Effect of slip on the linear stability of flow through a tube

A theoretical investigation of the linear stability of the flow of a Newtonian fluid through a tube is presented using an alternative boundary condition to the standard no-slip condition. The linear stability analysis is based on the classical method of infinitesimal axially symmetric harmonic perturbations super-imposed on the steady state solution. In this analysis the standard no-slip boundary condition is replaced with the Navier boundary condition, independently proposed over a hundred years ago by both Navier and Maxwell. This boundary condition contains an extra parameter, which may be regarded as a positive slip length. The aim of this analysis is to assess the effect of a nonzero slip length on the behavior of infinitesimal disturbances in the flow through a tube. It is demonstrated that for positive slip the rate of decay of the least damped disturbance is reduced, although the flow still remains stable to all infinitesimal disturbances of the type considered, as it does for the no-slip boundary condition.     

Effect of slip on the linear stability of flow through a tube

A theoretical investigation of the linear stability of the flow of a Newtonian fluid through a tube is presented using an alternative boundary condition to the standard no-slip condition. The linear stability analysis is based on the classical method of infinitesimal axially symmetric harmonic perturbations super-imposed on the steady state solution. In this analysis the standard no-slip boundary condition is replaced with the Navier boundary condition, independently proposed over a hundred years ago by both Navier and Maxwell. This boundary condition contains an extra parameter, which may be regarded as a positive slip length. The aim of this analysis is to assess the effect of a nonzero slip length on the behavior of infinitesimal disturbances in the flow through a tube. It is demonstrated that for positive slip the rate of decay of the least damped disturbance is reduced, although the flow still remains stable to all infinitesimal disturbances of the type considered, as it does for the no-slip boundary condition.     

AN ARTIFICIAL BOUNDARY CONDITION FOR THE INCOMPRESSIBLE VISCOUS FLOWS IN A NO-SLIP CHANNEL

1.IntroductionWhencomputingthenumericals0luti0nsofviscousfluidfl0wproblemsinallun-boundedd0main,0neoftenintroducesartificialboundaries,andsetsupanartificialbopundarycondition0nthem;thenthe0riginalproblemisreducedtoaproblemonab0undedc0mputationald0main.InordertoIimitthecomputatio11alcost,theseboundariesmustnotbet00farfromthedomainofinterest.Theref0re,theartificialboundaryc0nditi0nsmustbegoodapprotimationt0the"exact"boundaryconditions(sothatthes0lutionoftheproblemintheboundeddonlainisequaltothes…     

A nonlocal problem for singular linear systems of ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite half-interval. This system is supplemented by the boundedness condition for solutions and a nonlocal linear condition specified by the Stieltjes integral. A method for approximating the resulting problem by a problem posed on a finite interval is proposed, and the properties of the latter are investigated. A numerically stable method for solving this problem is examined. This method uses an auxiliary boundary value problem with separated boundary conditions.     

Diffraction by an acoustically transmissive or an electromagnetically dielectric half-plane

This work gives a mathematical model for an acoustically penetrable or electromagnetically dielectric half-plane. An approximate boundary condition is used that depends on the thickness of, and the material constants for, the half-plane. A solution is obtained, by using the approximate boundary condition, for the problem of a line source field diffracted by a penetrable/dielectric half-plane. The asymmetry of the approximate boundary condition results in a matrix Wiener–Hopf problem, which is solved explicitly.     

Spectral analysis of generalized Volterra equation

A generalized Volterra lattice with a nonzero boundary condition is considered by virtue of the inverse scattering transform. The two-sheeted Riemann surface associated with the boundary problem is transformed into the Riemann sphere by introducing a suitable variable transformation. The associated spectral properties of the lattice in single-valued variable was discussed. The constraint condition about the nonzero boundary condition and the scattering data is found.     

Pasternak地基上简支板振动问题的准格林函数方法

提出一种新的数值方法——准格林函数方法．以Pasternak地基上简支多边形薄板的振动问题为例,详细阐明了准格林函数方法的思想．即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将Pasternak地基上薄板自由振动问题的振型控制微分方程化为两个耦合的第二类Fredholm积分方程．边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性,最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率．数值方法表明,该方法具有较高的精度．     

Numerical solution of singular perturbation problems via deviating arguments

We present a new approach to numerically solving linear, singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in the implicit form is derived. Then, the outer region problem is solved as a two point boundary value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, a new inner region problem is obtained and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Some numerical examples have been solved to demonstrate the applicability of the method.     

On the convergence of solutions and eigenelements of a boundary value problem in a domain perforated along the boundary

We consider a boundary value problem in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the external boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. A limit (homogenized) problem is obtained. We prove the convergence of the solutions, eigenvalues, and eigenfunctions of the original problem to the solutions, eigenvalues, and eigenfunctions, respectively, of the limit problem.     

Boundary homogenization in domains with randomly oscillating boundary

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and establish error estimates.     

Spectral problem with Steklov condition on a thin perforated interface

A two-dimensional Steklov-type spectral problem for the Laplacian in a domain divided into two parts by a perforated interface with a periodic microstructure is considered. The Steklov boundary condition is set on the lateral sides of the channels, a Neumann condition is specified on the rest of the interface, and a Dirichlet and Neumann condition is set on the outer boundary of the domain. Two-term asymptotic expansions of the eigenvalues and the corresponding eigenfunctions of this spectral problem are constructed.     

An Iterative FEM for Solving Wave Problems of a Halfspace

An iterative finite element method for solving wave problems of a halfspace is presented in this paper. The halfspace is first truncated by introducing a fictive finite boundary on which some fictive boundary conditions must be imposed. A finite computational domain is in each iteration subjected to actual boundary conditions on real boundary and to fictive Dirichlet or Neumann boundary conditions on the fictive boundary. The radiation condition is satisfied by using DtN operator. The DtN operator is not introduce in the finite element formulation on the fictive boundary so any finite elements can be used. The method is simple and specially useful for computing higher harmonics.     

Solution of the skin effect problem with an arbitrary coefficient of specular reflection

An analytical solution of the skin effect problem in a metal with specular-diffuse boundary conditions is obtained. A new analytical method is developed that makes it possible to obtain a solution up to an arbitrary degree of accuracy. The method is based on the idea of representing not only the boundary condition on the field in the form of a source (which is conventional) but also the boundary condition on the distribution function. The solution is obtained in the form of a von Neumann series.     

二维不可压缩磁流体方程边界层分离的条件

该文研究了二维不可压缩磁流体方程的解,其中要求磁流体的速度满足Dirichlet边界条件、磁场在边界上的值与时间无关. 利用Taylor展开式和不可压缩流的结构分歧理论, 得到了磁流体方程发生边界层分离的条件, 它取决于外力、初值和磁场在边界上的取值, 并且该条件可以预测磁流体边界层分离发生的时间与地点.     

Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem

In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected.     

Application of a 14-point averaging operator in the grid method

The Dirichlet problem for Laplace’s equation in a rectangular parallelepiped is solved by applying the grid method. A 14-point averaging operator is used to specify the grid equations on the entire grid introduced in the parallelepiped. Given boundary values that are continuous on the parallelepiped edges and have first derivatives satisfying the Lipschitz condition on each parallelepiped face, the resulting discrete solution of the Dirichlet problem converges uniformly and quadratically with respect to the mesh size. Assuming that the boundary values on the faces have fourth derivatives satisfying the Hölder condition and the second derivatives on the edges obey an additional compatibility condition implied by Laplace’s equation, the discrete solution has uniform and quartic convergence with respect to the mesh size. The convergence of the method is also analyzed in certain cases when the boundary values are of intermediate smoothness.     

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

Abstract

Nonlinear systems are often subject to random influences. Sometimes the noise enters the system through physical boundaries and this leads to stochastic dynamic boundary conditions. A dynamic, as opposed to static, boundary condition involves the time derivative as well as spatial derivatives for the system state variables on the boundary. Although stochastic static (Neumann or Dirichet type) boundary conditions have been applied for stochastic partial differential equations, not much is known about the dynamical impact of stochastic dynamic boundary conditions. The purpose of this article is to study possible impacts of stochastic dynamic boundary conditions on the long term dynamics of the Cahn-Hilliard equation arising in the materials science. We show that the dimension estimation of the random attractor increases as the coefficient for the dynamic term in the stochastic dynamic boundary condition decreases. However, the dimension of the random attractor is not affected by the corresponding stochastic static boundary condition.     

Simultaneously developing flow and heat transfer of non-Newtonian fluids in equilateral triangular duct

A numerical investigation based on the Galerkin finite element method was carried out to solve the full three-dimensional governing equations for simultaneously developing steady laminar flow and heat transfer to a purely viscous non-Newtonian fluid described by a power law model flowing in equilateral triangular ducts. Two commonly used thermal boundary conditions, constant wall temperature (T boundary condition) and constant wall heat flux both axially and peripherally (H2 boundary condition) were examined. It is shown that the Nusselt number distribution along the walls is affected appreciably by the variation of the power law index. Results are presented and discussed for a wide range of power law indices and Prandtl numbers for T and H2 boundary conditions.     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.