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Networks & Heterogeneous Media

2008 , Volume 3 , Issue 1

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Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model
Raimund Bürger, Antonio García, Kenneth H. Karlsen and John D. Towers
2008, 3(1): 1-41 doi: 10.3934/nhm.2008.3.1 +[Abstract](128) +[PDF](1213.8KB)
The classical Lighthill-Whitham-Richards (LWR) kinematic traffic model is extended to a unidirectional road on which the maximum density $a(x)$ represents road inhomogeneities, such as variable numbers of lanes, and is allowed to vary discontinuously. The car density $\phi = \phi(x,t)$ is then determined by the following initial value problem for a scalar conservation law with a spatially discontinuous flux:

$\phi_t + (\phi v(\phi/{a(x)})_x = 0, \quad \phi(x,0)=\phi_0(x),\quad x \in \mathbb{R},\quad t\in (0,T),$ (*)

where $v(z)$ is the velocity function. We adapt to (*) a new notion of entropy solutions (Bürger, Karlsen, and Towers [Submitted, 2007]), which involves a Kružkov-type entropy inequality based on a specific flux connection $(A,B)$, and which we interpret in terms of traffic flow. This concept is consistent with both the driver's ride impulse and the desire of drivers to speed up.
    We prove that entropy solutions of type $(A,B)$ are unique. This solution concept also leads to simple, transparent, and unified convergence proofs for numerical schemes. Indeed, we adjust to (*) new variants of the Engquist-Osher (EO) scheme (Bürger, Karlsen, and Towers [Submitted, 2007]), and of the Hilliges-Weidlich (HW) scheme analyzed by the authors [ J. Engrg. Math., to appear]. It is proven that the EO and HW schemes and a related Godunov scheme converge to the unique entropy solution of type $(A,B)$ of (*). For the Godunov version, this is the first rigorous convergence and well-posedness result, since no unnecessarily restrictive regularity assumptions are imposed on the solution. Numerical experiments for first-order schemes and formally second-order MUSCL/Runge-Kutta versions are presented.

Multiphase modeling and qualitative analysis of the growth of tumor cords
Andrea Tosin
2008, 3(1): 43-83 doi: 10.3934/nhm.2008.3.43 +[Abstract](96) +[PDF](978.6KB)
In this paper a macroscopic model of tumor cord growth is developed, relying on the mathematical theory of deformable porous media. Tumor is modeled as a saturated mixture of proliferating cells, extracellular fluid and extracellular matrix, that occupies a spatial region close to a blood vessel whence cells get the nutrient needed for their vital functions. Growth of tumor cells takes place within a healthy host tissue, which is in turn modeled as a saturated mixture of non-proliferating cells. Interactions between these two regions are accounted for as an essential mechanism for the growth of the tumor mass. By weakening the role of the extracellular matrix, which is regarded as a rigid non-remodeling scaffold, a system of two partial differential equations is derived, describing the evolution of the cell volume ratio coupled to the dynamics of the nutrient, whose higher and lower concentration levels determine proliferation or death of tumor cells, respectively. Numerical simulations of a reference two-dimensional problem are shown and commented, and a qualitative mathematical analysis of some of its key issues is proposed.
Leaf superposition property for integer rectifiable currents
Luigi Ambrosio, Gianluca Crippa and Philippe G. Lefloch
2008, 3(1): 85-95 doi: 10.3934/nhm.2008.3.85 +[Abstract](80) +[PDF](171.1KB)
We consider the class of integer rectifiable currents without boundary in $\R^n\times\R$ satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.
Homogenization and correctors for the wave equation in non periodic perforated domains
Patrizia Donato and Florian Gaveau
2008, 3(1): 97-124 doi: 10.3934/nhm.2008.3.97 +[Abstract](95) +[PDF](285.9KB)
We consider here the wave equation in a (not necessarily periodic) perforated domain, with a Neumann condition on the boundary of the holes. Assuming $H^0$-convergence ([3]) on the elliptic part of the operator, we prove two main theorems: a convergence result and a corrector one. To prove the corrector result, we make use of a suitable family of elliptic local correctors given in [4] whose columns are piecewise locally square integrable gradients. As in the case without holes ([2]), some additional assumptions on the data are needed.
A network model of geometrically constrained deformations of granular materials
K. A. Ariyawansa, Leonid Berlyand and Alexander Panchenko
2008, 3(1): 125-148 doi: 10.3934/nhm.2008.3.125 +[Abstract](90) +[PDF](305.5KB)
We study quasi-static deformation of dense granular packings. In the reference configuration, a granular material is under confining stress (pre-stress). Then the packing is deformed by imposing external boundary conditions, which model engineering experiments such as shear and compression. The deformation is assumed to preserve the local structure of neighbors for each particle, which is a realistic assumption for highly compacted packings driven by small boundary displacements. We propose a two-dimensional network model of such deformations. The model takes into account elastic interparticle interactions and incorporates geometric impenetrability constraints. The effects of friction are neglected. In our model, a granular packing is represented by a spring-lattice network, whereby the particle centers correspond to vertices of the network, and interparticle contacts correspond to the edges. We work with general network geometries: periodicity is not assumed. For the springs, we use a quadratic elastic energy function. Combined with the linearized impenetrability constraints, this function provides a regularization of the hard-sphere potential for small displacements.
   When the network deforms, each spring either preserves its length (this corresponds to a solid-like contact), or expands (this represents a broken contact). Our goal is to study distribution of solid-like contacts in the energy-minimizing configuration. We prove that under certain geometric conditions on the network, there are at least two non-stretched springs attached to each node, which means that every particle has at least two solid-like contacts. The result implies that a particle cannot loose contact with all of its neighbors. This eliminates micro-avalanches as a mechanism for structural weakening in small shear deformation.
Modelling of wettability alteration processes in carbonate oil reservoirs
Liping Yu, Hans Kleppe, Terje Kaarstad, Svein M. Skjaeveland, Steinar Evje and Ingebret Fjelde
2008, 3(1): 149-183 doi: 10.3934/nhm.2008.3.149 +[Abstract](71) +[PDF](635.3KB)
Previous studies have shown that seawater may alter the wettability in the direction of more water-wet conditions in carbonate reservoirs. The reason for this is that ions from the salt (sulphat, magnesium, calsium, etc) can create a wettability alteration toward more water-wet conditions as salt is absorbed on the rock.
   In order to initiate a more systematic study of this phenomenon a 1-D mathematical model relevant for spontaneous imbibition is formulated. The model represents a core plug on laboratory scale where a general wettability alteration (WA) agent is included. Relative permeability and capillary pressure curves are obtained via interpolation between two sets of curves corresponding to oil-wet and water-wet conditions. This interpolation depends on the adsorption isotherm in such a way that when no adsorption of the WA agent has taken place, oil-wet conditions prevail. However, as the adsorption of this agent takes place, gradually there is a shift towards more water-wet conditions. Hence, the basic mechanism that adsorption of the WA agent is responsible for the wettability alteration, is naturally captured by the model.
   Conservation of mass of oil, water, and the WA agent, combined with Darcy's law, yield a 2x2 system of coupled parabolic convection-diffusion equations, one equation for the water phase and another for the concentration of the WA agent. The model describes the interactions between gravity and capillarity when initial oil-wet core experiences a wettability alteration towards more water-wet conditions due to the spreading of the WA agent by molecular diffusion. Basic properties of the model are studied by considering a discrete version. Numerical computations are performed to explore the role of molecular diffusion of the WA agent into the core plug, the balance between gravity and capillary forces, and dynamic wettability alteration versus permanent wetting states. In particular, a new and characteristic oil-bank is observed. This is due to incorporation of dynamic wettability alteration and cannot be seen for case with permanent wetting characteristics. More precisely, the phenomenon is caused by a cross-diffusion term appearing in capillary diffusion term.

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