Showing papers on "Mixture theory published in 2014"

A mixture theory for size and density segregation in shallow granular free-surface flows

Abstract: In the past ten years much work has been undertaken on developing mixture theory continuum models to describe kinetic sieving-driven size segregation. We propose an extension to these models that allows their application to bidisperse flows over inclined channels, with particles varying in density and size. Our model incorporates both a recently proposed explicit formula for how the total pressure is distributed among different species of particles, which is one of the key elements of mixture theory-based kinetic sieving models, and a shear rate-dependent drag. The resulting model is used to predict the range of particle sizes and densities for which the mixture segregates. The prediction of no segregation in the model is benchmarked by using discrete particle simulations, and good agreement is found when a single fitting parameter is used which determines whether the pressure scales with the diameter, surface area or volume of the particle.

General coupling of porous flows and hyperelastic formulations -- From thermodynamics principles to energy balance and compatible time schemes

Abstract: We formulate a general poromechanics model -- within the framework of a two-phase mixture theory -- compatible with large strains and without any simplification in the momentum expressions, in particular concerning the fluid flows. The only specific assumptions made are fluid incompressibility and isothermal conditions. Our formulation is based on fundamental physical principles -- namely, essential conservation and thermodynamics laws -- and we thus obtain a Clausius-Duhem inequality which is crucial for devising compatible constitutive laws. We then propose to model the solid behavior based on a generalized hyperelastic free energy potential -- with additional viscous effects -- which allows to represent a wide range of mechanical behaviors. The resulting formulation takes the form of a coupled system similar to a fluid-structure interaction problem written in an Arbitrary Lagrangian-Eulerian formalism, with additional volume-distributed interaction forces. We achieve another important objective by identifying the essential energy balance prevailing in the model, and this paves the way for further works on mathematical analyses, and time and space discretizations of the formulation.

On the coefficients of the interaction forces in a two-phase flow of a fluid infused with particles

Abstract: In this short paper we study the flow of a mixture of a fluid infused with particles in a channel. We use the classical mixture theory approach whereby constitutive relations are proposed for the stress tensor of each phase. For the interaction forces, the effect of different hindrance functions for the drag force is studied; moreover a generalized form of the expression for the coefficients of the interactions forces, also known as the hindrance functions, is suggested. For studying this two-component system numerically, a three-dimensional CFD solver based on OpenFOAM® has been developed. Applying this solver, a specific problem (blood flow) has been studied for which our numerical results and experimental data show good agreement.

Concentration driven phase transitions in multiphase porous media with application to methane oxidation in landfill cover layers

Abstract: This study focuses on a formulation within the theory of porus media for continuum multicomponent modeling of bacterial driven methane oxidation in a porous landfill cover layer which consists of a porous solid matrix (soil and bacteria) saturated by a liquid (water) and gas phase. The solid, liquid, and gas phases are considered as immiscible constituents occupying spatially their individual volume fraction. However, the gas phase is composed of three components, namely methane (CH4), oxygen (O2), and carbon dioxide (CO2). A thermodynamically consistent constitutive framework is derived by evaluating the entropy inequality on the basis of Coleman and Noll [8], which results in constitutive relations for the constituent stress and pressure states, interaction forces, and mass exchanges. For the final set of process variables of the derived finite element calculation concept we consider the displacement of the solid matrix, the partial hydrostatic gas pressure and osmotic concentration pressures. For simplicity, we assume a constant water pressure and isothermal conditions. The theoretical formulations are implemented in the finite element code FEAP by Taylor [29]. A new set of experimental batch tests has been created that considers the model parameter dependencies on the process variables; these tests are used to evaluate the nonlinear model parameter set. After presenting the framework developed for the finite element calculation concept, including the representation of the governing weak formulations, we examine representative numerical examples.

Orthotropic composite constitutive model of mineral–asphalt mix reinforced with grid and its approximation with an isotropic model

Abstract: A theoretical model for an elastic anisotropic composite of road mixtures reinforced with grids together with its isotropic approximation is proposed. In the considered models, some elements of the mechanics of fibrous composite materials and optimisation theory are used. A composite model is an energy model; that is, constitutive relationships result from the differentiation of the stored energy function arising from two summed parts corresponding to the matrix and reinforcement grid energy via mixture theory. The obtained model is incorporated into the commercial finite element program ABAQUS via a user subroutine written in Fortran. The correctness of the proposed composite constitutive model has been tested by a numerical solution (with the application of a finite element program) of the structure, in which the matrix is modelled as a three-dimensional continuum region and the reinforcement as working in one dimension a set of truss elements. In addition, in the paper the proposition of approximation ...

Spatial resolution of the electrical conductance of ionic fluids using a Green-Kubo method

Abstract: We present a Green-Kubo method to spatially resolve transport coefficients in compositionally heterogeneous mixtures. We develop the underlying theory based on well-known results from mixture theory, Irving-Kirkwood field estimation, and linear response theory. Then, using standard molecular dynamics techniques, we apply the methodology to representative systems. With a homogeneous salt water system, where the expectation of the distribution of conductivity is clear, we demonstrate the sensitivities of the method to system size, and other physical and algorithmic parameters. Then we present a simple model of an electrochemical double layer where we explore the resolution limit of the method. In this system, we observe significant anisotropy in the wall-normal vs. transverse ionic conductances, as well as near wall effects. Finally, we discuss extensions and applications to more realistic systems such as batteries where detailed understanding of the transport properties in the vicinity of the electrodes is of technological importance.

Formulation of thermo-hydro-mechanical coupling behavior of unsaturated soils based on hybrid mixture theory

Abstract: Thermo-Hydro-Mechanical (THM) coupling processes in unsaturated soils are very important in both theoretical researches and engineering applications. A coupled formulation based on hybrid mixture theory is derived to model the THM coupling behavior of unsaturated soils. The free-energy and dissipative functions for different phases are derived from Taylor’s series expansions. Constitutive relations for THM coupled behaviors of unsaturated soils, which include deformation, entropy change, fluid flow, heat conduction, and dynamic compatibility conditions on the interfaces, are then established. The number of field equations is shown to be equal to the number of unknown variables; thus, a closure of this coupling problem is established. In addition to modifications of the physical conservation equations with coupling effect terms, the constitutive equations, which consider the coupling between elastoplastic deformation of the soil skeleton, fluid flow, and heat transfer, are also derived.

A Multi-phase Mixture Theory for Debris Flows: Part III - Critical Angle of an Infinite Slope

Abstract: This paper presents theoretical formula, based on the theory of multi-phase mixture, for the prediction of critical slope angle for which the sediment deposits rested on an infinite slope will be initiated to be debris flows. The deposit sediment is assumed to be a mixture of four constituents, i.e., stone, mud, water and air. Two-dimensional steady uniform flow field is considered. Critical slope angles can be calculated from the consideration of momentum equations or static balance of driving force and resisting force. Comparison of the present results with those obtained by seepage flow theory, Takahashi theory and modified Takahashi theory shows that the theoretical formula derived from multi-phase mixture theory gives the same prediction as Takahashi theory and can be extended to the sediments containing more than two kinds of constituents. Variation of critical angle slope with volume fraction of gravel and sand for stony flow, mud flow and mixed flow is also conducted and the results are reasonable as expected.

A simple mixture theory for ν Newtonian and generalized Newtonian constituents

Abstract: This work presents the development of mathematical models based on conservation laws for a saturated mixture of ν homogeneous, isotropic, and incompressible constituents for isothermal flows. The constituents and the mixture are assumed to be Newtonian or generalized Newtonian fluids. Power law and Carreau–Yasuda models are considered for generalized Newtonian shear thinning fluids. The mathematical model is derived for a ν constituent mixture with volume fractions \({\phi_\alpha}\) using principles of continuum mechanics: conservation of mass, balance of momenta, first and second laws of thermodynamics, and principles of mixture theory yielding continuity equations, momentum equations, energy equation, and constitutive theories for mechanical pressures and deviatoric Cauchy stress tensors in terms of the dependent variables related to the constituents. It is shown that for Newtonian fluids with constant transport properties, the mathematical models for constituents are decoupled. In this case, one could use individual constituent models to obtain constituent deformation fields, and then use mixture theory to obtain the deformation field for the mixture. In the case of generalized Newtonian fluids, the dependence of viscosities on deformation field does not permit decoupling. Numerical studies are also presented to demonstrate this aspect. Using fully developed flow of Newtonian and generalized Newtonian fluids between parallel plates as a model problem, it is shown that partial pressures pα of the constituents must be expressed in terms of the mixture pressure p. In this work, we propose \({p_\alpha=\phi_\alpha p}\) and \({\sum_\alpha^ u p_\alpha = p}\) which implies \({\sum_\alpha^ u \phi_\alpha = 1}\) which obviously holds. This rule for partial pressure is shown to be valid for a mixture of Newtonian and generalized Newtonian constituents yielding Newtonian and generalized Newtonian mixture. Modifications of the currently used constitutive theories for deviatoric Cauchy stress tensor are proposed. These modifications are demonstrated to be essential in order for the mixture theory for ν constituents to yield a valid mathematical model when the constituents are the same. Dimensionless form of the mathematical models is derived and used to present numerical studies for boundary value problems using finite element processes based on a residual functional, that is, least squares finite element processes in which local approximations are considered in \({H^{k,p}\left(\bar{\Omega}^e\right)}\) scalar product spaces. Fully developed flow between parallel plates and 1:2 asymmetric backward facing step is used as model problems for a mixture of two constituents.

Comparisons between mixture theory and darcy law for two-phase flow in porous media using gitt and fluent simulations

Abstract: Water injection into oil formations has a very wide application in the oil industry. The injection of water has as main objective: to maintain formation pressure and provide the recovery of oil by water displacement. This work compares two different theoretical formula- tions used in reservoir simulations. One is the saturation differential equation and the other is the mixture theory equation applied to fluid flow in porous media. The saturation differential equation is widely used in reservoir simulations. This equation is found when Darcy's Law for each phase is applied in the continuity equation for each phase. However, there is another theoretical approach for multiphase flows in porous media. Mixture theory allows a local de- scription of the flow in a porous medium, supported by a thermodynamically consistent theory which generalizes the classical continuum mechanics. This article compares the results ob- tained by the equation based on the mixture theory with the saturation differential equation. To solve the saturation equation the Generalized Integral Transform Technique (GITT) was employed. The GITT has been successfully employed in various petroleum reservoir simula- tion problems. The numerical method for the solution of mixture theory was obtained using advanced, commercial, general-purpose CFD code: FLUENT.