Two-phase Heterogeneous Media Research Articles

Numerical Modeling of Wave Processes Accompanying Combustion of Inhomogeneously Distributed Composite Propellant

A mathematical model and a numerical method are proposed for studying the interior ballistic process related to combustion of inhomogeneously distributed propellant in the two-dimensional axisymmetric approximation. The gas–propellant mixture is modeled by a two-phase nonequilibrium heterogeneous medium consisting of a multicomponent gas phase of combustion products and a polydisperse solid phase of propellant granules. The mathematical model of nonequilibrium two-phase flow is based on the nonconservative Euler equations. A Godunov-type scheme with an approximate Riemann solver is developed for their solution. The propellant combustion is considered taking into account the motion of the projectile, which is modelled using the free-boundary method. Results are represented concerning the origin and evolution of the interior ballistic wave process proceeding during the combustion of an inhomogeneously distributed propellant charge and the motion of the projectile. A comparative analysis with the case of static (nonmoving) propellant is carried out.

Mathematical modeling of wide-range compressible two-phase flows

The paper considers flow modeling of two-phase heterogeneous medium. Each phase of the medium is considered as continuum, which is described by the compressible Euler equations. The phases are separated by the contact surface (interface) and are not mixed on the molecular level. Examples of such medium are: mixtures of solid particles and gas (in dense or dilute concentration of particles), liquids with small bubbles, solid porous materials filled with gas or liquid. The phase can be of connected structure (dense particles, porous solid, gas between dilute particles, etc.) or of non-connected structure (separated inclusions as gas bubbles, dilute particles, closed pores in solid, etc.). The connectivity of the phase is closely related to the propagation of acoustic perturbations. An attempt was made to consider all the cases of the phase connectivity in the framework of a unique approach. The paper presents an approach that couples the models designed for different cases of phase connectivity to the generalized hyperbolic and thermodynamically consistent form. The proposed model is applicable for simulating flows with change of the phase connectivity, e.g. dense-to-dilute two-phase flows. The model is verified on several problems of gas–solid granular medium flows. In numerical simulations the Godunov method with the HLLEM flux approximation on arbitrary moving Euler grids is used.

Disordered hyperuniform heterogeneous materials

Disordered hyperuniform many-body systems are distinguishable states of matter that lie between a crystal and liquid: they are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These systems play a vital role in a number of fundamental and applied problems: glass formation, jamming, rigidity, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, quantum systems, and pure mathematics. Much of what we know theoretically about disordered hyperuniform states of matter involves many-particle systems. In this paper, we derive new rigorous criteria that disordered hyperuniform two-phase heterogeneous materials must obey and explore their consequences. Two-phase heterogeneous media are ubiquitous; examples include composites and porous media, biological media, foams, polymer blends, granular media, cellular solids, and colloids. We begin by obtaining some results that apply to hyperuniform two-phase media in which one phase is a sphere packing in d-dimensional Euclidean space . Among other results, we rigorously establish the requirements for packings of spheres of different sizes to be ‘multihyperuniform’. We then consider hyperuniformity for general two-phase media in . Here we apply realizability conditions for an autocovariance function and its associated spectral density of a two-phase medium, and then incorporate hyperuniformity as a constraint in order to derive new conditions. We show that some functional forms can immediately be eliminated from consideration and identify other forms that are allowable. Specific examples and counterexamples are described. Contact is made with well-known microstructural models (e.g. overlapping spheres and checkerboards) as well as irregular phase-separation and Turing-type patterns. We also ascertain a family of autocovariance functions (or spectral densities) that are realizable by disordered hyperuniform two-phase media in any space dimension, and present select explicit constructions of realizations. These studies provide insight into the nature of disordered hyperuniformity in the context of heterogeneous materials and have implications for the design of such novel amorphous materials.

Filtration model of plastic flow

AbstractA filtration model for plastic flow based on the idea of a deformed material considered as a two-phase heterogeneous medium has been suggested. In this approach, the wave displacement is regarded as a shock transition in the medium. One of the phases (the excited one) is responsible for system restructuring, and the other phase (the normal one) is unrelated to structural transformations. The plastic wave is the result of the interaction of these two phases. The governing equations for the filtration model are obtained. They include the laws of momentum and mass conservation, as well as the filtration ratio of the phases.

A two-phase hyperelastic-viscoplastic constitutive model for semi-crystalline polymers: Application to polyethylene materials with a variable range of crystal fractions

Polyethylene-based polymers as biomedical materials can contribute to a wide range of biomechanical applications. Therefore, it is important to identify, analyse, and predict with precision their mechanical behaviour. Polyethylene materials are semi-crystalline systems consisting of both amorphous and crystalline phases interacting in a rather complex manner. When the amorphous phase is in the rubbery state, the mechanical behaviour is strongly dependent on the crystal fraction, therefore leading to essentially thermoplastic or elastomeric responses. In this study, the finite deformation stress–strain response of polyethylene materials is modelled by considering these semi-crystalline polymers as two-phase heterogeneous media in order to provide insight into the role of crystalline and amorphous phases on the macro-behaviour and on the material deformation resistances, i.e. intermolecular and network resistances. A hyperelastic–viscoplastic model is developed in contemplation of representing the overall mechanical response of polyethylene materials under large deformation. An evolutionary optimization procedure based on a genetic algorithm is developed to identify the model parameters at different strain rates. The identification results show good agreement with experimental data, demonstrating the usefulness of the proposed approach: the constitutive model, with only one set of identified parameters, allows reproducing the stress–strain behaviour of polyethylene materials exhibiting a wide range of crystallinities, the crystal content becoming the only variable of the model.

A model of averaged radiation transfer in two-phase heterogeneous medium

Presently employed for describing the radiation transfer in heterogeneous media is the radiation transfer equation (RTE) which is rigorously validated only for homogeneous media, although the hypothesis of the validity of one and the same model for both homogeneous and heterogeneous media is questionable. The local intensities of radiation in different phases of heterogeneous medium may significantly differ; therefore, the only averaged radiation intensity employed in the RTE model is insufficient. A model of radiation transfer is obtained for two-phase heterogeneous medium in the geometrical optics limit, which consists of two transfer equations for partial radiation intensities averaged in each phase separately. These equations resemble ordinary RTEs but include the exchange of radiation between the phases; therefore, they are referred to as the vector model of RTE. This model reduces to ordinary RTE if one of two phases is nontransparent or one phase prevails in the volume. It is demonstrated that the vector model of RTE in these two extreme cases does not contradict the known results of calculations by the ray optics and Monte-Carlo methods, as well as the experimental data. The suggested vector model is necessary if both phases are transparent or semitransparent and their volume fractions are comparable, because no adequate mathematical models are available in this case. The vector model describes the known results of Monte-Carlo simulation of packed beds of semitransparent spheres. The use of the vector model of RTE for experimental identification of the radiative properties is illustrated with the example of normal-directional reflectance of packed bed of semitransparent SiC particles. The results of numerical calculations confirm the general experimentally observed tendency for increase in reflectance with increasing angle of reflection. The main contribution to the error is made by the boundary conditions for vector RTEs; therefore, detailed analysis of the suggested model in boundary regions is required.

Homogenization of radiation transfer in two-phase media with irregular phase boundaries

The model currently applied to radiation transfer in heterogeneous media is the radiation transfer equation (RTE) rigorously derived for homogeneous media only. However, the hypothesis that heterogeneous media can be described by the same model as homogeneous media is questionable. Indeed, local radiation intensities in different phases of a heterogeneous medium can be of considerably different values, so that the unique homogenized radiation intensity employed in the RTE model is not sufficient. A mathematical model of radiation transfer is obtained for a two-phase heterogeneous medium in the limit of geometrical optics. It consists of two transport equations for the partial homogenized radiation intensities defined by averaging in each phase. The equations are similar to the conventional RTE but contain terms taking into account exchange of radiation between phases. This model is referred to as the vector RTE model. It reduces to the conventional RTE in the cases when one of the two phases is opaque or one phase prevails in volume. In these two limiting cases, the vector RTE model is shown not to contradict the known calculations by ray optics and Monte Carlo simulation and to agree with the known experimental data. The proposed vector RTE model is crucial when the two phases are transparent or semitransparent and their volume fractions are comparable because of the absence of satisfactory mathematical models for this case. It describes the known results of Monte Carlo simulation for packed beds of semitransparent spheres. The application of the vector RTE model for an experimental identification of radiative properties is illustrated by normal-directional reflectance of the packed bed of semitransparent SiC particles. The numerical calculations confirm the general experimental tendency of increasing the reflectance with the reflection angle. The principal uncertainty arises from the boundary conditions applied to the vector RTE equations, so that a detailed analysis of the model near the boundaries would be worthwhile.

Mathematical Modeling of the Hydrodynamics over Permeable Surfaces

A mathematical model is developed to describe the flow of a two-phase heterogeneous medium over the surface of a porous body of arbitrary shape, through which there is filtration. The conservation equations of the mechanics of heterogeneous media are written in an orthogonal coordinate system and are solved by the method of surfaces of equal flow rate. The characteristics of the flows over the surface of a flat porous plate and over the inner surface of a rotating tapered porous rotor are numerically calculated.

Analytical calculation of the conductivity of highly heterogeneous media with allowance for percolation phenomena

Purely analytical methods without employing any computer-empirical relations or parameters have been used to construct a simple model of the conductivity of a two-phase heterogeneous medium composed of grains in the form of oriented ellipsoids. The model is applicable at any, up to infinite, contrast of the conductivities of the phases and in the entire range of their possible concentrations. Moreover, it “automatically” predicts the existence of a percolation threshold, the value of which depends on the grain shape.

Heat Conduction Equations for Two-Phase Heterogeneous Media

Heat Conduction Equations for Two-Phase Heterogeneous Media

Toward a theory of transport in heterogeneous media

A unified equation is obtained for description of nonsteady state heat and mass transport in two-phase heterogeneous media in the low- and high-frequency approximations. Heating of a granular bed by a solid wall is considered as an example.