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Networks & Heterogeneous Media

2007 , Volume 2 , Issue 3

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Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary
Shijin Deng, Weike Wang and Shih-Hsien Yu
2007, 2(3): 383-395 doi: 10.3934/nhm.2007.2.383 +[Abstract](78) +[PDF](174.8KB)
In this paper, we consider an initial-boundary value problem for the Broadwell model with a supersonic physical boundary. By using the Green’s function established in [6] and weighted energy estimates, we show that the solution converges pointwise to the equilibrium state when the perturbations are sufficiently small.
Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem
Andro Mikelić, Giovanna Guidoboni and Sunčica Čanić
2007, 2(3): 397-423 doi: 10.3934/nhm.2007.2.397 +[Abstract](95) +[PDF](344.6KB)
This is a study of the fluid-structure interaction between the stationary Stokes flow of an incompressible, Newtonian viscous fluid filling a three-dimensional, linearly elastic, pre-stressed hollow tube. The main motivation comes from the study of blood flow in human arteries. Most literature on fluid-structure interaction in blood flow utilizes thin structure models (shell or membrane) to describe the behavior of arterial walls. However, arterial walls are thick, three-dimensional structures with the wall thickness comparable to the vessel inner radius. In addition, arteries in vivo exhibit residual stress: when cut along the radius, arteries spring open releasing the residual strain. This work focuses on the implications of the two phenomena on the solution of the fluid-structure interaction problem, in the parameter regime corresponding to the blood flow in medium-to-large human arteries. In particular, it is assumed that the aspect ratio of the cylindrical structure $\epsilon = R/L$ is small. Using asymptotic analysis and ideas from homogenization theory for porous media flows, an effective, closed model is obtained in the limit as both the thickness of the vessel wall and the radius of the cylinder approach zero, simultaneously. The effective model satisfies the original three-dimensional, axially symmetric problem to the $\epsilon^2$-accuracy. Several novel properties of the solution are obtained using this approach. A modification of the well-known "Law of Laplace'' is derived, holding for thick elastic cylinders. A calculation of the effective longitudinal displacement is obtained, showing that the leading-order longitudinal displacement is completely determined by the external loading. Finally, it is shown that the residual stress influences the solution only at the $\epsilon$-order. More precisely, it is shown that the only place where the residual stress influences the solution of this fluid-structure interaction problem is in the calculation of the $\epsilon$-correction of the longitudinal displacement.
Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks
Serge Nicaise and Julie Valein
2007, 2(3): 425-479 doi: 10.3934/nhm.2007.2.425 +[Abstract](166) +[PDF](509.3KB)
In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.
Enskog-like discrete velocity models for vehicular traffic flow
Michael Herty, Lorenzo Pareschi and Mohammed Seaïd
2007, 2(3): 481-496 doi: 10.3934/nhm.2007.2.481 +[Abstract](86) +[PDF](727.9KB)
We consider an Enskog-like discrete velocity model which in the limit yields the viscous Lighthill-Whitham-Richards equation used to describe vehicular traffic flow. Consideration is given to a discrete velocity model with two speeds. Extensions to the Aw-Rascle system and more general discrete velocity models are also discussed. In particular, only positive speeds are allowed in the discrete velocity equations. To numerically solve the discrete velocity equations we implement a Monte Carlo method using the interpretation that each particle corresponds to a vehicle. Numerical results are presented for two practical situations in vehicular traffic flow. The proposed models are able to provide accurate solutions including both, forward and backward moving waves.
Equilibria for data networks
Alessia Marigo
2007, 2(3): 497-528 doi: 10.3934/nhm.2007.2.497 +[Abstract](91) +[PDF](642.9KB)
This paper investigates equilibrium solutions for data flows on a network. We consider a fluid dynamic model based on conservation laws. The dynamics at nodes is solved by FIFO policy combined with through flux maximization. We first link the dimension of the equilibria space to topological properties of the graph associated to the network. Then we focus on regular plane tilings with square or triangular cells. For various networks, we completely determine the characteristics of periodic equilibria and, in some cases, of all equilibria. The obtained results are expected to play a role both in the analysis of asymptotic behavior of network load and for security issues in case of node failures.
Homogenization approach to filtration through a fibrous medium
Mohamed Belhadj, Eric Cancès, Jean-Frédéric Gerbeau and Andro Mikelić
2007, 2(3): 529-550 doi: 10.3934/nhm.2007.2.529 +[Abstract](129) +[PDF](401.5KB)
We study the flow through fibrous media using homogenization techniques. The fibre network under study is the one already used by M. Briane in the context of heat conduction of biological tissues. We derive and justify the effective Darcy equation and the permeability tensor for such fibrous media. The theoretical results on the permeability are illustrated by some numerical simulations. Finally, the low solid fraction limit is considered. Applying results by G. Allaire to our setting, we justify rigorously the leading order term in the empirical formulas for the effective permeability used in engineering. The results are also confirmed by a direct numerical calculation of the permeability, in which the small diameter of the fibres requires high accuracy approximations.
A derivation of linear elastic energies from pair-interaction atomistic systems
Andrea Braides, Margherita Solci and Enrico Vitali
2007, 2(3): 551-567 doi: 10.3934/nhm.2007.2.551 +[Abstract](83) +[PDF](231.3KB)
Pair-interaction atomistic energies may give rise, in the framework of the passage from discrete systems to continuous variational problems, to nonlinear energies with genuinely quasiconvex integrands. This phenomenon takes place even for simple harmonic interactions as shown by an example by Friesecke and Theil [19]. On the other hand, a rigorous derivation of linearly elastic energies from energies with quasiconvex integrands can be obtained by $\Gamma$-convergence following the method by Dal Maso, Negri and Percivale [14]. We show that the derivation of linear theories by $\Gamma$-convergence can be obtained directly from lattice interactions in the regime of small deformations. Our proof relies on a lower bound by comparison with the continuous result, and on a direct Taylor expansion for the upper bound. The computation is carried over for a family of lattice energies comprising interactions on the triangular lattice in dimension two.

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