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Networks & Heterogeneous Media

2010 , Volume 5 , Issue 2

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Homogenization of variational functionals with nonstandard growth in perforated domains
Brahim Amaziane, Leonid Pankratov and Andrey Piatnitski
2010, 5(2): 189-215 doi: 10.3934/nhm.2010.5.189 +[Abstract](336) +[PDF](360.6KB)
The aim of the paper is to study the asymptotic behavior of solutions to a Neumann boundary value problem for a nonlinear elliptic equation with nonstandard growth condition of the form

-div(|$\nabla$ uε | pε (x)-2 $\nabla$ uε )+ (| uε | pε (x)-2 uε = f(x)

in a perforated domain Ωε , ε being a small parameter that characterizes the microscopic length scale of the microstructure. Under the assumption that the functions pε(x) converge uniformly to a limit function $p_0(x)$ and that $p_0$ satisfy certain logarithmic uniform continuity condition, it is shown that uε converges, as ε$ \to 0$, to a solution of homogenized equation whose coefficients are calculated in terms of local energy characteristics of the domain Ωε . This result is then illustrated with periodic and locally periodic examples.

A mathematical model for dynamic wettability alteration controlled by water-rock chemistry
Steinar Evje and Aksel Hiorth
2010, 5(2): 217-256 doi: 10.3934/nhm.2010.5.217 +[Abstract](408) +[PDF](622.5KB)
Previous experimental studies of spontaneous imbibition on chalk core plugs have shown that seawater may change the wettability in the direction of more water-wet conditions in chalk reservoirs. One possible explanation for this wettability alteration is that various ions in the water phase (sulphate, calcium, magnesium, etc.) enter the formation water due to molecular diffusion. This creates a non-equilibrium state in the pore space that results in chemical reactions in the aqueous phase as well as possible water-rock interaction in terms of dissolution/precipitation of minerals and/or changes in surface charge. In turn, this paves the way for changes in the wetting state of the porous media in question. The purpose of this paper is to put together a novel mathematical model that allows for systematic investigations, relevant for laboratory experiments, of the interplay between (i) two-phase water-oil flow (pressure driven and/or capillary driven); (ii) aqueous chemistry and water-rock interaction; (iii) dynamic wettability alteration due to water-rock interaction.
   In particular, we explore in detail a 1D version of the model relevant for spontaneous imbibition experiments where wettability alteration has been linked to dissolution of calcite. Dynamic wettability alteration is built into the model by defining relative permeability and capillary pressure curves as an interpolation of two sets of end point curves corresponding to mixed-wet and water-wet conditions. This interpolation depends on the dissolution of calcite in such a way that when no dissolution has taken place, mixed-wet conditions prevail. However, gradually there is a shift towards more water-wet conditions at the places in the core where dissolution of calcite takes place. A striking feature reflected by the experimental data found in the literature is that the steady state level of oil recovery, for a fixed temperature, depends directly on the brine composition. We demonstrate that the proposed model naturally can explain this behavior by relating the wettability change to changes in the mineral composition due to dissolution/precipitation. Special attention is paid to the effect of varying, respectively, the concentration of $\text{SO}_4^{2-}$ ions and $\text{Mg}^{2+}$ ions in seawater like brines. The effect of changing the temperature is also demonstrated and evaluated in view of observed experimental behavior.
Rate-independent phase transitions in elastic materials: A Young-measure approach
Alice Fiaschi
2010, 5(2): 257-298 doi: 10.3934/nhm.2010.5.257 +[Abstract](348) +[PDF](416.6KB)
A quasistatic evolution problem for a phase transition model with nonconvex energy density is considered in terms of Young measures. We focus on the particular case of a finite number of phases. The new feature consists in the usage of suitable regularity arguments in order to prove an existence result for a notion of evolution presenting some improvements with respect to the one defined in [13], for infinitely many phases.
Stars of vibrating strings: Switching boundary feedback stabilization
Martin Gugat and Mario Sigalotti
2010, 5(2): 299-314 doi: 10.3934/nhm.2010.5.299 +[Abstract](497) +[PDF](214.4KB)
We consider a star-shaped network consisting of a single node with $N\geq 3$ connected arcs. The dynamics on each arc is governed by the wave equation. The arcs are coupled at the node and each arc is controlled at the other end. Without assumptions on the lengths of the arcs, we show that if the feedback control is active at all exterior ends, the system velocity vanishes in finite time.
   In order to achieve exponential decay to zero of the system velocity, it is not necessary that the system is controlled at all $N$ exterior ends, but stabilization is still possible if, from time to time, one of the feedback controllers breaks down. We give sufficient conditions that guarantee that such a switching feedback stabilization where not all controls are necessarily active at each time is successful.
Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks
Zhong-Jie Han and Gen-Qi Xu
2010, 5(2): 315-334 doi: 10.3934/nhm.2010.5.315 +[Abstract](450) +[PDF](606.0KB)
A kind of planar network of strings with non-collocated terms in boundary feedback controls is considered. Suppose that the network is constituted by $n$ non-uniform strings, connected by one vibrating point mass. The displacements of these strings are continuous at the common vertex. The non-collocated terms are contained in feedback controls at exterior nodes. The well-posedness of the corresponding closed-loop system is proved. A complete spectral analysis is carried out and the asymptotic expression of the spectrum of this system operator is obtained, which implies that the asymptotic behavior of the spectrum is independent of these non-collocated terms. Then the Riesz basis property of the (generalized) eigenvectors of the system operator is proved. Thus, the spectrum determined growth condition holds. Finally, the exponential stability of a special case of this kind of network is gotten under certain conditions. In order to support these results, a numerical simulation is given.
Electromagnetic circuits
Graeme W. Milton and Pierre Seppecher
2010, 5(2): 335-360 doi: 10.3934/nhm.2010.5.335 +[Abstract](334) +[PDF](291.1KB)
The electromagnetic analog of an elastic spring-mass network is constructed. These electromagnetic circuits offer the promise of manipulating electromagnetic fields in new ways, and linear electrical circuits correspond to a subclass of them. The electromagnetic circuits consist of thin triangular magnetic components joined at the edges by cylindrical dielectric components. Some of the edges can be terminal edges to which electric fields are applied. The response is measured in terms of the real or virtual free currents that are associated with the terminal edges. The relation between the terminal electric fields and the terminal free currents is governed by a symmetric complex matrix $\W$. In the case where all the terminal edges are disjoint, and the frequency is fixed, a complete characterization is obtained of all possible response matrices $\W$ both in the lossless and lossy cases. This is done by introducing a subclass of electromagnetic circuits, called electromagnetic ladder networks, which can realize the response matrix $\W$ of any other type of electromagnetic circuit with disjoint terminal edges. It is sketched how an electromagnetic ladder network, structured as a cubic network, can have a macroscopic electromagnetic continuum response which is non-Maxwellian, and novel.
Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain
Mamadou Sango
2010, 5(2): 361-384 doi: 10.3934/nhm.2010.5.361 +[Abstract](459) +[PDF](317.1KB)
We investigate the Neumann problem for a nonlinear elliptic operator $Au^{( s) }=-\sum_{i=1}^{n}\frac{\partial }{ \partial x_{i}}( a_{i}( x,\frac{\partial u^{( s) }}{ \partial x})) $ of Leray-Lions type in the domain $\Omega ^{( s) }=\Omega \backslash F^{( s) }$, where $\Omega $ is a domain in $\mathbf{R}^{n}$($n\geq 3$), $F^{( s) }$ is a closed set located in the neighbourhood of a $(n-1)$-dimensional manifold $ \Gamma $ lying inside $\Omega $. We study the asymptotic behaviour of $ u^{( s) }$ as $s\rightarrow \infty $, when the set $F^{( s) }$ tends to $\Gamma $. Under appropriate conditions, we prove that $ u^{( s) }$ converges in suitable topologies to a solution of a limit boundary value problem of transmission type, where the transmission conditions contain an additional term.

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