?CStokes?CFourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313?C1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn?CHilliard?CNavier?CStokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn?CHilliard?CNavier?CStokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).     

Incompressible rate type fluids with pressure and shear-rate dependent material moduli

Rate type constitutive theories are developed for describing the response of inhomogeneous fluids whose material properties can depend upon the shear rate and the mean normal stress, within a general thermodynamic setting. The classical Navier–Stokes fluid and the power-law fluid are special subclasses of the rate type fluids that have been developed. The models that have been obtained are particularly useful in describing the behavior of biological and geological fluids, and food products in view of their inherent inhomogeneity.     

A numerical method for the motion of a mixture of two viscous incompressible fluids

A finite difference technique for the simulation of the motion of a mixture of two viscous incompressible fluids in a closed basin is presented. The mathematical model which has been discretized is the closed system deduced from the general equations, governing the motion of the mixture. The numerical scheme is based on the marker and cell method [4] extended to consider the molecular diffusion process. Computational examples are described and discussed at the end of the paper.     

A “Density-Based” Algorithm for Cluster Analysis Using Species Sampling Gaussian Mixture Models

We propose a new model for cluster analysis in a Bayesian nonparametric framework. Our model combines two ingredients, species sampling mixture models of Gaussian distributions on one hand, and a deterministic clustering procedure (DBSCAN) on the other. Here, two observations from the underlying species sampling mixture model share the same cluster if the distance between the densities corresponding to their latent parameters is smaller than a threshold; this yields a random partition which is coarser than the one induced by the species sampling mixture. Since this procedure depends on the value of the threshold, we suggest a strategy to fix it. In addition, we discuss implementation and applications of the model; comparison with more standard clustering algorithms will be given as well. Supplementary materials for the article are available online.     

Modeling of the microwave drying process of aqueous dielectrics

We propose a thermodynamic framework for describing the microwave drying process of aqueous dielectrics based on Maxwell-Lorentz Field equations and mixture theory. Several issues are discussed such as the form of entropy equation; the constitutive relations for the macroscopic electric polarization vectors, Cauchy stresses, heat fluxes, internal momentum supplies, etc., for each component of the mixture: porous solid, water and gas in different regions; and the interfacial jump conditions between different regions in the mixture. A brief examination of the status of material frame indifference within the context of our framework is presented.     

A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients

We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution. In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus Leray [J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math. 63 (1934) 193–248] and Caffarelli, Kohn and Nirenberg [L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771–831] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.     

Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids

Science China Mathematics - We investigate a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids with the goal to prove for it the existence of...