Analytical and Numerical Method of Symplectic System for Stokes Flow in the Two-Dimensional Rectangular Domain

XU Xin-sheng; WANG Ga-ping; SUN Fa-ming

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2008 > 29(6): 639-648

XU Xin-sheng, WANG Ga-ping, SUN Fa-ming. Analytical and Numerical Method of Symplectic System for Stokes Flow in the Two-Dimensional Rectangular Domain[J]. Applied Mathematics and Mechanics, 2008, 29(6): 639-648.

Citation:

PDF( 790 KB)

Analytical and Numerical Method of Symplectic System for Stokes Flow in the Two-Dimensional Rectangular Domain

State Key Laboratory of Structure Analysis for Industrial Equipment;Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, P. R. China

Received Date: 2008-02-04
Rev Recd Date: 2008-04-17
Publish Date: 2008-06-15

Abstract

Abstract

A new analytical method of symplectic system, Hamiltonian system, was introduced for solving the problem of the Stokes flow in two-dimensional rectangular domain. In the system, the fundamental problem was reduced to eigenvalue and eigensolution problem, and the solution and boundary conditions can be expanded by eigensolutions employing adjoint relationships of the symplectic ortho-normalization between the eigensolutions. The close method of the symplectic enginsolution was presented based on the completeness of the symplectic eigensolution space. The results explain that fundamental flows can be described by zero eigenvalue eigensolutions and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in rectangular domain and show the effectiveness of the method for solving a variety of problems. Meanwhile, the method is a path for solving other problems.
- Hamiltonian system,
- symplectic eigenvalues,
- symplectic eigensolutions,
- Stokes flow,
- rectangular domain

FullText(HTML)

References(22)

References

[1]	Burggraf O R. Analytical and numerical studies of the structure of steady separated flows[J].J Fluid Mech,1966,24(1):113-151. doi: 10.1017/S0022112066000545
[2]	Pan F, Acrivos A. Steady flows in rectangular cavities[J].J Fluid Mech,1967,28(4):643-655. doi: 10.1017/S002211206700237X
[3]	Kelmanson M A, Lonsdale B. Eddy genesis in the double-lid-driven cavity[J].Q J Mech Appl Math,1996,49(4):633-655.
[4]	林长圣.双板驱动矩形空腔STOKES流动的数值模拟[J].南京工程学院学报(自然科学版),2004,2(4):29-35.
[5]	Joseph D D, Sturges L. The convergence of biorthogonal series for biharmonic and Stokes flow edge problems: Part Ⅱ[J].SIAM J Appl Math,1978,34(1):7-26. doi: 10.1137/0134002
[6]	Smith R C T. The bending of a semi-infinite strip[J].Austral J Sci Res,1952,5(2):227-237.
[7]	Sturges L D. Stokes flow in a two-dimensional cavity with moving end walls[J].Phys Fluids,1986,29(5):1731-1734. doi: 10.1063/1.866008
[8]	Shankar P N. The eddy structure in Stokes flow in a cavity[J].J Fluid Mech,1993,250(1):371-383. doi: 10.1017/S0022112093001491
[9]	Gaskell P H, Savage M D, Summers J L,et al.Stokes flow in closed, rectangular domains[J].Appl Math Model,1998,22(9):727-743. doi: 10.1016/S0307-904X(98)10060-4
[10]	Khuri S A. Biorthogonal series solution of Stokes flow problems in sectorial regions[J].SIAM J Appl Math,1996,56(1):19-39. doi: 10.1137/0156002
[11]	Meleshko V V. Steady Stokes flow in a rectangular cavity[J].Proc Roy Soc London,Ser A,1996，452(1952):1999-2022. doi: 10.1098/rspa.1996.0106
[12]	Meleshko V V. Gomilko A M. Infinite systems for a biharmonic problem in a rectangle[J].Proc Roy Soc London,Ser A,1997,453(1965):2139-2160. doi: 10.1098/rspa.1997.0115
[13]	Srinivasan R. Accurate solutions for steady plane flow in the driven cavity —Ⅰ:Stokes flow[J].Zeitschrift für Angewandte Mathematik und Physik,1995,46(4):524-545. doi: 10.1007/BF00917442
[14]	Weiss R F, Florsheim B H. Flow in a cavity at low Reynolds number[J].Phys Fluids,1965,8(9):1631-1635. doi: 10.1063/1.1761474
[15]	Munson B R, Sturges L D.Low Reynolds number flow in a rotating tank with barriers[J].Phys Fluids,1983,26(5):1173-1176. doi: 10.1063/1.864281
[16]	Shen C, Floryan J M. Low Reynolds number flow over cavities[J].Phys Fluids,1985,28(11):3191-3202. doi: 10.1063/1.865366
[17]	Taneda S. Visualization of separating Stokes flows[J].J Phys Soc Jpn,1979,46(6):1935-1942. doi: 10.1143/JPSJ.46.1935
[18]	O'Brien V. Closed streamlines associated with channel flow over a cavity[J].Phys Fluids,1972,15(12):2089-2097. doi: 10.1063/1.1693840
[19]	Wang C Y. Flow over a surface with parallel grooves[J].Phys Fluids,2003,15(5):1114-1121. doi: 10.1063/1.1560925
[20]	Zhong W X.Duality System in Applied Mechanics and Optimal Control[M].New York:Kluwer Academic Publishers,2004,188-191.
[21]	张鸿庆, 阿拉坦仓, 钟万勰. Hamilton体系与辛正交系的完备性[J].应用数学和力学,1997,18(3):217-221.
[22]	徐新生,王尕平.Stokes 流问题中的辛本征解方法[J].力学学报,2006,38(5):682-687.