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Networks & Heterogeneous Media

2009 , Volume 4 , Issue 4

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Kinetic models for polymers with inertial effects
Pierre Degond and  Hailiang Liu
2009, 4(4): 625-647 doi: 10.3934/nhm.2009.4.625 +[Abstract](44) +[PDF](299.6KB)
Novel kinetic models for both Dumbbell-like and rigid-rod like polymers are derived, based on the probability distribution function $f(t, x, n, \dot n)$ for a polymer molecule positioned at $x$ to be oriented along direction $n$ while embedded in a $\dot n$ environment created by inertial effects. It is shown that the probability distribution function of the extended model, when converging, will lead to well accepted kinetic models when inertial effects are ignored such as the Doi models for rod like polymers, and the Finitely Extensible Non-linear Elastic (FENE) models for Dumbbell like polymers.
Homogenization of a model of displacement with unbounded viscosity
Catherine Choquet and  Ali Sili
2009, 4(4): 649-666 doi: 10.3934/nhm.2009.4.649 +[Abstract](25) +[PDF](215.6KB)
We discuss the homogenization of a model problem describing the transport of heat and mass by a compressible miscible flow in a highly heterogeneous porous medium. The flow is governed by a nonlinear system of degenerate parabolic type coupling the pressure and the temperature. Using the technique of two-scale convergence and compensated compactness arguments, we prove some stability in the homogenization process.
Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers
Marco Cicalese , Antonio DeSimone and  Caterina Ida Zeppieri
2009, 4(4): 667-708 doi: 10.3934/nhm.2009.4.667 +[Abstract](50) +[PDF](444.1KB)
In the framework of linear elasticity, we study the limit of a class of discrete free energies modeling strain-alignment-coupled systems by a rigorous coarse-graining procedure, as the number of molecules diverges. We focus on three paradigmatic examples: magnetostrictive solids, ferroelectric crystals and nematic elastomers, obtaining in the limit three continuum models consistent with those commonly employed in the current literature. We also derive the correspondent macroscopic energies in the presence of displacement boundary conditions and of various kinds of applied external fields.
Spectrum analysis of a serially connected Euler-Bernoulli beams problem
Denis Mercier
2009, 4(4): 709-730 doi: 10.3934/nhm.2009.4.709 +[Abstract](46) +[PDF](257.1KB)
In this article we analyse the eigenfrequencies of a hyperbolic system which corresponds to a chain of Euler-Bernoulli beams. More precisely we show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds as soon as the roots of a function denoted by $f_{\infty}$ (and giving the asymptotic behaviour of the eigenvalues) are all simple. For a chain of $N$ different beams, this assumption on the multiplicity of the roots of $f_{\infty}$ is proved to be satisfied. A direct consequence of this result is that we obtain the exact controllability of an associated boundary controllability problem. It is well-known that the spectral gap is a important key point in order to get the exact controllabilty of these one-dimensional problem and we think that the new method developed in this paper could be applied in other related problems.
A non-Markovian model of rill erosion
Michael Damron and  C. L. Winter
2009, 4(4): 731-753 doi: 10.3934/nhm.2009.4.731 +[Abstract](33) +[PDF](1656.2KB)
We introduce a new model for rill erosion. We start with a network similar to that in the Discrete Web [7, 11] and instantiate a dynamics which makes the process highly non-Markovian. The behavior of nodes in the streams is similar to the behavior of Polya urns with time-dependent input. In this paper we use a combination of rigorous arguments and simulation results to show that the model exhibits many properties of rill erosion; in particular, nodes which are deeper in the network tend to switch less quickly.
A mathematical model relevant for weakening of chalk reservoirs due to chemical reactions
Steinar Evje , Aksel Hiorth , Merete V. Madland and  Reidar I. Korsnes
2009, 4(4): 755-788 doi: 10.3934/nhm.2009.4.755 +[Abstract](44) +[PDF](894.8KB)
In this work a mathematical model is proposed for modeling of coupled dissolution/precipitation and transport processes relevant for the study of chalk weakening effects in carbonate reservoirs. The model is composed of a number of convection-diffusion-reaction equations, representing various ions in the water phase, coupled to some stiff ordinary differential equations (ODEs) representing species in the solid phase. More precisely, the model includes the three minerals $\text{CaCO}_3$ (calcite), $\text{CaSO}_4$ (anhydrite), and $\text{MgCO}_3$ (magnesite) in the solid phase (i.e., the rock) together with a number of ions contained in the water phase and essential for describing the dissolution/precipitation processes. Modeling of kinetics is included for the dissolution/precipitation processes, whereas thermodynamical equilibrium is assumed for the aqueous chemistry. A numerical discretization of the full model is presented. An operator splitting approach is employed where the transport effects (convection and diffusion) and chemical reactions (dissolution/precipitation) are solved in separate steps. This amounts to switching between solving a system of convection-diffusion equations and a system of ODEs. Characteristic features of the model is then explored. In particular, a first evaluation of the model is included where comparison with experimental behavior is made. For that purpose we consider a simplified system where a mixture of water and $\text{MgCl}_2$ (magnesium chloride) is injected with a constant rate in a core plug that initially is filled with pure water at a temperature of $T=130^{\circ}$ Celsius. The main characteristics of the resulting process, as predicted by the model, is precipitation of $\text{MgCO}_3$ and a corresponding dissolution of $\text{CaCO}_3$. The injection rate and the molecular diffusion coefficients are chosen in good agreement with the experimental setup, whereas the reaction rate constants are treated as parameters. In particular, by a suitable choice of reaction rate constants, the model produces results that agree well with experimental profiles for measured ion concentrations at the outlet. Thus, the model seems to offer a sound basis for further systematic investigations of more complicated precipitation/dissolution processes relevant for increased insight into chalk weakening effects in carbonate reservoirs.
On the derivation of linear elasticity from atomistic models
Bernd Schmidt
2009, 4(4): 789-812 doi: 10.3934/nhm.2009.4.789 +[Abstract](49) +[PDF](304.4KB)
We derive linear elastic energy functionals from atomistic models as a $\Gamma$-limit when the number of atoms tends to infinity, respectively, when the interatomic distances tend to zero. Our approach generalizes a recent result of Braides, Solci and Vitali [2]. In particular, we study mass spring models with full nearest and next-to-nearest pair interactions. We also consider boundary value problems where a part of the boundary is free.
A novel model for intersections of vehicular traffic flow
Michael Herty , J.-P. Lebacque and  S. Moutari
2009, 4(4): 813-826 doi: 10.3934/nhm.2009.4.813 +[Abstract](88) +[PDF](277.8KB)
This paper deals with intersections' modeling for vehicular traffic flow governed by the Lighthill $\&$ Whitham [24] and Richards [26] model. We present a straightforward reformulation of recent intersections' models, introduced in [19] and [4], using a description in terms of supply and demand functions [22, 6]. This formulation is used to state the new model which takes into account a possible storage capacity of an intersection as seen in roundabouts or highway on-ramps. We discuss the Riemann problem at the junction and present numerical simulations.

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