Corner Layer Research Articles

Singularly Perturbed Solution of Non-Fourier Temperature Field with Mixed Boundary Value Dual-Phase Delay with Thermal Conductivity Jumps

Based on the properties of laminates, a class of nonlinear singularly perturbed mixture equations with discontinuous coefficients over bounded domains is constructed by using a dual-phase delayed heat conduction model. First, the singular perturbation expansion method is used, combined with the corresponding boundary conditions, the partial differential equation method and the Laplace transform method are used to obtain the external solution, the boundary layer, and the corner layer. Secondly, the time-varying temperature field at the discontinuity is obtained, which leads to the asymptotic expansion of the solution. Finally, the consistent validity of the asymptotic solution is obtained through residual estimation.

Extraction, Purification, Bioactivities and Application of Matrix Proteins From Pearl Powder and Nacre Powder: A Review.

Natural pearls are formed when sand or parasites (irritants) accidentally enter into the oyster body and form pearls under the cover of the nacre layer. Pearl powder is a powdery substance by grinding pearls into small grains, however, the nacre powder is the inner layer of outer corner layer and middle prism layer. Pearl medicine in China has a history of more than 2,000 years, pearl has the effects of calming the mind, clearing the eyes, detoxifying the muscle and so on. In this paper, the researches on the extraction of pearl powder and nacre powder, the isolation and purification of matrix protein and the various biological activities (osteogenic activity, antioxidant, anti-inflammatory, anti-apoptotic, promoting the migration of fibroblasts, and so on) are reviewed in detail. To provide readers with a faster understanding, the method of extraction and purification and the application of nacre powder and pearl powder are clearly presented in the form of figures and tables. In line with the concept of waste or by-product, there are more reports of nacre extract than pearl extract, due to the expensive and limited in origin of pearls. Mainly on the direct use of nacre powder and pearl powder or on the use of extracts (mainly water soluble proteins) through experiments in vivo or in vitro, and shows whether it is effective through the results of various indexes. There is no further study on substances other than extracts, and the structural analysis of extracts needs further exploration.

Revisiting steady viscous flow of a generalized Newtonian fluid through a slender elastic tube using shell theory

AbstractA flow vessel with an elastic wall can deform significantly due to viscous fluid flow within it, even at vanishing Reynolds number (no fluid inertia). Deformation leads to an enhancement of throughput due to the change in cross‐sectional area. The latter gives rise to a non‐constant pressure gradient in the flow‐wise direction and, hence, to a nonlinear flow rate–pressure drop relation (unlike the Hagen–Poiseuille law for a rigid tube). Many biofluids are non‐Newtonian, and are well approximated by generalized Newtonian (say, power‐law) rheological models. Consequently, we analyze the problem of steady low Reynolds number flow of a generalized Newtonian fluid through a slender elastic tube by coupling fluid lubrication theory to a structural problem posed in terms of Donnell shell theory. A perturbative approach (in the slenderness parameter) yields analytical solutions for both the flow and the deformation. Using matched asymptotics, we obtain a uniformly valid solution for the tube's radial displacement, which features both a boundary layer and a corner layer caused by localized bending near the clamped ends. In doing so, we obtain a “generalized Hagen–Poiseuille law” for soft microtubes. We benchmark the mathematical predictions against three‐dimensional two‐way coupled direct numerical simulations (DNS) of flow and deformation performed using the commercial computational engineering platform by ANSYS. The simulations show good agreement and establish the range of validity of the theory. Finally, we discuss the implications of the theory on the problem of the flow‐induced deformation of a blood vessel, which is featured in some textbooks.

Alternating direction numerical scheme for singularly perturbed 2D degenerate parabolic convection-diffusion problems

In this article, we study the numerical solution of singularly perturbed 2D degenerate parabolic convection-diffusion problems on a rectangular domain. The solution of this problem exhibits parabolic boundary layers along x=0,y=0 and a corner layer in the neighborhood of (0, 0). First, we use an alternating direction implicit finite difference scheme to discretize the time derivative of the continuous problem on a uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problems, we apply the upwind finite difference scheme on a piecewise-uniform Shishkin mesh. We derive error estimate for the proposed numerical scheme, which shows that the scheme is ε-uniformly convergent of almost first-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.

Boundary layer mesh generation on arbitrary geometries

SummaryVolume boundary layer mesh generation on non‐smooth geometries with multiple normals is considered in this work. A new highlight is given to corner and ridge boundary layer mesh generation through the generalized Voronoi diagram in general, and the spherical Voronoi diagram in particular. This provides the keystone to allow for the first time boundary layer mesh generation at arbitrary corner configurations. The work proposed in is revisited and compared with the new approach. A detailed description of the new algorithm is provided. The spherical Voronoi diagram delivers geometrically the optimal normals as well as topologically the optimal connectivities, as far as normality is concerned. Numerical examples illustrate the accuracy and robustness of the method. This approach seems to handle arbitrary geometry boundary layer mesh generation, in theory as well as in practice. Copyright © 2017 John Wiley & Sons, Ltd.

Characterizing spatial distribution of the adsorbed water in wood cell wall of Ginkgo biloba L. by μ-FTIR and confocal Raman spectroscopy

Abstract The adsorbed water influences significantly, the physical and mechanical properties of wood. In the present paper, the spatial distribution of adsorbed water in wood cell walls has been studied by μ-Fourier transform infrared (μ-FTIR) and confocal Raman spectroscopy. In situ μ-FTIR spectra were collected from three randomly selected areas in different cell wall regions, which were exposed to an environment with 0% to 96% relative humidity (RH). The water adsorption sites were easily detectable based on OH, C=O, and C-O group vibrations and it was shown that the adsorbed water concentration was not uniform in different regions. Confocal Raman spectroscopy images were collected from the cell corner (CC) and middle layer of the secondary wall (S2) and the non-uniformity of water distribution could also be confirmed by this approach. It was demonstrated that both μ-FTIR and confocal Raman spectroscopy provide valuable information about the spatial distribution of adsorbed water in morphologically distinct cell wall regions.

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

We study the ground state which minimizes a Gross–Pitaevskii energy with general non-radial trapping potential, under the unit mass constraint, in the Thomas–Fermi limit where a small parameter \({\varepsilon}\) tends to 0. This ground state plays an important role in the mathematical treatment of recent experiments on the phenomenon of Bose–Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrödinger equations. Many of these applications require delicate estimates for the behavior of the ground state near the boundary of the condensate, as \({\varepsilon\to 0}\), in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the condensate, to understand the superfluid flow around an obstacle, and it also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the classical Thomas–Fermi approximation and accurately approximate the layer by the Hastings–McLeod solution of the Painlevé-II equation. This settles an open problem (cf. Aftalion in Vortices in Bose Einstein Condensates. Birkhäuser Boston, Boston, 2006, pg. 13 or Open Problem 8.1), answered very recently only for the special case of the model harmonic potential (Gallo and Pelinovsky in Asymptot Anal 73:53–96, 2011). In fact, we even improve upon previous results that relied heavily on the radial symmetry of the potential trap. Moreover, we show that the ground state has the maximal regularity available, namely it remains uniformly bounded in the \({\frac{1}{2}}\)-Hölder norm, which is the exact Hölder regularity of the singular limit profile, as \({\varepsilon\to 0}\). Our study is highly motivated by an interesting open problem posed recently by Aftalion, Jerrard, and Royo-Letelier (J Funct Anal 260:2387–2406 2011), and an open question of Gallo and Pelinovsky (J Math Anal Appl 355:495–526, 2009), concerning the removal of the radial symmetry assumption from the potential trap.

Initial–boundary layer associated with the nonlinear Darcy–Brinkman system

We study the interaction of initial layer and boundary layer in the nonlinear Darcy–Brinkman system in the vanishing Darcy number limit. In particular, we show the existence of a function of corner layer type (so-called initial–boundary layer) in the solution of the nonlinear Darcy–Brinkman system. An approximate solution is constructed by the method of multiple scale expansion in space and in time. We establish the optimal convergence rates in various Sobolev norms via energy method.

THE ASYMPTOTIC ESTIMATION AND APPLICATION FOR A CLASS OF SEMI-LINEAR SINGULARLY PERTURBED PROBLEMS

In this paper,a class of semi-linear singularly perturbed boundary value problems is considered.The left and the right boundary layer functions and the internal corner layer function with algebraic type characteristic are constructed.Using the theory of differential inequalities,the existence of solution for the problem is proved and the asymptotic estimation of solution is obtained.The general conclusion of this problem is given.And the result is applied to a model of combustion problem and the corresponding asymptotic estimation is obtained.The related problem and the corresponding result are generalized.

Particle Image Velocimetry Measurement of Laminar Boundary Layer in a Streamwise Corner

Measurement of velocity profiles along a laminar boundary layer in a streamwise corner was carried out by using particle image velocimetry in contrast to conventional hot-wire anemometry. To make the measurement successful, the laser light source for particle image velocimetry was placed at a far-downstream station, and a light sheet was emitted in parallel to the corner intersection line. The platewas tilted at very small angles relative to the freestream to realize the laminar flow starting from the corner leading edge. Discussions on the characteristic features of the velocity profile in the bisector plane are given based on the measurement data. It was found that the measured velocity profiles correspond to the Blasius branch solution of the theoretical corner layer equations.

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.

A singularly perturbed semilinear reaction–diffusion problem in a polygonal domain

The semilinear reaction–diffusion equation − ε 2 Δ u + b ( x , u ) = 0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the “reduced equation” b ( x , u 0 ( x ) ) = 0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation − Δ z + f ( z ) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.

Influence of corner layers in the variational determination of bubble solutions of the constrained Allen–Cahn equation

A steady-state bubble solution to the constrained mass conserving Allen–Cahn equation in a two-dimensional domain is constructed in the limit of small diffusivity. The solution is asymptotically constant inside a circle of radius rb centred at some unknown location x0 and has a sharp interface at the bubble radius that allows for a transition to a different asymptotically constant state outside the bubble. In a study by M. J. Ward (Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56, 1996, 247–1279), the bubble centre was determined by a limiting solvability condition. The solution found by Ward suggests the existence of a corner type boundary layer where the normal derivative of the bubble solution readjusts to satisfy the no-flux condition at the boundary of the domain. This work is concerned with the details of the readjustment. A variational approach similar to the one of W. L. Kath, C. Knessl and B. J. Matkowsky (A variational approach to nonlinear singularly perturbed boundary-value problems. Stud. Appl. Math. 77, 1987, 61–88) shows the formation of a corner layer (for the derivative of the solution) which influences as a high-order correction the available determination of the bubble centre. This corner layer describes to leading order the readjustment of the level lines of the bubble to lines parallel to the boundary of the container; moreover, it provides to leading order a smooth solution across the corner layer.

PML absorbing boundary conditions for the linearized and nonlinear Euler equations in the case of oblique mean flow

AbstractFor the case of uniform mean flow in an arbitrary direction, perfectly matched layer (PML) absorbing boundary conditions are presented for both the linearized and nonlinear Euler equations. Although linear perfectly matched side layers with an oblique mean flow have been studied in previous works, we propose in the present paper a construction of corner layer equations that are dynamically stable. Stability issues are investigated by examining the dispersion relations of linear waves supported by the corner layer equations. For increased efficiency, a pseudo mean flow is included in the derivation of the PML equations for the nonlinear case. Numerical examples are given to support the validity of the proposed equations. Specifically, the linear PML formulation is tested for the case of acoustic, vorticity, and entropy waves traveling with an oblique mean flow. The nonlinear formulation is tested with an isentropic vortex moving diagonally with a constant velocity. Copyright © 2008 John Wiley & Sons, Ltd.

Computations of the first eigenpairs for the Schrödinger operator with magnetic field

This paper is devoted to computations of eigenvalues and eigenvectors for the Schrödinger operator with constant magnetic field in a domain with corners, as the semi-classical parameter h tends to 0. The eigenvectors corresponding to the smallest eigenvalues concentrate in the corners: They have a two-scale structure, consisting of a corner layer at scale h and an oscillatory term at scale h. The high frequency oscillations make the numerical computations particularly delicate. We propose a high order finite element method to overcome this difficulty. Relying on such a discretization, we illustrate theoretical results on plane sectors, squares, and other straight or curved polygons. We conclude by discussing convergence issues.

Ray Solution of a Singularly Perturbed Elliptic PDE with Applications to Communications Networks

We analyze a second order, linear, elliptic PDE with mixed boundary conditions. This problem arose as a limiting case of a Markov‐modulated queueing model for data handling switches in communications networks. We use singular perturbation methods to analyze the problem. In particular we use the ray method to solve the PDE in the limit where convection dominates diffusion. We show that there are both interior and boundary caustics, as well as a cusp point where two caustics meet, an internal layer, boundary layers, and a corner layer. Our analysis leads to approximate formulas for the queue length (or buffer content) distribution at the switch.

The Chapman–Enskog procedure for an age-structured population model: initial, boundary and corner layer corrections

We consider a mathematical model of an age-structured population of some fisheries (for example, anchovies, sardines or soles). Two time scales are involved in the problem: the fast time scale for the migration dynamics and the slow time scale for the demographic process. At a first step, we study the so called ‘aggregated’ system by means of the semigroups theory. Then, we study the asymptotic behaviour of the model by using the Chapman–Enskog procedure. In particular, we study initial, boundary and corner layer effects in order to obtain the exact initial and boundary conditions the approximated solution has to satisfy.

Mechanism and large eddy simulation of dust devils

Large Eddy Simulations (LES) of vertical convective vortices and dust devils in the terrestrial Convective Boundary Layer (CBL) are presented. A CBL‐scale simulation and a dust‐devil‐scale simulation are preformed at different resolutions. The CBL‐scale simulation is performed to study the generation of vertical vortices and the dust‐devil‐scale simulation is performed to study the detailed structures and stages of dust devil development. The simulation results show that dust devils undergo three stages of development as the swirl ratio increases: a weak vortex stage, a single‐celled vortex stage and a two‐celled vortex stage. The structure of a dust devil can be divided into four regions: outflow, core, corner and inflow layer. The different structures and characteristics of the modelled dust devil during various stages of development and the influence of the surface on the dust devil profile are described using some of the simulation results as illustrations. The causes of the different structures and characteristics are analysed through the interplay between the rotation, the radial pressure gradient, the buoyancy and the axial pressure gradient. Dust devils are a type of concentrated vortex similar to tornadoes. The differences in the structure and characteristics of tornadoes and the modelled dust devils are discussed in this paper. The carrying aloft of sand particles within the modelled dust devil is explored using a Lagrangian coordinate system. Sand particles can be transported by the updraft with particles of varying diameters follow different tracks.

Analysis of the streamline-diffusion finite element method on a piecewise uniform mesh for a convection-diffusion problem with exponential layers

On the unit square, we consider a singularly perturbed convection-diffusion boundary value problem whose solution has two exponential boundary layers. We apply the streamline-diffusion finite element method with piecewise bilinear trial functions on a Shishkin mesh of O ( N 2 ) points and show that the error in the discrete space between the computed solution and the interpolant of the true solution is, uniformly in the diffusion parameter ɛ , of order ɛ 1/2 N –1 ln N + N – 3/2 in the usual streamline-diffusion norm. This includes an L 2 -norm error estimate of order O ( N – 3/2 ) in the convection–dominated case ɛ ⩽ N – 1 ln –2 N . As a corollary we prove that the method is convergent of order N –1/2 ln 3/2 N (again uniformly in ɛ ) in the local L ∞ norm on the fine part of the mesh (i.e., inside the boundary layers). This local L ∞ estimate within the layers can be improved to order ɛ 1/2 N –1/2 ln 3/2 N + N –1 ln 1/2 N , uniformly in ɛ , away from the corner layer.

On laminar natural convection in a vertical corner

The three-dimensional laminar natural convection boundary layer steady flow problem in a vertical corner (corner layer) is formulated and the appropriate boundary conditions are derived. Numerical results pertaining to equations defining the two-points boundary-value problem at the corner layer side-edge show that the boundary layer equations proper have a dual solution.