变形介质中流体流动的数值方法
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摘要
多孔介质是一种固体介质，其中含有任意分布的、彼此连通且大小不一的孔隙．
多孔介质承受很高的压力和温度时，会发生部分或完全的不可逆形变，如果多孔介质具有了变形性质，我们称这之为变形介质．
在造纸业等工业应用中，变形介质中的动态流动是十分重要的．造纸的挤压阶段可以抽象为变形介质中的流体流动问题，认识并改进工作机理可以提高效率，节约能耗[1]．前人已做了很大的努力开发造纸业中的数值模拟工具来预测和控制造纸湿压过程中的去水，例如El-Hosseiny(1991)阐述了湿压过程中水流的动态特性．
本文针对变形介质中的过滤和湿压过程中的动态特性[2]，在原坐标z的基础上引入了坐标y，导出了孔隙比随时间t和空间y的抛物型方程．进一步，我们给出了方程的Galerkin有限元数值解法，并进行了误差分析，获得了孔隙比和位置函数z的先验误差估计．[3]-[9]中的作者讨论了各种方程的Galerkin有限元数值解法．
全文共分三章．
第一章描述了物理背景、基本概念和数学模型．第一节给出了一些基本的物理概念，第二节给出了变形介质流体流动一维问题的数学模型和一些记号．
第二章讨论了变形介质问题的半离散有限元格式和收敛性分析．第一节给出了变形介质问题的变分形式和半离散格式，第二节给出了半离散问题的收敛性分析和误差估计第三节给出了位置函数z的半离散形式的误差估计
第三章讨论了变形介质问题的全离散格式及收敛性分析．第一节给出了变形介质问题的有限元全离散格式，第二节给出了全离散格式的收敛性分析，第三节给出了位置函数z的全离散形式的误差估计．
The solid medium consist of holes connected each other and located randomly with different shape and size is called porous medium.
Due to the very high pressure and temperature the porous media endured,the porous media tends to change in shape completely and partly. So the porous media has the property of deformation. We name this kind of media compressible porous media.
Knowledge about the dynamic flow conditions of porous media is of the utmost importance in industrial application such as Papermaking. Increasing the efficiency of the press section of paper machines is an attractive route to achieving energy savings,which is recognized as the problem of fluid flow in deformed porous media[1]. Great efforts have been made to develop numerical simulation tools for papermaking to predict and control the water removal during the wet pressing of paper webs. A comprehensive review of articles treating the dynamics of water flow during wet pressing has been published by El-Hosseiny(1991).
This article deals with the dynamic behavious during filtration and wet pressing of compressible porous media[2]. this article introduces a new coordinate system y on the basis of previous coordinate system z,which deforms with the medium. An parabolic equation is developed which void ratio changes with time tand spaccy Furthermore,we give Galerkin numerical solution of the equation and error estimates of the finite element solution and location function z have been shown. Galerkin numerical solution of all kinds of equations have been discussed in [3]-[9].
The whole dissertation consists of three chapters.
In the chapter 1 ,we discussed the physical background、basic concepts and mathematical model.The section 1 disussed some basic physical concepts.The section 2 discussed mathematical model of one-dimensional problem of fluid flow in porous media.and some marks.
In the chapter 2,we discussed semi-discrete form and convergence analysis of deformed porous media problem.The section 1 disussed the variational form and semi-discrete form of deformed porous media problem.In section 2,we give conver-gence analysis and error estimate of semi-discrete form.
Im section 3,we give error estimate of semi-discrete form of location function z,
In the chapter 3,we discussed fully discrete form and convergence analysis ofde-formed porous media problem.The section 1 disussed fully discrete form of deformed porous media problem.In section 2,we give convergence analysis and error estimate of fully discrcte form.In section 3,wc give error estimate of fully discrete form of location function z.
多孔介质承受很高的压力和温度时，会发生部分或完全的不可逆形变，如果多孔介质具有了变形性质，我们称这之为变形介质．
在造纸业等工业应用中，变形介质中的动态流动是十分重要的．造纸的挤压阶段可以抽象为变形介质中的流体流动问题，认识并改进工作机理可以提高效率，节约能耗[1]．前人已做了很大的努力开发造纸业中的数值模拟工具来预测和控制造纸湿压过程中的去水，例如El-Hosseiny(1991)阐述了湿压过程中水流的动态特性．
本文针对变形介质中的过滤和湿压过程中的动态特性[2]，在原坐标z的基础上引入了坐标y，导出了孔隙比随时间t和空间y的抛物型方程．进一步，我们给出了方程的Galerkin有限元数值解法，并进行了误差分析，获得了孔隙比和位置函数z的先验误差估计．[3]-[9]中的作者讨论了各种方程的Galerkin有限元数值解法．
全文共分三章．
第一章描述了物理背景、基本概念和数学模型．第一节给出了一些基本的物理概念，第二节给出了变形介质流体流动一维问题的数学模型和一些记号．
第二章讨论了变形介质问题的半离散有限元格式和收敛性分析．第一节给出了变形介质问题的变分形式和半离散格式，第二节给出了半离散问题的收敛性分析和误差估计第三节给出了位置函数z的半离散形式的误差估计
第三章讨论了变形介质问题的全离散格式及收敛性分析．第一节给出了变形介质问题的有限元全离散格式，第二节给出了全离散格式的收敛性分析，第三节给出了位置函数z的全离散形式的误差估计．
The solid medium consist of holes connected each other and located randomly with different shape and size is called porous medium.
Due to the very high pressure and temperature the porous media endured,the porous media tends to change in shape completely and partly. So the porous media has the property of deformation. We name this kind of media compressible porous media.
Knowledge about the dynamic flow conditions of porous media is of the utmost importance in industrial application such as Papermaking. Increasing the efficiency of the press section of paper machines is an attractive route to achieving energy savings,which is recognized as the problem of fluid flow in deformed porous media[1]. Great efforts have been made to develop numerical simulation tools for papermaking to predict and control the water removal during the wet pressing of paper webs. A comprehensive review of articles treating the dynamics of water flow during wet pressing has been published by El-Hosseiny(1991).
This article deals with the dynamic behavious during filtration and wet pressing of compressible porous media[2]. this article introduces a new coordinate system y on the basis of previous coordinate system z,which deforms with the medium. An parabolic equation is developed which void ratio changes with time tand spaccy Furthermore,we give Galerkin numerical solution of the equation and error estimates of the finite element solution and location function z have been shown. Galerkin numerical solution of all kinds of equations have been discussed in [3]-[9].
The whole dissertation consists of three chapters.
In the chapter 1 ,we discussed the physical background、basic concepts and mathematical model.The section 1 disussed some basic physical concepts.The section 2 discussed mathematical model of one-dimensional problem of fluid flow in porous media.and some marks.
In the chapter 2,we discussed semi-discrete form and convergence analysis of deformed porous media problem.The section 1 disussed the variational form and semi-discrete form of deformed porous media problem.In section 2,we give conver-gence analysis and error estimate of semi-discrete form.
Im section 3,we give error estimate of semi-discrete form of location function z,
In the chapter 3,we discussed fully discrete form and convergence analysis ofde-formed porous media problem.The section 1 disussed fully discrete form of deformed porous media problem.In section 2,we give convergence analysis and error estimate of fully discrcte form.In section 3,wc give error estimate of fully discrete form of location function z.
引文
[1] K. Ann-Sofi J(o|¨)nsson and Bengt T. L. J(o|¨)nsson, Fluid Flow in Compressible Porous Media: I: Steady-State Conditions, AIChE Journal., 38(1992), 1340-1348.
[2] K. Ann-Sofi Jonsson and Bengt T. L. Jonsson, Fluid Flow in Compressible Porous Media: I: Dynamic Behavior, AIChE Journal., 38(1992), 1349-1356.
[3] 韦达·托梅，抛物问题Galerkin有限元法，吉林大学出版社，吉林，1986．
[4] J. Douglas Jr. and T. Dupont,Galerkin methods for parabolic equations,SIAM J. Numer. Anal.,7(1970),575-626.
[5] M. F. Wheeler, A priori L2 error estimates for Galerkin approximations toparabolic partial differential equations, SIAM J. Numer. Anal.,10(1973),723-759.
[6] P. G. Ciarlet, The Finite Element Method for Elliptic Problem, North Holland, Amsterdam(1978). eqquations. SIAM J. Numer. Anal.,10(1973),723-759.
[7] M. Luskin, A Galerkin method for nonlinear parabolic equations withnonlinear bouudcry conditions. STAM J. Numer. Anal.,16(1979),284-299.
[8] H. H. Rachford, Jr., Two-level discrete-time Galerkin approximations forsecond order nonlinear parabolic partial differential equations. SIAM J.Numer. Anal.,10(1973),1010-1026.
[9] V.Thomee and L.B.Wahlbin, On Galerkin methods in semiliear parabolicproblems. SIAM J. Numer. Anal.,12(1975),378-389.
[2] K. Ann-Sofi Jonsson and Bengt T. L. Jonsson, Fluid Flow in Compressible Porous Media: I: Dynamic Behavior, AIChE Journal., 38(1992), 1349-1356.
[3] 韦达·托梅，抛物问题Galerkin有限元法，吉林大学出版社，吉林，1986．
[4] J. Douglas Jr. and T. Dupont,Galerkin methods for parabolic equations,SIAM J. Numer. Anal.,7(1970),575-626.
[5] M. F. Wheeler, A priori L2 error estimates for Galerkin approximations toparabolic partial differential equations, SIAM J. Numer. Anal.,10(1973),723-759.
[6] P. G. Ciarlet, The Finite Element Method for Elliptic Problem, North Holland, Amsterdam(1978). eqquations. SIAM J. Numer. Anal.,10(1973),723-759.
[7] M. Luskin, A Galerkin method for nonlinear parabolic equations withnonlinear bouudcry conditions. STAM J. Numer. Anal.,16(1979),284-299.
[8] H. H. Rachford, Jr., Two-level discrete-time Galerkin approximations forsecond order nonlinear parabolic partial differential equations. SIAM J.Numer. Anal.,10(1973),1010-1026.
[9] V.Thomee and L.B.Wahlbin, On Galerkin methods in semiliear parabolicproblems. SIAM J. Numer. Anal.,12(1975),378-389.