Continual mechanics of monodisperse suspensions. rheological equations of state for suspensions of moderate concentration : PMM vol. 37, n6, 1973, pp. 1059–1077

Rheological relationships linking mean and moment stresses and, also, the force and moment of interphase reaction in a macroscopic flow of small solid sphere suspension with the kinematic characteristics of the flow are derived. This makes it possible to close the system of equations of suspension hydrodynamics. Coefficients of viscosity and of moment viscosity of a suspension are obtained and calculated.The equations of conservation of mass, momentum and moment of momentum of suspension and of its phases, considered (from the macroscopic point of view) to be coexistent continuous media, were formulated in a general form in [1]. These equations contain unknown vectors and tensors which define the interaction between the considered continuous media and, also, stresses and moment stresses appearing when these are in motion. To close the equations of conservation it is necessary to express all these quantities in terms of unknown variables of these equations (mean concentration of suspension, pressure in the fluid phases, and phase velocities). This problem is the second of the fundamental problems of hydromechanics of suspensions indicated in [1].Here this problem is solved with the use of a kind of self-consistent field theory, which is essentially an extension and generalization of methods developed in [2 – 7]. Expressions for all of the quantities mentioned above are derived. They can be considered to be rheological equations of state for suspensions. Expressions for the various coefficients of these equations and their dependence on parameters of phases and on the flow frequency spectrum are also considered.     

Analysis of chemotactical biofilm growth in evolving microstructures

The work is concerned with the growth of biofilms made by chemotactical bacteria within a two-dimensional saturated porous media. The increase of a biomass on the surface of the solid matrix changes the porosity and impede the flow through the pores. By formal periodic homogenization an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients given by the evolving microstructure at the pore-scale was derived in [3]. Based on the assumption of uniform evolve of the underlying pore geometry and slight self-diffusivity of the bacteria, solvability in a weak sense global in time or at least up to a possible clogging phenomenon can be shown. Furthermore, by assuming sufficient regularity on the data we prove boundedness of the solution. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of biofilm growth in porous media

Biofilms are very important in controlling pollution in aquifers. The bacteria may either consume the contaminant or form biobarriers to limit its spread. In this paper we review the mathematical modeling of biofilm growth at the microscopic and macroscopic scales, together with a scale-up technique. At the pore-scale, we solve the Navier-Stokes equations for the flow, the advection-diffusion equation for the transport, together with equations for the biofilm growth. These results are scaled up using network model techniques, in order to have relations between the amount and distribution of the biomass, and macroscopic properties such as permeability and porosity. A macroscopic model is also presented. We give some results.     

On the Coefficients of Pore Tortuosity in the Effective Biot Model

In order to justify the effective Biot model of a porous medium, a precise expression for the density of kinetic energy is established and studied. In investigating this expression, the displacements and their velocities are averaged and the dispersion of normalized velocities in the fluid phase is introduced. The coefficients of pore tortuosity occurring in the Biot equations are expressed in terms of the dispersion, rendering the Biot equations of a porous continuous medium justified. Bibliography: 6 titles.     

Numerical simulation of flow past a sphere in vertical motion within a stratified fluid

In the present study we consider a viscous fluid, stratified by a diffusive saline agent and compute numerically the flow produced by a solid sphere moving vertically and uniformly. The governing equations describing this situation are solved on a variational grid. The results show the dependence of the boundary-layer separation point and the vanishing of vortices behind the sphere as the stratification increases at moderate Reynolds number flows. Details of the flow, density and pressure fields near the sphere are also shown. Important quantities for engineering use (drags, pressure and skin coefficients) are also computed and displayed in the Richardson vs. Reynolds number space. Comparison with experimental evidence shows and excellent agreement.     

Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.     

Consistent macroscopic tangent computation for FFT-based computational homogenization at finite strains

The present work addresses the efficient computation of effective properties of periodic microstructures by the use of Fast Fourier Transforms. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of an RVE, the computation of the associated moduli is not straight-forward. The contribution of the present paper is thus the derivation and implementation of an algorithmically consistent macroscopic tangent operator that comprises the effective properties of the RVE. In contrast to finite-difference based approaches, an exact solution for the macroscopic tangent is derived by means of the classical Lippmann-Schwinger equation. The problem then reduces to the solution of a system of linear equations even for nonlinear material behaviour. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Tartar Equation for Homogenization of a Model of the Dynamics of Fine-Dispersion Mixtures

We consider a mathematical model describing a nonstationary Stokes flow in a fine-dispersion mixture of viscous incompressible fluids with rapidly oscillating initial data. We perform homogenization of the model as the frequency of oscillations tends to infinity; this leads to the problem of finding effective coefficients of the averaged equations. To solve this problem, we propose and implement a method which bases on supplementing the averaged system with the Cauchy problem for the kinetic Tartar equation whose unique solution is the Tartar H-measure. Thereby we construct a correct closed model for describing the motion of a homogeneous mixture, because the effective coefficients of the averaged equations are uniquely expressed in terms of the H-measure.     

Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach

In the present study, we propose a novel multiphysics model that merges two time-dependent problems – the Fluid-Structure Interaction (FSI) and the ultrasonic wave propagation in a fluid-structure domain with a one directional coupling from the FSI problem to the ultrasonic wave propagation problem. This model is referred to as the “eXtended fluid-structure interaction (eXFSI)” problem. This model comprises isothermal, incompressible Navier–Stokes equations with nonlinear elastodynamics using the Saint-Venant Kirchhoff solid model. The ultrasonic wave propagation problem comprises monolithically coupled acoustic and elastic wave equations. To ensure that the fluid and structure domains are conforming, we use the ALE technique. The solution principle for the coupled problem is to first solve the FSI problem and then to solve the wave propagation problem. Accordingly, the boundary conditions for the wave propagation problem are automatically adopted from the FSI problem at each time step. The overall problem is highly nonlinear, which is tackled via a Newton-like method. The model is verified using several alternative domain configurations. To ensure the credibility of the modeling approach, the numerical solution is contrasted against experimental data.     

On methods of deriving equations describing effective models of layered media

Methods of deriving equations describing effective models of layered periodic media are presented. Elastic and fluid media, as well as porous Biot media, may be among these media. First, effective models are derived by a rigorous method, and then some operations in the derivation are replaced by simpler ones providing correct results. As a consequence, a comparatively simple and justified method of deriving equations of an effective model is established. In particular, this method allows us to simplify to a degree and justify the derivation of an effective model for media containing Biot layers; this method also produces equations of an effective model of a porous layered medium intersected by fractures with slipping contacts. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998 pp. 219–243. Translated by L. A. Molotkov.     

Propagation of Normal Waves in an Isolated Porous Fluid-Saturated Biot Layer

For a porous fluid-saturated Biot layer with boundaries free from stresses and pressure, the wave field is found and dispersion equations are derived. The roots of the dispersion equations and the dependence of the phase velocities of the normal waves on the wave number are investigated by analytic methods. It is shown that the phase velocities of most of the normal waves decrease with increasing wave number. Special investigations are conducted in the case of bend and plate waves and their phase velocities for high and low frequencies. It is also shown that on the boundary of a porous Biot half-space, the Rayleigh wave does not always originate, and conditions for the existence of such a wave are established. Bibliography: 7 titles.     

Numerical simulation of three-dimensional incompressible fluid in a box flow passage considering fluid–structure interaction by differential quadrature method

A numerical simulation scheme of 3D incompressible viscous fluid in a box flow passage is developed to solve Navier–Stokes (N–S) equations by firstly taking fluid–structure interaction (FSI) into account. This numerical scheme with FSI is based on the polynomial differential quadrature (PDQ) approximation technique, in which motions of both the fluid and the solid boundary structures are well described. The flow passage investigated consists of four rectangular plates, of which two are rigid, while another two are elastic. In the simulation the elastic plates are allowed to vibrate subjected to excitation of the time-dependent dynamical pressure induced by the unsteady flow in the passage. Meanwhile, the vibrating plates change the flow pattern by producing many transient sources and sinks on the plates. The effects of FSI on the flow are evaluated by running numerical examples with the incoming flow’s Reynolds numbers of 3000, 7000 and 10,000, respectively. Numerical computations show that FSI has significant influence on both the velocity and pressure fields, and the DQ method developed here is effective for modelling 3D incompressible viscous fluid with FSI.     

Explicit jump immersed interface method for virtual material design of the effective elastic moduli of composite materials

Virtual material design is the microscopic variation of materials in the computer, followed by the numerical evaluation of the effect of this variation on the material’s macroscopic properties. The goal of this procedure is an in some sense improved material. Here, we give examples regarding the dependence of the effective elastic moduli of a composite material on the geometry of the shape of an inclusion. A new approach on how to solve such interface problems avoids mesh generation and gives second order accurate results even in the vicinity of the interface. The Explicit Jump Immersed Interface Method is a finite difference method for elliptic partial differential equations that works on an equidistant Cartesian grid in spite of non-grid aligned discontinuities in equation parameters and solution. Near discontinuities, the standard finite difference approximations are modified by adding correction terms that involve jumps in the function and its derivatives. This work derives the correction terms for two dimensional linear elasticity with piecewise constant coefficients, i.e. for composite materials. It demonstrates numerically convergence and approximation properties of the method.      

Electroosmosis law via homogenization of electrolyte flow equations in porous media

By the homogenization approach we justify a two-scale model of ion transport in porous media for one-dimensional horizontal steady flows driven by a pressure gradient and an external horizontal electrical field. By up-scaling, the electroosmotic flow equations in horizontal nanoslits separated by thin solid layers are approximated by a homogenized system of macroscale equations in the form of the Poisson equation for induced vertical electrical field and Onsager's reciprocity relations between global fluxes (hydrodynamic and electric) and forces (horizontal pressure gradient and external electrical field). In addition, the two-scale approach provides macroscopic mobility coefficients in the Onsager relations.     

An explicitly solvable kinetic model for vehicular traffic and associated macroscopic equations

In the present paper, a kinetic model for vehicular traffic is presented and investigated in detail. For this model, the stationary distributions can be determined explicitly. A derivation of associated macroscopic traffic flow equations from the kinetic equation is given. The coefficients appearing in these equations are identified from the solutions of the underlying stationary kinetic equation and are given explicitly. Moreover, numerical experiments and comparisons between different macroscopic models are presented.     

Dead-core solutions for slightly non-isothermal diffusion-reaction problems with power-law kinetics

The paper deals with dead-core solutions to a non-isothermal reaction- diffusion problem with power-law kinetics for a single reaction that takes place in a catalyst pellet along with mass and heat transfer from the bulk phase to the outer pellet surface. The model boundary value problem for two coupled non-linear diffusion-reaction equations is solved using the semi-analytical method. The exact solutions are established under the assumption of a small temperature gradient in the pellet. The nonlinear algebraic expressions are derived for the critical Thiele modulus, dead-zone length, reactant concentration, and temperature profiles in catalyst pellets of planar geometry. The effects of the reaction order, Arrhenius number, energy generation function, Thiele modulus, and Biot numbers are investigated on the concentration and temperature profiles, dead-zone length, and critical Thiele modulus.     

Biot Stress and Strain in Thin-Plate Theory for Large Deformations

We propose a theory of nonlinear deformation of a plate on the basis of an energetically conjugate pair of the Biot stress tensors and the right stretch tensor. When the dimensionality of the problem is reduced from three to two, the classical Kirchhoff conjectures are used, the linear part is retained in the expansion of the right stretch tensor with respect to a degenerate coordinate, and no additional simplifications are assumed. Connection is obtained between the asymmetric and symmetric components of the Biot tensor; the equivalence is demonstrated of the virtual work principle with the equilibrium equations, the natural boundary conditions, and additional conditions for the dependence of asymmetric stress moment resultants on symmetric moments.     

Homogenization of a model for water–gas flow through double‐porosity media

This paper presents a study of immiscible compressible two‐phase, such as water and gas, flow through double porosity media. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the standard Darcy–Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, that is, where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium becomes discontinuous. Consequently, the model involves highly oscillatory characteristics. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. We obtain the convergence of the solutions, and a macroscopic model of the problem is constructed using the notion of two‐scale convergence combined with the dilatation technique. Copyright © 2015 John Wiley & Sons, Ltd.     

Self-similar Solutions of Nonlinear Elasticity and Poroelasticity Equations

Recent advances in nonlinear wave propagation in elastic and porous elastic (poro-elastic) material have presented new nonlinear evolutionary equations. The derivation of these equations in three-dimensional space is based on the semilinear Biot theory. The nonlinear elastodynamic equations are derived form the more general model of poro-elastodynamic using consistency arguments. For simplicity, we discuss and carry out the analysis for the nonlinear elastic model. It is found in this article that the methods of symmetry groups and self-similar solutions can furnish solutions to the nonlinear elastodynamic wave equation. It is also found that these models lead to shock wave development in finite time. Necessary conditions for the existence of the solution are given and well-posedness of the Cauchy problem is discussed.