Showing papers on "Method of matched asymptotic expansions published in 2020"
TL;DR: In this article, the authors studied the null geodesics extending from the near-horizon region out to the far region in the background of the Schwarzschild and the singly spinning Myers-Perry black holes in the large dimension limit.
Abstract: We study the null geodesics extending from the near-horizon region out to the far region in the background of the Schwarzschild and the singly spinning Myers-Perry black holes in the large dimension limit. We find that in this limit the radial integrals of these geodesics can be obtained by using the method of matched asymptotic expansions. If the motion of the photon is confined to the equator plane, then all geodesic equations are solvable analytically. The study in this paper may provide a toy model to analyze the observables relevant to the electromagnetic phenomena occurring near the black holes.
19 citations
TL;DR: In this paper, the authors used matched asymptotic expansions to solve the equations of force-free electrodynamics in a perturbative expansion valid at small black hole spin.
Abstract: The Blandford-Znajek mechanism is the continuous extraction of energy from a rotating black hole via plasma currents flowing on magnetic field lines threading the horizon. In the discovery paper, Blandford and Znajek demonstrated the mechanism by solving the equations of force-free electrodynamics in a perturbative expansion valid at small black hole spin. Attempts to extend this perturbation analysis to higher order have encountered inconsistencies. We overcome this problem using the method of matched asymptotic expansions, taking care to resolve all of the singular surfaces (light surfaces) in the problem. Working with the monopole field configuration, we show explicitly how the inconsistencies are resolved in this framework and calculate the field configuration to one order higher than previously known. However, there is no correction to the energy extraction rate at this order. These results confirm the basic consistency of the split monopole at small spin and lay a foundation for further perturbative studies of the Blandford-Znajek mechanism.
18 citations
TL;DR: In this paper, the spreading of viscous fluid injected under an elastic sheet is studied, which is driven by gravity and by elastic bending and tension forces and resisted by viscous forces.
Abstract: We study the spreading of viscous fluid injected under an elastic sheet, which is driven by gravity and by elastic bending and tension forces and resisted by viscous forces. The injected fluid forms a large blister and spreads outwards analogously to a viscous gravity current or a capillary droplet. The relative strengths of the three driving forces are determined by how the horizontal length scales of the system compare with three key transition length scales. Bending is dominant on small length scales, tension is dominant on intermediate length scales and gravity is dominant on large length scales. We show how to use the method of matched asymptotic expansions to predict the spreading rate and thickness profile of the blister of fluid in the seven possible asymptotic regimes, for both two-dimensional and axisymmetric geometries. Consideration of different physical effects at the fluid front increases the number of regimes yet further.
16 citations
TL;DR: In this paper, the authors employ the matched asymptotic expansions to model Helmholtz resonators, with thermoviscous effects incorporated starting from first principles and with the lumped parameters characterizing the neck and cavity geometries precisely defined and provided explicitly for a wide range of geometry.
12 citations
TL;DR: In this article, the authors study the problem of wave scattering by periodic arrays of Neumann scatterers in the plane, where the characteristic length scale of the scatterer is considered small relative to the lattice period.
11 citations
TL;DR: In this paper, the authors theoretically investigate the motion of an arbitrarily shaped particle in a linear density stratified fluid with weak stratification and negligible inertia, and calculate the hydrodynamic force and torque experienced by the particle using the method of matched asymptotic expansions.
Abstract: In this work, we theoretically investigate the motion of an arbitrarily shaped particle in a linear density stratified fluid with weak stratification and negligible inertia. We calculate the hydrodynamic force and torque experienced by the particle using the method of matched asymptotic expansions. We analyse our results for two classes of particles (non-skew and skew) depending on whether the particle possesses a centre of hydrodynamic stress. For both classes, we derive general expressions for the modified resistance tensors in the presence of stratification. We demonstrate the application of our results by considering some specific examples of particles settling in a direction parallel to the density gradient by considering both the limits of high () Peclet numbers. We find that presence of stratification causes a slender body to rotate and settle along the broader side due to the contribution of the hydrostatic torque. Our work sheds light on the impact of stratification on the transport of arbitrarily shaped particles in density stratified environments in low-Reynolds-number regimes.
10 citations
TL;DR: In this article, the authors derived the mean velocity profile in the convective atmospheric boundary layer (CBL) using the method of matched asymptotic expansions and derived the MM scaling from first principles.
Abstract: The mean velocity profile in the convective atmospheric boundary layer (CBL) is derived analytically. The shear-stress budget equations and the mean momentum equations are employed in the derivation. The multi-point Monin–Obukhov similarity (MMO) recently proposed and analytically derived by Tong & Nguyen (J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) and Tong & Ding (J. Fluid Mech., vol. 864, 2019, pp. 640–669) provides the scaling properties of the statistics in the shear-stress budget equations. Our previous and present studies have shown that the CBL is mathematically a singular perturbation problem. Therefore, we obtain the mean velocity profile using the method of matched asymptotic expansions. Three scaling layers are identified: the outer layer, which includes the mixed layer, the inner-outer layer and the inner-inner layer, which includes the roughness layer. There are two overlapping layers, the local-free-convection layer and the log layer, respectively. Two new velocity-defect laws are discovered: the mixed-layer velocity-defect law and the surface-layer velocity-defect law. The local-free-convection mean profile is obtained by asymptotically matching the expansions in the first two layers. The log law is obtained by matching the expansions in the last two layers. The von Karman constant is obtained using velocity and length scales, and therefore has a physical interpretation. A new friction law, the convective logarithmic friction law, is obtained. The present work provides an analytical derivation of the mean velocity profile hypothesized in the Monin–Obukhov similarity theory, and is part of a comprehensive derivation of the MMO scaling from first principles.
10 citations
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TL;DR: In this paper, the authors study the problem of wave scattering by periodic arrays of Neumann scatterers in the plane, where the characteristic length scale of the scatterer is considered small relative to the lattice period.
Abstract: We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on topical examples from topological wave physics.
9 citations
TL;DR: In this paper, the authors developed asymptotic approximations in the near contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry.
Abstract: Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations.
8 citations
TL;DR: In this paper, the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et al.
Abstract: In a singularly perturbed limit, we analyse the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci., 20, Suppl., (2010)], which models urban crime with police intervention. In our extended RD model, the field variables are the attractiveness field for burglary, the criminal population density and the police population density. Our model includes a scalar parameter that determines the strength of the police drift towards maxima of the attractiveness field. For a special choice of this parameter, we recover the ‘cops-on-the-dots’ policing strategy of Jones et. al., where the police mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocal eigenvalue problems (NLEPs) that characterise the linear stability of these hotspot steady states to (1) timescale instabilities. For a cops-on-the-dots policing strategy, we prove that a multi-hotspot steady state is linearly stable to synchronous perturbations of the hotspot amplitudes. Alternatively, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical–numerical method is used to construct linear stability phase diagrams in the police vs. criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model, this provides a parameter range where the effect of a cops-on-the-dots policing strategy is to only displace crime temporally between neighbouring spatial regions. Our hybrid approach to study the NLEPs combines rigorous spectral results with a numerical parameterisation of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.
8 citations
TL;DR: In this paper, an AFM-based nanoindentation of a spherical particle absorbed on a corrugated surface of rigid substrate is considered in the framework of continuum mechanics, and the corresponding problem of unilateral frictionless contact with multiple contact zones is formulated using isotropic infinitesimal elasticity.
TL;DR: In this paper, the authors considered the scattering of plane longitudinal monochromatic waves from a heterogeneous inclusion of arbitrary shape in an infinite poroelastic medium, and formulated the scattering problem in terms of the volume integral equations for displacements of the solid skeleton and fluid pressure in the pore space.
TL;DR: In inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer-Meinhardt reaction-diffusion system the structure and linear stability of selected one- and two-spike patterns are investigated.
Abstract: The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction-diffusion equations has been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer-Meinhardt reaction-diffusion system. We show that these boundary conditions necessitate the formation of spikes that concentrate in a boundary layer near the domain boundaries. Using the method of matched asymptotic expansions we construct boundary layer spikes and derive a new class of shifted Nonlocal Eigenvalue Problems for which we rigorously prove partial stability results. Moreover by using a combination of asymptotic, rigorous, and numerical methods we investigate the structure and linear stability of selected one- and two-spike patterns. In particular we find that inhomogeneous Neumann boundary conditions increase both the range of parameter values over which asymmetric two-spike patterns exist and are stable.
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TL;DR: In this paper, the authors employ the matched asymptotic expansions to model Helmholtz resonators, with thermoviscous effects incorporated starting from first principles and with the lumped parameters characterizing the neck and cavity geometries precisely defined and provided explicitly for a wide range of geometry.
Abstract: We systematically employ the method of matched asymptotic expansions to model Helmholtz resonators, with thermoviscous effects incorporated starting from first principles and with the lumped parameters characterizing the neck and cavity geometries precisely defined and provided explicitly for a wide range of geometries. With an eye towards modeling acoustic metasurfaces, we consider resonators embedded in a rigid surface, each resonator consisting of an arbitrarily shaped cavity connected to the external half-space by a small cylindrical neck. The bulk of the analysis is devoted to the problem where a single resonator is subjected to a normally incident plane wave; the model is then extended using "Foldy's method" to the case of multiple resonators subjected to an arbitrary incident field. As an illustration, we derive critical-coupling conditions for optimal and perfect absorption by a single resonator and a model metasurface, respectively.
TL;DR: In this paper, the authors used matched asymptotic expansions to solve the equations of force-free electrodynamics in a perturbative expansion valid at small black hole spin.
Abstract: The Blandford-Znajek mechanism is the continuous extraction of energy from a rotating black hole via plasma currents flowing on magnetic field lines threading the horizon. In the discovery paper, Blandford and Znajek demonstrated the mechanism by solving the equations of force-free electrodynamics in a perturbative expansion valid at small black hole spin. Attempts to extend this perturbation analysis to higher order have encountered inconsistencies.We overcome this problem using the method of matched asymptotic expansions, taking care to resolve all of the singular surfaces (light surfaces) in the problem. Working with the monopole field configuration, we show explicitly how the inconsistencies are resolved in this framework and calculate the field configuration to one order higher than previously known. However, there is no correction to the energy extraction rate at this order. These results confirm the basic consistency of the split monopole at small spin and lay a foundation for further perturbative studies of the Blandford-Znajek mechanism.
TL;DR: Reduced models to approximate the solution of the electromagnetic scattering problem in an unbounded domain which contains a small perfectly conducting sphere are developed based on the method of matched asymptotic expansions.
TL;DR: In this article, a biased velocity jump process with excluded-volume interactions for chemotaxis was considered, where the size of each particle was taken into account and a nonlinear kinetic model using two different approaches was derived.
Abstract: In this paper we consider a biased velocity jump process with excluded-volume interactions for chemotaxis, where we account for the size of each particle. Starting with a system of N individual hard rod particles in one dimension, we derive a nonlinear kinetic model using two different approaches. The first approach is a systematic derivation for small occupied fraction of particles based on the method of matched asymptotic expansions. The second approach, based on a compression method that exploits the single-file motion of hard core particles, does not have the limitation of a small occupied fraction but requires constant tumbling rates. We validate our nonlinear model with numerical simulations, comparing its solutions with the corresponding noninteracting linear model as well as stochastic simulations of the underlying particle system.
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TL;DR: A class of cell-bulk coupled PDE-ODE models, motivated by quorum and diffusion sensing phenomena in microbial systems, that characterize communication between localized spatially segregated dynamically active signaling compartments that have a permeable boundary are analyzed.
Abstract: We analyze a class of cell-bulk coupled PDE-ODE models, motivated by quorum and diffusion sensing phenomena in microbial systems, that characterize communication between localized spatially segregated dynamically active signaling compartments that have a permeable boundary. Each cell secretes a signaling chemical into the bulk region at a constant rate and receives a feedback of the bulk chemical from the entire collection of cells. This global feedback, which activates signaling pathways within the cells, modifies the intracellular dynamics according to the external environment. The cell secretion and global feedback are regulated by permeability parameters across the cell membrane. For arbitrary reaction-kinetics within each cell, the method of matched asymptotic expansions is used in the limit of small cell radius to construct steady-state solutions of the PDE-ODE model, and to derive a globally coupled nonlinear matrix eigenvalue problem (GCEP) that characterizes the linear stability properties of the steady-states. In the limit of large bulk diffusivity an asymptotic analysis of the PDE-ODE model leads to a limiting ODE system for the spatial average of the concentration in the bulk region that is coupled to the intracellular dynamics within the cells. Results from the linear stability theory and ODE dynamics are illustrated for Sel'kov reaction-kinetics, where the kinetic parameters are chosen so that each cell is quiescent when uncoupled from the bulk medium. For various specific spatial configurations of cells, the linear stability theory is used to construct phase diagrams in parameter space characterizing where a switch-like emergence of intracellular oscillations can occur through a Hopf bifurcation.
01 Jan 2020
TL;DR: In this article, a semi-analytical approach for predicting crack onset at sharp notches, bi-material junctions and openings in fiber-reinforced laminates is proposed.
Abstract: The objective of this work is to provide an efficient semi-analytical approach for predicting crack onset at sharp notches, bi-material junctions and openings in fibre-reinforced laminates. On the one hand, this is achieved employing a complex potential approach that accurately captures stress concentrations and enables a precise calculation of generalised stress intensity factors. On the other hand, the method of matched asymptotic expansions is applied interpreting a newly nucleated crack as a small perturbation parameter within a singular perturbation problem. This methodology enables an efficient evaluation of the coupled stress and energy criterion forming a necessary and sufficient condition for fracture. Predictions relying on the asymptotic approach are validated against fully numerical reference data as well as experimental findings from literature.
TL;DR: In this article, the authors present a different method of constructing asymptotic solutions of the inner region equations and study the convergence properties of the new and old solutions, and they show that there are not only regimes of excellent agreement with straight-through methods, but also regimes of strong disagreement.
Abstract: A recent publication presents a detailed prescription for constructing global eigenfunctions and complex growth rates for resistive instabilities in axisymmetric toroidal systems, using the method of matched asymptotic expansions. At each singular surface in the plasma, the small-x asymptotic solutions of the ideal MHD outer-region equations are matched to the large-x asymptotic solutions of the resistive inner-region equations, with x being the distance from the singular surface. The success of this method depends upon accurate asymptotic solutions of the resistive inner region equations. Extensive studies have shown that there are not only regimes of excellent agreement with straight-through methods, without asymptotic matching, but also regimes of strong disagreement. In an effort to find the cause of this discrepancy, we present here a different method of constructing asymptotic solutions of the inner region equations and we study the convergence properties of the new and old solutions.
TL;DR: In this paper, the authors considered the problem of a free liquid film vertically bounded by solid walls and being under the influence of gravity and thermocapillary forces in a thin-layer approximation.
Abstract: Problem of a free liquid film vertically bounded by solid walls and being under the influence of gravity and thermocapillary forces in a thin-layer approximation is considered. A solution in which the film thickness is constant and temperature is a linear function of the longitudinal coordinate is investigated for stability analytically using the method of matched asymptotic expansions and numerically using the orthogonalization method for various values of acceleration of gravity. The results obtained analytically and numerically are in good agreement. It is shown that the solution is unstable, but the increment of perturbations is small even under terrestrial gravity.
TL;DR: In this paper, a strongly anisotropic magnetic trap whose trapping potential is much higher in the direction z than in the transverse direction is considered, where the condensate takes the form of a plane disk.
Abstract: Within the framework of the Gross–Pitaevskii model, we deduce a system of equations that describes the motion of quantized vortices in Bose–Einstein condensates. We consider a strongly anisotropic magnetic trap whose trapping potential is much higher in the direction z than in the transverse direction. Under the action of this potential, the condensate takes the form of a plane disk. The transition to the two-dimensional case enables us to apply the method of matched asymptotic expansions and obtain the equations of vortex motion in the explicit form. We take into account the rotation of the condensate as a whole and the effect of dissipative processes as a result of which the system of vortices comes to the equilibrium state. Some examples of the vortex motion are presented for different values of the external parameters.