Local Well-posedness for Linearized Degenerate MHD Boundary Layer Equations in Analytic Setting

In this note we derive MHD boundary layer equations according to viscosity and resistivity coefficients. Especially, when these viscosity and resistivity coefficients are of different orders, it leads to degenerate MHD boundary layer equations. We prove these degenerate boundary layers are stable around a steady solution.     

Response of heat transfer from a moving flat plate in a parabolic flow

This paper deals with the solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations and the study of the response of heat transfer when there is a parabolic flow over a moving flat plate. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse thermal boundary layer equations. A general analysis is made from which we deduce (i) Solutions of two-dimensional thermal boundary layer on a moving flat plate, (ii) Solutions of thermal boundary layer on a yawed flat plate, (iii) Solutions of thermal boundary layer when there is a parabolic flow over a moving flat plate by giving different values to β and Cx. Solutions are developed for large and small times and curves are drawn representing the variations of heat transfer from the plate with time for all the cases. The limiting time is also calculated.     

Passing from bulk to bulk-surface evolution in the Allen–Cahn equation

In this paper we formulate a boundary layer approximation for an Allen–Cahn-type equation involving a small parameter ${\varepsilon}$ . Here, ${\varepsilon}$ is related to the thickness of the boundary layer and we are interested in the limit ${\varepsilon \to 0}$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L²-gradient flow with the corresponding Allen–Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Γ- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.     

Quasi-neutral and zero-viscosity limits of Navier–Stokes–Poisson equations in the half-space

The present paper is concerned with the quasi-neutral and zero-viscosity limits of Navier–Stokes–Poisson equations in the half-space. We consider the Navier-slip boundary condition for velocity and Dirichlet boundary condition for electric potential. By means of asymptotic analysis with multiple scales, we construct an approximate solution of the Navier–Stokes–Poisson equations involving two different kinds of boundary layer, and establish the linear stability of the boundary layer approximations by conormal energy estimate.     

Alternating boundary layer type solutions of some singularly perturbed periodic parabolic equations with Dirichlet and Robin boundary conditions

In this paper, we continue the analysis of alternating boundary layer type solutions to certain singularly perturbed parabolic equations for which the degenerate equations (obtained by setting small parameter multiplying derivatives equal to zero) are algebraic equations that have three roots. Here, we consider spatially one-dimensional equations. We address special cases where the following are true: (a) boundary conditions are of the Dirichlet type with different values of unknown functions specified at different endpoints of the interval of interest; (b) boundary conditions are of the Robin type with an appropriate power of a small parameter multiplying the derivative in the conditions. We emphasize a number of new features of alternating boundary layer type solutions that appear in these cases. One of the important applications of such equations is related to modeling certain types of bioswitches. Special choices of Dirichlet and Robin type boundary conditions can be used to tune up such bioswitches. This article was submitted by the authors in English.     

Effective Boundary Conditions for the Heat Equation with Interior Inclusion

Of concern is the scenario of a heat equation on a domain that contains a thin layer, on which the thermal conductivity is drastically different from that in the bulk. The multi-scales in the spatial variable and the thermal conductivity lead to computational difficulties, so we may think of the thin layer as a thickless surface, on which we impose "effective boundary conditions"(EBCs). These boundary conditions not only ease the computational burden, but also reveal the effect of the inclusion. In this paper, by considering the asymptotic behavior of the heat equation with interior inclusion subject to Dirichlet boundary condition, as the thickness of the thin layer shrinks, we derive, on a closed curve inside a two-dimensional domain, EBCs which include a Poisson equation on the curve, and a non-local one. It turns out that the EBCs depend on the magnitude of the thermal conductivity in the thin layer,compared to the reciprocal of its thickness.     

A numerical study of boundary layer flow of viscous incompressible fluid past an inclined stretching sheet and heat transfer using nonpolynomial spline method

In the present paper, we study the boundary layer flow of viscous incompressible fluid over an inclined stretching sheet with body force and heat transfer. Considering the stream function, we convert the boundary layer equation into nonlinear third-order ordinary differential equation together with appropriate boundary conditions in an infinite domain. The nonlinear boundary value problem has been linearized by using the quasilinearization technique. Then, we develop a nonpolynomial spline method, which is used to solve the flow problem. The convergence analysis of the method is also discussed. We study the velocity function for different angles of inclination and Froude number with the help of various graphs and tables. Then using these in heat convection flow, we obtain the expression for temperature field. Skin friction is also calculated. The various results have been given in tables. At last, we calculated the Nusselt number.     

一类双参数三阶半线性方程边值问题的奇摄动

研究一类双参数三阶半线性方程边值问题的奇摄动，讨论了摄动解随两参数的不同量级所呈现的不同性态的边界层现象，并给出了解的一致有效的渐近展开式．     

Zero viscosity and diffusion vanishing limit of the incompressible magnetohydrodynamic system with perfectly conducting wall

This work investigates the incompressible viscous and diffusive MHD system with perfectly conducting wall boundary condition. For different horizontal and vertical viscosities and magnetic diffusions, when they go to zero with different or same speeds, we establish the convergence to the ideal inviscid MHD system and the anisotropic inviscid MHD system by constructing the exact boundary layer functions.     

Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum

In this paper, we study the well-posedness of the thermal boundary layer equation in two-dimensional incompressible heat conducting flow. The thermal boundary layer equation describes the behavior of thermal layer and viscous layer for the two-dimensional incompressible viscous flow with heat conduction in the small viscosity and heat conductivity limit. When the initial datum are analytic, with respect to the tangential variable of the boundary, and without the monotonicity condition of the tangential velocity, by using the Littlewood-Paley theory, we obtain the local-in-time existence and uniqueness of solution to this thermal boundary layer problem.     

抛物型偏微分方程奇异摄动问题的边界层方法

本文讨论抛物型偏微分方程奇异摄动问题,通常,为了使边界层的特性不致丧失,在边界层附近必须减小网格,当网格足够小时需要很大的运算量。我们提出边界层格式,在边界层附近不必取很细的网格,数值例子表明采用中等步长即可满足精度。     

Solutions for the problem of boundary layer formation in a pseudo-plastic fluid: The case of gradual acceleration

We consider the development of the nonstationary boundary layer about a body that gradually starts to move in a resting fluid. Under certain conditions, we construct the solutions for the problem of formation of boundary layer in a pseudo-plastic fluid. The method used here is mainly based on a transformation which reduces the boundary layer system to a boundary value problem for a single quasilinear parabolic equation.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

含开边界二维Stokes问题的Galerkin边界元解法

本文推导了含有开边界的二维有限域上Stokes问题的边界积分方程, 得出基于单层位势的第一类间接边界积分方程.对与之等价的边界变分方程用Galerkin边界元求解以得出单层位势的向量密度. 对于含有开边界端点的边界单元,采用特别的插值函数, 以模拟其固有的奇异性.论文用若干数值算例模拟了含有开边界的有限区域上不可压缩粘性流体的绕流.       

两互相垂直的平面间的层流边界层

本文得到了两互相垂直的平面间的层流边界层的三级近似解.在边界层中,边界层方程中的粘性项和惯性项具有相同的数量级[3].本文则首先假定惯性项大于粘性项去求解边界层方程;然后,令粘性项大于贯性项.最后,取二者的平均值作为边界层方程的真实解.本文所得一级及二级近似解和文献[1]的结果相同.本文的三级近似解则较[1]的结果更精确.     

Boundary layers for stress diffusive perturbation in viscoelastic fluids

In this paper, we study a stress diffusive perturbation of the system describing a viscoelastic flow. We analyse the boundary layer which arises near the boundary and we observe in particular that there is no boundary layer on the velocity at the first order.     

关于边界层方法

本文指出传统的边界层方法(包括匹配法和Vi?ik—Lyusternik方法)的不足:不能作出边界层项的渐近展开式.提出多重尺度构造边界层项的方法,得到符合实情的结果.又与Levinson所用的方法比较,本方法能更简单地导出后一方法给出的边界层项的渐近展开式.又应用此方法研究现有的关于奇异摄动的某些成果,指出这些成果的局限性,并在一般情况下作出解的渐近展开式.     

Zero dissipation limit of the 1D linearized Navier-Stokes equations for a compressible fluid

In this paper we study the asymptotic limiting behavior of the solutions to the initial boundary value problem for linearized one-dimensional compressible Navier-Stokes equations. We consider the characteristic boundary conditions, that is we assume that an eigenvalue of the associated inviscid Euler system vanishes uniformly on the boundary. The aim of this paper is to understand the evolution of the boundary layer, to construct the asymptotic ansatz which is uniformly valid up to the boundary, and to obtain rigorously the uniform convergence to the solution of the Euler equations without the weakness assumption on the boundary layer.     

On the numerical approximations of stiff convection–diffusion equations in a circle

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a $P_1$ classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.