Perturbation of eigenvalues due to gaps in two-dimensional boundaries

Davis, A. M. J.; Smith, Stefan G. Llewellyn

doi:10.1098/rspa.2006.1796

Cited by 11 publications

(4 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that (3.1) is the lined condition (2.2), except that we now include an impulse term on the right. This is precisely the form of Green's function used by Davis & Llewellyn Smith (2007), and earlier in a different context by LeBlond & Mysak (1978). It then follows by superposition that…”

Section: The Green's Function and Integral Equation Formulationmentioning

confidence: 52%

Eigenmodes of lined flow ducts with rigid splices

Davis¹,

Peake²

2011

J. Fluid Mech.

View full text Add to dashboard Cite

This paper presents an analytic expression for the acoustic eigenmodes of a cylindrical lined duct with rigid axially running splices in the presence of flow. The cylindrical duct is considered to be uniformly lined except for two symmetrically positioned axially running rigid liner splices. An exact analytic expression for the acoustic pressure eigenmodes is given in terms of an azimuthal Fourier sum, with the Fourier coefficients given by a recurrence relation. Since this expression is derived using a Green’s function method, the completeness of the expansion is guaranteed. A numerical procedure is described for solving this recurrence relation, which is found to converge exponentially with respect to number of Fourier terms used and is in practice quick to compute; this is then used to give several numerical examples for both uniform and sheared mean flow. An asymptotic expression is derived to directly calculate the pressure eigenmodes for thin splices. This asymptotic expression is shown to be quantitatively accurate for ducts with very thin splices of less than 1 % unlined area and qualitatively helpful for thicker splices of the order of 6 % unlined area. A thin splice is in some cases shown to increase the damping of certain acoustic modes. The influences of thin splices and thin boundary layers are compared and found to be of comparable magnitude for the parameters considered. Trapped modes at the splices are also identified and investigated.

show abstract

Section: The Green's Function and Integral Equation Formulationmentioning

confidence: 52%

Eigenmodes of lined flow ducts with rigid splices

Davis¹,

Peake²

2011

J. Fluid Mech.

View full text Add to dashboard Cite

show abstract

“…Our results show that, to within the three-term asymptotic approximation, the principal eigenvalue λ(ε) is related to the average MFPTv by λ ∼ 1/(Dv). Related eigenvalue perturbation and optimization problems for the Laplacian in two-dimensional domains with localized interior traps, or with traps on the domain boundary, are studied in [4], [7], [8], [9], and [24] (see also the references therein).…”

mentioning

confidence: 99%

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Cheviakov¹,

Ward²,

Straube³

2010

Multiscale Model. Simul.

177

263

View full text Add to dashboard Cite

Abstract. The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in R 3 that contains N small nonoverlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The threeterm asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere. Numerical results, based on global optimization methods, are given for both the optimum discrete energy and the arrangements of the centers {x 1 , . . . , x N } of N circular traps on the boundary of the sphere. A scaling law for the optimum discrete energy, valid for N 1, is also derived.Key words. narrow escape, mean first passage time, matched asymptotic expansions, surface Neumann Green's functions, discrete variational problem, logarithmic switchback terms AMS subject classifications. 35B25, 35C20, 35P15, 35J05, 35J08DOI. 10.1137/100782620 1. Introduction. The narrow escape problem concerns the motion of a Brownian particle confined in a bounded domain Ω ∈ R d (d = 2, 3) whose boundary ∂Ω = ∂Ω r ∪ ∂Ω a is almost entirely reflecting (∂Ω r ), except for small absorbing windows, or traps, labeled collectively by ∂Ω a , through which the particle can escape. Denoting the trajectory of the Brownian particle by X(t), the mean first passage time (MFPT) v(x) is defined as the expectation value of the time τ taken for the Brownian particle to become absorbed somewhere in ∂Ω a starting initially fromThe calculation of v(x) becomes a narrow escape problem in the limit when the measure of the absorbing set |∂Ω a | = O(ε d−1 ) is asymptotically small, where 0 < ε 1 measures the dimensionless radius of an absorbing window. Since the MFPT diverges as ε → 0, the calculation of the MFPT v(x) constitutes a singular perturbation problem.The narrow escape problem has many applications in biophysical modeling (see [2], [16], [19], [39] and the references therein). For the case of a two-dimensional domain, the narrow escape problem has been studied with a variety of analytical methods in [19], [42], [43], [20], and Part I of this paper [30]. In this paper, we use

show abstract

“…The mean escape time corresponds to the reciprocal of the first eigenvalue of the Laplacian with mixed Dirichlet-Neumann boundary conditions, where the size of the Dirichlet part is small. The development of narrow escape theory has thus benefited from the study of singularly perturbed eigenvalue problems both from an applied [10][11][12][13][14] and a formal [15,16] perspective. Alternatively, in certain applications one may be interested in the mean time for the fastest diffusing particle among many to arrive at a small target, a problem first formulated in Weiss et al [17].…”

Section: Introductionmentioning

confidence: 99%

Modelling and asymptotic analysis of the concentration difference in a nanoregion between an influx andoutflux diffusion acrossnarrow windows

Paquin-Lefebvre

Holcman

2021

Proc. R. Soc. A.

View full text Add to dashboard Cite

When a flux of Brownian particles is injected in a narrow window located on the surface of a bounded domain, these particles diffuse and can eventually escape through a cluster of narrow windows. At steady state, we compute asymptotically the distribution of concentration between the different windows. The solution is obtained by solving Laplace’s equation using Green’s function techniques and second-order asymptotic analysis, and depends on the influx amplitude, the diffusion properties, as well as the geometrical organization of all the windows, such as their distances and the mean curvature. We explore the range of validity of the present asymptotic expansions using numerical simulations of the mixed boundary value problem. Finally, we introduce a length scale to estimate how deep inside a domain a local diffusion current can spread.

show abstract

Perturbation of eigenvalues due to gaps in two-dimensional boundaries

Cited by 11 publications

References 18 publications

Eigenmodes of lined flow ducts with rigid splices

Eigenmodes of lined flow ducts with rigid splices

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Modelling and asymptotic analysis of the concentration difference in a nanoregion between an influx andoutflux diffusion acrossnarrow windows

Contact Info

Product

Resources

About