基于不动点方法求解非线性Falkner-Skan流动方程

许丁; 谢公南

doi:10.3879/j.issn.1000-0887.2015.01.007

基于不动点方法求解非线性Falkner-Skan流动方程

doi: 10.3879/j.issn.1000-0887.2015.01.007

许丁¹,
谢公南²

¹机械结构强度与振动国家重点实验室(西安交通大学), 西安 710049;²西北工业大学工程仿真与宇航计算实验室, 西安 710072

基金项目: 国家自然科学基金(11102150)；中央高校基本科研业务费专项资金

详细信息

作者简介:
许丁(1980—)，男，西安人，讲师，博士(通讯作者. E-mail: dingxu@mail.xjtu.edu.cn).

中图分类号: O351;TB126
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出版历程
- 收稿日期: 2014-07-16
- 刊出日期: 2015-01-15

Application of the Fixed Point Method to Solve the Nonlinear Falkner-Skan Flow Equation

XU Ding¹,
XIE Gong-nan²

¹State Key Laboratory for Strength and Vibration of Mechanical Structures(Xi’an Jiaotong University), Xi’an 710049, P.R.China;²Engineering Simulation and Aerospace Computing (ESAC),Northwestern Polytechnical University, Xi’an 710072, P.R.China

Funds: The National Natural Science Foundation of China(11102150)

摘要

摘要: Falkner-Skan流动方程描述绕楔面的流动，该方程具有很强的非线性.首先通过引入变换式，将原半无限大区域上的流动问题转化为有限区间上的两点边值问题.接着基于泛函分析中的不动点理论，采用不动点方法求解两点边值问题从而得到FalknerSkan流动方程的解.最后将不动点方法给出的结果和文献中的数值结果相比较，发现不动点方法得到的结果具有很高的精度，并且解的精度很容易通过迭代而不断得到提高.表明不动点方法是一种求解非线性微分方程行之有效的方法.
- Falkner-Skan流动 /
- 不动点方法 /
- 非线性微分方程 /
- 边值问题
Abstract: The Falkner-Skan flow equation is a strongly nonlinear differential equation, which describes the flow around a wedge. In order to overcome the difficulties originated from the semi-infinite interval and asymptotic boundary condition in this flow problem, transformations were simultaneously conducted for both the independent variable and the correponding function to convert the problem to a 2-point boundary value one within a finite interval. The deduced new-form nonlinear differential equation was subsequently solved with the fixed point method (FPM). The present analytical results obtained with the FPM agree well with the previous referential numerical ones. The accuracy of the present solution is conveniently improved through iteration under the FPM framework, which shows that the FPM makes a promising tool for nonlinear differential equations.
- Falkner-Skan flow /
- fixed point method /
- nonlinear differential equation /
- boundary value problem

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参考文献(23)

[1]	Wang C Y. Exact solutions of the unsteady Navier-Stokes equations[J].Applied Mechanics Reviews,1989,42(11S): S269-S282.
[2]	Wang C Y. Exact solutions of the steady-state Navier-Stokes equations[J].Annual Review of Fluid Mechanics,1991,23: 159-177.
[3]	Falkner V M, Skan S W. Some approximate solutions of the boundary layer equations[J].Philosophical Magazine,1931,12: 865-896.
[4]	White F M.Viscous Fluid Flow [M]. New York: McGraw-Hill, 1991.
[5]	Schlichting H, Gersten K.Boundary-Layer Theory [M]. Springer Verlag, 2000.
[6]	Blasius H. Grenzschichten in flüssigkeiten mit kleiner reibung[J].Z Math Phys,1908,56: 1-37.
[7]	Hiemenz K. Die grenzchicht an einem in den gleichformingen flussigkeitsstrom eingetauchten geraden kreiszylinder[J].Dinglers Polytech J,1911,326: 321-410.
[8]	Hartree D R. On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer[J].Mathematical Proceedings of the Cambridge Philosophical Society,1937,33(2): 223-239.
[9]	Asaithambi A. A finite-difference method for the Falkner-Skan equation[J].Applied Mathematics and Computation,1998,92(2/3): 135-141.
[10]	Fazio R. Blasius problem and Falkner-Skan model: T-pfer’s algorithm and its extension[J].Computers & Fluids,2013,〖STHZ〗73: 202-209.
[11]	Boyd J P. The Blasius function: computations before computers, the value of tricks, undergraduate projects, and open research problems[J].SIAM Review,2008,50(4): 791-804.
[12]	Boyd J P. The Blasius function in the complex plane[J].Experimental Mathematics,1999,8(4): 381-394.
[13]	LIAO Shi-jun. A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate[J].Journal of Fluid Mechanics,1999,385: 101-128.
[14]	Motsa S S, Sibanda P. An efficient numerical method for solving Falkner-Skan boundary layer flows[J].International Journal for Numerical Methods in Fluids,2012,69(2): 499-508.
[15]	Marinca V, Ene R D, Marinca B. Analytic approximate solution for Falkner-Skan equation[J].The Scientific World Journal,2014,2014: 617453.
[16]	Yun B I. New approximate analytical solutions of the Falkner-Skan equation[J].Journal of Applied Mathematics,2012,2012: 170802.
[17]	XU Ding, XU Jing-lei, XIE Gong-nan. Revisiting Blasius flow by fixed point method[J].Abstract and Applied Analysis,2014,2014: 953151.
[18]	Zeidler E.Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems[M]. Springer, 1986.
[19]	XU Ding, GUO Xin. Fixed point analytical method for nonlinear differential equations[J].Journal of Computational and Nonlinear Dynamics,2013,8(1): 011005.
[20]	XU Ding, GUO Xin. Application of fixed point method to obtain semi-analytical solution to Blasius flow and its variation[J].Applied Mathematics and Computation,2013,224: 791-802.
[21]	XU Ding, WANG Xian, XIE Gong-nan. Spectral fixed point method for nonlinear oscillation equation with periodic solution[J].Mathematical Problems in Engineering,2013,2013: 538716.
[22]	郭欣, 王娴, 许丁, 谢公南. 混合层无粘稳定性分析的Legendre级数解[J]. 应用数学和力学, 2013,34(8): 782-794.(GUO Xin, WANG Xian, XU Ding, XIE Gong-nan. A Legendre series solution to Rayleigh stability equation of mixing layer[J].Applied Mathematics and Mechanics,2013,34(8): 782-794.(in Chinese))
[23]	赵学端, 廖其奠. 粘性流体力学[M]. 北京: 机械工业出版社,1992.(ZHAO Xue-duan, LIAO Qi-dian.Visocus Fluid Flow [M]. Beijing: China Machine Press, 1992.(in Chinese))