Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors

Ward, Michael J.; Keller, Joseph B.

doi:10.1002/sapm19918511

Cited by 18 publications

(10 citation statements)

References 12 publications

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“…Homogenization procedure. At least formally, one can homogenize the finegrained boundary condition by studying a boundary layer near ∂Ω and matching the boundary layer solution with the solution in the interior of the domain (for formal asymptotic studies of related problems, see, e.g., [34,38,39]). Given a point x 0 ∈ ∂Ω, we choose the coordinates (x, y, z) to be the rescaled Cartesian coordinates in the vicinity of x 0 aligned with the normal ν to ∂Ω at x 0 :…”

Section: Problem Formulationmentioning

confidence: 99%

Boundary Homogenization for Periodic Arrays of Absorbers

Muratov

Shvartsman

2008

Multiscale Model. Simul.

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We introduce a homogenization procedure for reaction-diffusion equations in domains whose boundary consists of small alternating regions with prescribed Dirichlet and Neumann data of comparable areas. The homogenized problem is shown to satisfy an effective Dirichlet boundary condition which depends on the geometry of the small-scale boundary structure. This problem is also related to finding the effective trapping rate for a Brownian particle next to a surface with a periodic array of perfect absorbers. We use the method of optimal geometric grids to numerically solve the unit cell problem of homogenization. The geometric homogenization factor is obtained for a number of cell geometries (stripes, square and hexagonal arrays of disk-shaped absorbers or emitters) as a function of the surface area fraction occupied by the absorbers. Empirical analytical expressions that give excellent fits to data for the entire range of area fractions and correct asymptotic behaviors in the limits of small and large absorber area fractions are proposed.

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Section: Problem Formulationmentioning

confidence: 99%

Boundary Homogenization for Periodic Arrays of Absorbers

Muratov

Shvartsman

2008

Multiscale Model. Simul.

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“…[1], [11], [13], [14], [16]) devoted to characterizing the sensitivity of a simple fold point to various classes of perturbations of (1.1). Specifically, for various perturbed forms of (1.1), asymptotic expansions for the location of the perturbed simple fold point have been derived.…”

Section: A = U^dx (L-lc)mentioning

confidence: 99%

“…In the inner region, we expand the solution as v = uo(xo) + evi + e 2 V2 H , where vi and V2 satisfy (5.10a,b), (5.11). From [14] the far field behaviors of vi and V21 in analogy with (5.12a,b), are…”

Section: Three Dimensionsmentioning

confidence: 99%

On perturbations of cubic fold points for nonlinear eigenvalue problems

Ward¹,

Wolf²

1994

Methods and Applications of Analysis

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ABSTRACT. We consider fold point behavior for nonlinear eigenvalue problems of the form Aito + XQF(UO,P) = 0, x e D; d u uo + buo = 0, x e dD.Here b and 0 are positive parameters. In circular cylindrical or spherical geometries, we assume that F (u, (3) is such that this problem has multiple radially symmetric solutions for some range of AQ only when 0 < /3 < (3Q. As (3 -»• (3Q from below, two simple fold points are assumed to coalesce producing a cubic fold point at AQ = A c o when (3 = j3o. We calculate the change in the cubic fold point location resulting from various classes of perturbations of this problem. The perturbations we consider are a small change in the boundary condition maintained on the boundary of a circular cylindrical or a spherical domain, a small distortion of the boundary of a circular cylindrical domain, and the removal of a small hole from a domain. In each case, we derive asymptotic expansions for the location of the perturbed cubic fold point in terms of a small parameter e measuring the size of the perturbation. A numerical scheme is then formulated to evaluate the coefficients in these expansions and the method is illustrated for the combustion nonlinearity F(u,(3) = exp[w/(l + /3u)]. In certain cases, we compare our expansions for the perturbed cubic fold point location with some previous results and with results obtained from full numerical solutions to the perturbed problems. The significance of these results for the perturbed cubic fold point location are that they can provide the first step in characterizing, at least locally, the alteration of the boundary in parameter space that separates regions where the nonlinear eigenvalue problem has multiple solutions from regions where the problem has only unique solutions.

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“…An asymptotic theory to determine Ac(e) for (5.1), and for more general strong localized perturbations, was initiated in Ward & Keller (1991) and was extended and validated numerically in W ard & Van de Velde (1991). The asymptotic theory presented there typically provided only the first correction to the location of the unperturbed fold point.…”

Section: Strong Localized Perturbations: Higher Order Theorymentioning

confidence: 99%

“…Here we determine Ac when a small subdomain De of ' radius ' e is removed from D and a boundary condition imposed on the resulting hole. The theory to determine Ac in this case was initiated in W ard & Keller (1991) and was extended and validated numerically in W ard & Van de Velde (1991). In §5 we further extend the theory by deriving a two term expansion for Ac in the limit of small-hole radius.…”

Section: Introductionmentioning

confidence: 98%

Criticality in reactors under domain or external temperature perturbations

1991

Proc. R. Soc. Lond. A

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The conditions for the onset of thermal runaway in reactors with small non-uniformities is investigated. The reaction is modelled by an Arrhenius heat generation term with a finite activation energy and the dimensionless temperature, u 0 , is taken to satisfy a nonlinear equation of the form Δ u 0 + λ 0 F ( u 0 ) = 0, x ∈ D ; ∂ v u 0 + bu 0 = 0. x ϵ∂ D . We investigate three classes of perturbations of this problem. First, we treat a small temperature variation maintained on the boundary of the domain. Secondly, we consider a small distortion of the boundary of a circular cylindrical domain, and thirdly, we analyse the effect of a small hole in the domain. In each case we derive asymptotic expansions for the critical Frank-Kamenetskii parameter, λ c ( ϵ ), where ϵ is a measure of the size of the perturbation. A numerical scheme is then used to determine numerical values for the coefficients in the asymptotic expansion of λ c . Finally, some of the asymptotic results are compared with corresponding numerical results obtained from a full numerical solution of the perturbed problem.

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Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors

Cited by 18 publications

References 12 publications

Boundary Homogenization for Periodic Arrays of Absorbers

Boundary Homogenization for Periodic Arrays of Absorbers

On perturbations of cubic fold points for nonlinear eigenvalue problems

Criticality in reactors under domain or external temperature perturbations

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