Vanishing viscous limits for the 2D lake equations with Navier boundary conditions

The vanishing viscosity limit is considered for the viscous lake equations with Navier friction boundary conditions. We prove that the inviscid limit satisfies the inviscid lake equations, and the results include flows generated by Lp initial vorticity with 1<p?∞.     

Characteristic boundary layers in real vanishing viscosity limits

In this paper, we study the vanishing viscosity limit of initial boundary value problems for one-dimensional mixed hyperbolic-parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems: in particular, we assume that an eigenvalue of the inviscid system vanishes uniformly. We prove the stability of boundary layers expansions in small time (i.e before shocks for the inviscid system) as long as the amplitude of the boundary layers remains sufficiently small. In particular, by using Lagrangian coordinates, we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous Dirichlet condition for the velocity at the boundary.     

Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition

This work investigates the solvability, regularity and vanishing viscosity limit of the 3D viscous magnetohydrodynamic system in a class of bounded domains with a slip boundary condition.     

Vanishing viscosity limit for the 3D magnetohydrodynamic system with generalized Navier slip boundary conditions

In this paper, we investigate the vanishing viscosity limit problem for the 3D incompressible magnetohydrodynamic (MHD) system in a general bounded smooth domain of R ³ with the generalized Navier slip boundary conditions. We also obtain rates of convergence of the solution of viscous MHD to the corresponding ideal MHD. Copyright © 2016 John Wiley & Sons, Ltd.     

Far field boundary conditions for incompressible flows computation

Many far field boundary conditions are proposed in the literature to solve Navier-Stokes equations. It is necessary to distinguish the streamwise or outlet boundary conditions and the spanwise boundary conditions. In the first case the flow crosses the artificial frontier and it is required to avoid reflections that can change significantly the flow. In the second case the Navier-slip boundary condition is often used but if the frontier is not far enough the boundary is both inlet and outlet. Thus the Navier-slip boundary condition is not well suited as it imposes no flux through the frontier. The aim of this work is to compare some well-known boundary conditions, to quantify to which extend the artificial frontier can be close to the bodies in two- and three-dimensions and to take into account the flow rate through the spanwise directions.     

Simple finite element method in vorticity formulation for incompressible flows

A very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation. This method relies on recasting the traditional finite element method in the spirit of the high order accurate finite difference methods introduced by the authors in another work. Optimal accuracy of arbitrary order can be achieved using standard finite element or spectral elements. The method is convectively stable and is particularly suited for moderate to high Reynolds number flows.

     

Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension

We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate the regularity of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates c...     

Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is cν1/2.     

Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition

In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition. We establish the local well‐posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established, and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition. Copyright © 2017 John Wiley & Sons, Ltd.     

具有无穷边界值的二次非线性奇摄动边值问题的双边界层

研究了具有无穷边界值的二次非线性奇摄动边值问题的双边界层,利用边界层校正函数,构造其渐近解,并利用微分不等式理论,给出了一致有效渐近估计.最后给出算例验证了相关结论的正确性.     

Global behavior of spherically symmetric Navier-Stokes equations with density-dependent viscosity

In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in L^∞-norm and weighted H¹-norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics.     

Free boundary value problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in this paper. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero also at an algebraic time-rate as the time tends to infinity.     

The vanishing viscosity limit for some symmetric flows

The focus of this paper is on the analysis of the boundary layer and the associated vanishing viscosity limit for two classes of flows with symmetry, namely, Plane-Parallel Channel Flows and Parallel Pipe Flows. We construct explicit boundary layer correctors, which approximate the difference between the Navier–Stokes and the Euler solutions. Using properties of these correctors, we establish convergence of the Navier–Stokes solution to the Euler solution as viscosity vanishes with optimal rates of convergence. In addition, we investigate vorticity production on the boundary in the limit of vanishing viscosity. Our work significantly extends prior work in the literature.     

Implementation of near-wall boundary conditions for modeling boundary layers with free-stream turbulence

The paper is devoted to the extension of the near-wall domain decomposition, earlier developed in some previous works by the authors, to modeling flat-plate boundary layers undergoing laminar-to-turbulent bypass transition. The steady-state wall boundary layers at high-intensity free-stream turbulence are studied on the basis of differential turbulence models with the use of non-overlapping domain decomposition. In the approach the near-wall resolution is replaced by the interface boundary conditions of Robin type. In contrast to the previous studies, the main attention is paid to the laminar–turbulent transitional regime. With the use of modified turbulence models we study an effect of free-stream parameters on the development of dynamic processes in the boundary layer including a transitional regime and fully developed turbulent flow. In addition, for the first time a full scale domain decomposition is realized via iterations between the inner and outer subregions until a convergence. The computational profiles of the velocity and intensity of the turbulence kinetic energy are compared with experimental data. A possible range of location of the near-wall interface boundary is found.     

一个新的边界层方程及其在形状控制中的应用

本文给出固壁边界上(即一个二维流形上) 的流体速度梯度和压力的二阶偏微分方程, 从而也给出边界上法向应力, 以及流体中运动物体所受的阻力和升力的计算公式. 本方法的创新在于边界上法向速度梯度不是通过在边界层内速度梯度的数值微分达到, 而是通过它与其他变量一起作为一组偏微分方程的解而得到, 证明边界层方程组的适定性问题, 并且给出解关于边界形状的Gâteaux 导数所满足的偏微分方程. 本文将本方法应用于飞机外形的形状最优控制, 给出阻力泛函关于形状第一变分的可计算形式. 数值例子表明, 用本方法得到的阻力精度比通用程序得到要高.     

Existence of solutions with interior transition layers touching the boundary

For the case of Dirichlet boundary conditions, the existence of a solution with interior transition layer touching the boundary is proved.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 80–92.Original Russian Text Copyright © 2005 by I. V. Nedelko.This revised version was published online in April 2005 with a corrected issue number.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232     

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.     

The initial boundary value problem for Navier-Stokes equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC ³ boundary, under the assumption that |a| _L ²(Θ) + |f| _L ¹(0,∞;L ²(Θ)) or |∇a| _L ²(Θ) + |f| _L ²(0,∞;L ²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed. This work is supported by foundation of Institute of Mathematics, Academia Sinica     

Reducing slip boundary value problems from the half to the whole space. Applications to inviscid limits and to non-Newtonian fluids

The study of a very large class of linear and non-linear, stationary and evolutive partial differential problems in the half-space (or similar) under the slip boundary condition is reduced here to the much simpler study of the corresponding results for the same problem in the whole space. The approach is particularly suitable for proving new results in strong norms. To determine whether this extension is available, turns out to be a simple exercise. The verification depends on a few general features of the functional space X related to the space variables. Hence, we present an approach as much as possible independent of the particular space X. We appeal to a reflection technique. Hence a crucial assumption is to be in the presence of flat boundaries (see below). Instead of stating “general theorems” we rather prefer to illustrate how to apply our results by considering a couple of interesting problems. As a main example, we show that sharp vanishing viscosity limit results that hold for the evolution Navier-Stokes equations in the whole space can be extended to the slip boundary value problem in the half-space. We also show some applications to non-Newtonian fluid problems.