Reinforcement of a thin plate by a thin layer

We study the bending of a thin plate, stiffened with a thin elastic layer, of thickness δ. We describe the complete construction of an asymptotic expansion with respect to δ of the solution of the Kirchhoff–Love model and give optimal estimates for the remainder. We identify approximate boundary conditions, which take into account the effect of the stiffener at various orders. Thanks to the tools of multi‐scale analysis, we give optimal estimates for the error between the approximate problems and the original one. We deal with a layer of constant stiffness, as well as with a stiffness in δ^?1. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust and high-order effective boundary conditions for perfectly conducting scatterers coated by a thin dielectric layer

A new approach is considered to construct effective boundaryconditions for the solution of problems related to the scatteringof electromagnetic waves by perfectly conducting cylinders coatedby a thin dielectric shell. These boundary conditions aim tobe both robust and of high order while remaining set in termsof surface differential operators involving at most second-orderderivatives. Error estimates yield a theoretical basis for theuse of these boundary conditions in practical computations.Some numerical experiments at frequencies beyond the range ofvalidity of the usual impedance boundary conditions validatethe efficiency of the approach.     

Mathematical modelling of the catalyst layer of a polymer electrolyte fuel cell

In this paper, we derive a mathematical model for the cathodecatalyst layer of a polymer electrolyte fuel cell. The modelexplicitly incorporates the restriction placed on oxygen inreaching the reaction sites, capturing the experimentally observedfall in the current density to a limiting value at low cellvoltages. Temperature variations and interfacial transfer ofO₂ between the dissolved and gas phases are also included. Boundson the solutions are derived from which we provide a rigourousproof that the model admits a solution. Of particular interestare the maximum and minimum attainable values. We perform anasymptotic analysis in several limits inherent in the problemby identifying important groupings of parameters. This analysisreveals a number of key relationships between the solutions,including the current density, and the composition of the layer.A comparison of numerically computed solutions and asymptoticsolutions shows very good agreement. Implications of the resultsare discussed and future work is outlined.     

Generalized impedance boundary condition at high frequency for a domain with thin layer: the circular case

Consider a conducting disk surrounded by a thin dielectric layer submitted to an electric field at the pulsation ω. The conductivity of the layer grows like ω^1?γ, γ∈[0,1], when the pulsation ω?tends to infinity. Using a pseudodifferential approach on the torus, we build an equivalent boundary condition with the help of an appropriate factorization of Helmholtz operator in the layer. This generalized impedance condition approximates the thin membrane in the high frequency limit for small thickness of the layer. L ²-error estimates are given and we illustrate our results with numerical simulations. This work extends, in the circular geometry, previous works of Lafitte and Lebeau (Lafitte O. Lebeau G. 1993, Équations de Maxwell et opérateur d’impédance sur le bord d’un obstacle convexe absorbant. Comptes Rendus de l ' Académic dis Science, Paris, Série I, Mathématiques, 316(11), 1177–1182); (Lafitte O.D., 1999, Diffraction in the high frequency regime by a thin layer of dielectric material. I. The equivalent impedance boundary condition. SIAM Journal on Applied Mathematics, 59(3), 1028–1052 (electronic)) in which γ?identically equals zero.     

Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms

This paper deals with the blow-up properties of the solution to a semilin-ear parabolic system with localized nonlinear reaction terms, subject to the null Dirichlet boundary condition. We first give sufficient conditions for that the classical solution blows up in the finite time, secondly give necessary conditions and a sufficient condition for that two components blow up simultaneously, and then obtain the uniform blow-up profiles in the interior. Finally we describe the asymptotic behavior of the blow-up solution in the boundary layer.     

Asymptotics for steady‐state voltage potentials in a bidimensional highly contrasted medium with thin layer

We study the behaviour of steady‐state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively, α) surrounded by a thin membrane of thickness h and of complex permittivity α (respectively, 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady‐state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter α is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady‐state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so‐called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and α with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviours of the potentials. Copyright © 2007 John Wiley & Sons, Ltd.     

Perturbation analysis of a magnetohydrodynamic boundary layer flow

This paper presents a perturbation analysis study of the flow of an electrically conducting power-law fluid in the presence of a uniform transverse magnetic field over a stretching sheet. The perturbation solutions for small and large values of the mixed convection parameter are explored. The asymptotic behavior of the solutions was examined for different values of the power-law index and the magnetic parameter.     

Asymptotic behavior of the solution to a convection-diffusion problem with bulk chemical reaction in the wake of a particle

The problem of steady convective mass transfer between a particle and a continuum with a nonlinear bulk chemical reaction is considered for the case when the Peclet number Pe and the rate constant k _v of the reaction tend to infinity. An asymptotic solution with respect to a small parameter ε = Pe^?1/3 is obtained in the wake of the particle.     

Convective thermal conditions and the thermophoretic effect's impacts on a hybrid nanofluid over a moving thin needle

This research focuses on the heat source/sink, chemical reaction, and thermophoretic particle deposition in the influence of hybrid nanofluid over a moving thin needle subjected to a magnetic field. Using the appropriate transformations, a group of nonlinear partial differential equations may be converted to ordinary differential equations. Additionally, with the aid of computational software, the RKF-45 approach is used for the numerical assessment, as well as the shooting operation. It should be mentioned that the results' approval demonstrates a strong association with the previous findings. The resulting graphs mainly explain the fundamental characteristics of hybrid nanofluids and nanofluids, as well as the consequences of different restrictions. An increase in needle size enhances the velocity profile, temperature profile, and concentration profile. The radial and axial velocity profiles are reduced when the magnetic constraint is increased, whereas the thermal and concentration patterns are reversed. Improved heat source-sink as well as Biot number values will enhance the thermal profile. The concentration profiles will decrease due to reaction rate restrictions and thermophoretic limits. The inclusion of solid volume fraction reduces surface drag forces while increasing the rate of mass transfer. In most circumstances, hybrid nanofluid plays a prominent role than nanofluid.     

一类非线性奇摄动非局部反应扩散系统

Recently the nonlinear singularly perturbed problem has been investigated in theinternational academic circles[1 ,2 ] .Approximation methods have been developed andrefined,including the method of averaging,boundary layer method,matched asymptoticexpanision method and multiple scales method.Many scholars such as O' Malley,Jr.[3] ,Butuzov,Nefedov and Schneider[4] ,Kelley[5] ,Mizoguchi,Yanagida and Life[6] have done agreat deal of work.Using the method of differential inequality and other met…     

Numerical generation of transparent boundary conditions on the side surface of a vertical transverse isotropic layer

We consider elastodynamics in transversely isotropic media with vertical symmetry axis. The governing equations are the two-dimensional second-order system for displacements. A numerical method for generating transparent boundary conditions on the cylindrical surface is proposed. The correspondent operator is non-local in both z-direction and time: it handles low-frequency spatial harmonics of the solution convolving their Fourier coefficients with sums-of-exponentials kernels with respect to time. Test calculations show high accuracy, efficiency, and stability of the proposed non-reflecting conditions even for those media parameters where PML fails.     

Fast reaction limit of a three-component reaction-diffusion system

We consider a three-component reaction-diffusion system with a reaction rate parameter, and investigate its singular limit as the reaction rate tends to infinity. The limit problem is given by a free boundary problem which possesses three regions separated by the free boundaries. One component vanishes and the other two components remain positive in each region. Therefore, the dynamics is governed by a system of two equations.     

BOUNDARY LAYER ESTIMATION OF A SINGULAR PROBLEM WITH LIMIT EQUATION OF ORDER 2

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A new stabilized finite element method for advection‐diffusion‐reaction equations

In this article, a new stabilized finite element method is proposed and analyzed for advection‐diffusion‐reaction equations. The key feature is that both the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number are reasonably incorporated into the newly designed stabilization parameter. The error estimates are established, where, up to the regularity‐norm of the exact solution, the explicit‐dependence of the diffusivity, advection, reaction, and mesh size (or the dependence of the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number) is revealed. Such dependence in the error bounds provides a mathematical justification on the effectiveness of the proposed method for any values of diffusivity, advection, dissipative reaction, and mesh size. Numerical results are presented to illustrate the performance of the method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 616–645, 2016     

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation

A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.     

Numerical solution of a nonlinear reaction diffusion equation

In this paper, the authors propose a numerical method to compute the solution of a Cauchy problem with blow-up of the solution. The problem is split in two parts: a hyperbolic problem which is solved by using Hopf and Lax formula and a parabolic problem solved by a backward linearized Euler method in time and a finite element method in space. It is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self-similar solution remains bounded. The convergence of the scheme is obtained.     

Modeling of individual coherent structures in wall region of a turbulent boundary layer

Models for individual coherent structures in the wall region of a turbulent boundary layer are proposed. Method of numerical simulations is used to follow the evolution of the structures. It is found that the proposed model does bear many features of coherent structures found in experiments. Project supported by the National Natural Science Foundation of China (Grant No. 19732005) and National Climbing Project.     

Classification of blow-up with nonlinear diffusion and localized reaction

We study the behaviour of nonnegative solutions of the reaction-diffusion equation

     

Boundary layer to a system of viscous hyperbolic conservation laws

In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for n × n hyperbolic system of conservation laws with artificial viscosity in the half line （0, ∞）. We first show that a boundary layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an elementary energy method.