Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

Periodic Boundary Problem and Cauchy Problem for the Fluid Dynamic Equation in Geophysics

We study periodic boundary problem and Cauchy problem for the fluid dynamic equation in geophysics. The generalized and classical global solution of the mentioned problems are established. The method employed in this paper is Galerkin approximation and integral estimates.     

Boundary integral method for scattering problems with cracks buried in a piecewise homogeneous medium

We consider the scattering of time‐harmonic acoustic plane waves by a crack buried in a piecewise homogeneous medium. The integral representation for a solution is obtained in the form of potentials by using Green's formula. The density in potentials satisfies the uniquely solvable Fredholm integral equation. Then we obtain the existence and uniqueness of the solution. Copyright © 2011 John Wiley & Sons, Ltd.     

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

Boundary integral equation method in thermoelasticity part III: uncoupled thermoelasticity

This paper presents conversion of the volume integral of temperature gradients to a surface integral in the integral formulation of boundary value problems of uncoupled thermoelasticity. Particular classes of problems such as thermal stresses, quasi-static problems of uncoupled thermoelasticity, and stationary problems of thermoelasticity are considered. The improved formulation makes numerical computation more accurate and less formidable. The integral formulations in uncoupled thermoelasticity are given in the Laplace transform domain as well as in the time domain.     

A New Boundary Element Method for the Biharmonic Equation with Dirichlet Boundary Conditions

We consider a scalar boundary integral formulation for the biharmonic equation based on the Almansi representation. This formulation was derived by the first author in an earlier paper. Our aim here is to prove the ellipticity of the integral operator and hence establish convergence of and error bounds for Galerkin boundary element methods. The theory applies both in two and three dimensions, but only for star-shaped domains. Numerical results in two dimensions confirm our analysis.     

Strong solutions for the nonhomogeneous Navier–Stokes equations in unbounded domains

We show the existence of strong solutions for the nonhomogeneous Navier–Stokes equations in three‐dimensional domains with boundary uniformly of class C³. Under suitable assumptions, uniqueness is also proved. Copyright © 2009 John Wiley & Sons, Ltd.     

Interface boundary value problems of Robin‐transmission type for the Stokes and Brinkman systems on n‐dimensional Lipschitz domains: applications

In this paper, we describe a layer potential analysis in order to show an existence result for an interface boundary value problem of Robin‐transmission type for the Stokes and Brinkman systems on Lipschitz domains in Euclidean setting, when the given boundary data belong to some L^p or Sobolev spaces associated to such domains. Applications related to an exterior three‐dimensional Stokes flow past two concentric porous spheres with stress jump conditions on the fluid‐porous interface are also considered. Copyright © 2013 John Wiley & Sons, Ltd.     

Solution of the Navier–Stokes problem

A new a priori estimate for solutions to Navier–Stokes equations is derived. Uniqueness and existence of these solutions in ${\mathbb{R}}^3$ for all $t>0$ is proved in a class of solutions locally differentiable in time with values in $H^1\left({\mathbb{R}}^3\right)$, where $H^1\left({\mathbb{R}}^3\right)$ is the Sobolev space. By the solution a solution to an integral equation is understood. No smallness restrictions on the data are imposed.     

A symmetric boundary element method for the Stokes problem in multiple connected domains

We consider a symmetric Galerkin boundary element method for the Stokes problem with general boundary conditions including slip conditions. The boundary value problem is reformulated as Steklov–Poincaré boundary integral equation which is then solved by a standard approximation scheme. An essential tool in our approach is the invertibility of the single layer potential which requires the definition of appropriate factor spaces due to the topology of the domain. Here we describe a modified boundary element approach to solve Dirichlet boundary value problems in multiple connected domains. A suitable extension of the standard single layer potential leads to an operator which is elliptic on the original function space. Copyright © 2003 John Wiley & Sons, Ltd.     

Existence and uniqueness of solutions for singular integral equation

Using the mixed monotone method we establish existence and uniqueness results for a singular integral equation. The theorem obtained is very general and complements previous known results. The work was supported by the National Natural Science Foundation of China (No.10571021 and No.10701020) and Key Laboratory for Applied Statistics of MOE(KLAS) and Subject Foundation of Harbin University (No. HXK200714).     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

The method of integral equation for scattering problem with a mixed crack

We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R² as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd.     

An integral equation method for the electromagnetic scattering from cavities

Consider a time‐harmonic electromagnetic plane wave incident on a cavity in a ground plane. The physical process is modelled by Maxwell's equations. In this paper, integral representations of the solutions to the model problem in both fundamental polarizations are derived and studied. Existence and uniqueness of the solutions for the integral equations are established. The integral equations approach forms a basis for numerical solution of the model problem. In particular, for each fundamental polarization, an integral formulation with Gårding‐type estimates is derived. These formulations provide a basis for variational boundary element methods for solving the cavity problem. The Gårding‐type estimates imply convergence results for conforming boundary element methods. Copyright © 2000 John Wiley & Sons, Ltd     

Existence and uniqueness results for a fractional two‐times evolution problem with constraints of purely integral type

This paper deals with a fractional two‐times evolution equation associated with initial and purely boundary integral conditions. The existence and uniqueness of generalized solution are proved. The classical functional method based on a priori estimates and density used by many authors in the case of nonfractional differential equations is applied for the time fractional case. Copyright © 2015 John Wiley & Sons, Ltd.     

关于Helmholtz外问题的边界积分方程解的唯一性问题

本文用能量分析的观点探讨了用边界积分方程描述Helmholtz外问题时,解的唯一性不能保持的原因.文中证明了,当利用积分方程来描述问题时,实际上将无穷远处的Sommerfeld条件改成了既适合于外向波(辐射波),又适合于内向波(吸收波),即整个系统的能量保持守恒.并根据此观点解释了保持唯一性的算法.     

Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition

We consider the numerical solution for the Helmholtz equation in R~2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.     

Effective medium model for flow through beds of porous cylindrical fibres

We have used effective medium model for beds of circular cylindrical porous fibres in order to estimate the overall bed permeability (OBP). It is assumed that a representative circular porous cylindrical fibre is inside a fluid envelope beyond which effective medium is used. Both inside the cylindrical fibre and in the effective medium, Brinkman equation is used, however of different permeabilities and in the fluid envelope Stokes equation is used. The OBP corresponding to the porous fibres is estimated when the flow direction is perpendicular to the axis of the cylindrical fibres as well as parallel to the fibres. This in turn is used to estimate the OBP corresponding to a collection of porous cylindrical fibres that are randomly oriented. We have compared the results with some existing literature.