Exact Analytical Solution of the Magnetohydrodynamic Sink Flow
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摘要: 就一个特殊的磁流体动力学(MHD)流动,即速度幂指数为-1时的汇流,得到著名的Falkner-Skan方程精确的解析解.解析解是封闭的,并有多重解分支.分析了磁场参数和壁面伸长参数的影响.发现了有趣的速度分布现象:即使壁面固定,回流区域依然出现.在一个罕见的Falkner-Skan MHD流动中,得到了一组解,以精确封闭的解析公式表示,极大地丰富了著名的Falkner-Skan方程的解析解,也加深了对这重要又有趣方程的理解.Abstract: An exact analytical solution of the famous Falkner-Skan equation for magneto-hydro-dynamic (MHD) flow was obtained for a special case,namely the sink flow with a velocity power index of -1.The solution was given in a closed form.Multiple solution branches were observed.The effects of the magnetic parameter and the wall stretching parameter were analyzed.Interesting velocity profiles were observed with reversal flow regions even for a stationary wall.These solutions provide a rare case of the Falkner-Skan MHD flow with exact analytical closed form formula and greatly enrich the analytical solution to the celebrated Falkner-Skan equation and the understanding of this important and interesting equation.
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[1] Falkner V M, Skan S W. Some approximate solutions of the boundary layer equations[J]. Phil Mag, 1931, 12: 865-896. [2] Hartree D R. On an equation occurring in Falkner and Skan’s approximate treatment of the  ̄equations of the boundary layer[J]. Proc of the Cambridge Philosophical Society, 1937, 33(2): 223-239. doi: 10.1017/S0305004100019575 [3] Weyl H. On the differential equation of the simplest boundary-layer problems[J]. Ann of Math, 1942, 43: 381-407. doi: 10.2307/1968875 [4] Rosehead L. Laminar Boundary Layers[M]. London: Oxford University Press, 1963. [5] Hartman P. Ordinary Differential Equations[M]. New York: John Wiley & Sons Inc, 1964. [6] Stewartson K. Further solutions of the Falkner-Skan equations[J]. Proc of the Cambridge Phil Soc Math and Phys Sciences, 1954, 50: 454-465. doi: 10.1017/S030500410002956X [7] Libby P A, Liu T M. Further solutions of the Falkner-Skan equation[J]. AIAA J, 1967, 5(5): 1040-1042. doi: 10.2514/3.4130 [8] Zaturska M B, Banks W H H. A new solution branch of the Falkner-Skan equation[J]. Acta Mechanica, 2001, 152(1/4): 197-201. doi: 10.1007/BF01176954 [9] Schlichting H, Bussmann K. Exakte Losungen für die laminare Grenzschicht mit Absaugung und Ausblasen[J]. Schr Deutsch Akad Luftfahrtforschung Ser B, 1943, 7(2): 25-69.(in German) [10] Nickel K. Eine einfache Abschatzung fur Grenzschichten[J]. Ing Arch Bd, 1962, 31(2): 85-100. doi: 10.1007/BF00531419 [11] Yang H T, Chien L C. Analytic solutions of the Falkner-Skan equation when β=-1 and  ̄γ=0[J]. SIAM J Appl Math, 1975, 29(3): 558-569. doi: 10.1137/0129047 [12] Sakiadis B C. Boundary-layer behavior on continuous solid surface Ⅰ—boundary-layer equations for two-dimensional and axisymmetric flow[J]. J AIChe, 1961, 7(1):26-28. doi: 10.1002/aic.690070108 [13] Sakiadis B C. Boundary-layer behavior on continuous solid surface Ⅱ—boundary-layer equations for two-dimensional and axisymmetric flow[J]. J AIChe, 1961, 7: 221-225. doi: 10.1002/aic.690070211 [14] Klemp J P, Acrivos A. A method for integrating the boundary-layer equations through a region of reverse flow[J]. J Fluid Mech, 1972, 53(1): 177-191. doi: 10.1017/S0022112072000096 [15] Vajravelu K, Mohapatra R N. On fluid dynamic drag reduction in some boundary layer flows[J]. Acta Mechanica, 1990, 81(1/2): 59-68. doi: 10.1007/BF01174555 [16] Fang T. Further study on a moving-wall boundary-layer problem with mass transfer[J]. Acta Mechanica, 2003, 163(3/4): 183-188. [17] Weidman P D, Kubitschek D G, Davis A M J. The effect of transpiration on self-similar boundary layer flow over moving surfaces[J]. International Journal of Engineering Science, 2006, 44(11/12): 730-737. doi: 10.1016/j.ijengsci.2006.04.005 [18] Riley N, Weidman P D. Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary[J]. SIAM J Appl Math, 1989, 49(5): 1350-1358. doi: 10.1137/0149081 [19] Liao S J. A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate[J]. J Fluid Mech, 1999, 385:101-128. doi: 10.1017/S0022112099004292 [20] Sachdev P L, Kudenatti R B, Bujurke N M. Exact analytic solution of a boundary value problem for the Falkner-Skan equation[J]. Studies in Applied Mathematics, 2008, 120(1): 1-16. [21] Fang T, Zhang J. An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching[J]. Int J Non-Linear Mech, 2008, 43(6): 1000-1006. doi: 10.1016/j.ijnonlinmec.2008.05.006 [22] Yao B. Approximate analytical solution to the Falkner-Skan wedge flow with the permeable wall of uniform suction[J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(8): 3320-3326. doi: 10.1016/j.cnsns.2009.01.014 [23] Yao B, Chen J. Series solution to the Falkner-Skan equation with stretching boundary[J]. Applied Mathematics and Computation, 2009, 208(1): 156-164. doi: 10.1016/j.amc.2008.11.028 [24] Magyari E. Falkner-Skan flows past moving boundaries: an exactly solvable case[J]. Acta Mechanica, 2009, 203(1/2): 13-21. doi: 10.1007/s00707-008-0031-9 [25] Sutton G W, Sherman A. Engineering Magnetohydrodynamics[M]. New York: McGraw-Hill, 1965. [26] Cobble M H. Magnetofluiddynamic flow with a pressure-gradient and fluid injection[J]. Journal of Engineering Mathematics, 1977, 11(3): 249-256. doi: 10.1007/BF01535969 [27] Soundalgekar V M, Takhar H S, Singh M. Velocity and temperature field in MHD Falkner-Skan flow[J]. Journal of the Physical Society of Japan, 1981, 50: 3139-3143. doi: 10.1143/JPSJ.50.3139 [28] Abbasbandy S, Hayat T. Solution of the MHD Falkner-Skan flow by Hankel-Pade method[J]. Physics Letters A, 2009, 373(7): 731-734. doi: 10.1016/j.physleta.2008.12.045 [29] Abbasbandy S, Hayat T. Solution of the MHD Falkner-Skan flow by homotopy analysis method[J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(9/10): 3591-3598. doi: 10.1016/j.cnsns.2009.01.030 [30] Ishak A, Nazar R, Pop I. MHD boundary-layer flow past a moving wedge[J]. Magnetohydrodynamics, 2009, 45(1): 103-110. [31] Schlichting H, Gersten K. Boundary Layer Theory[M]. 8th Revised and Enlarged Edition. Springer, 2000: 171-174. [32] Pohlhausen K. Zur nherungsweisen integration der differentialgleichung der laminaren grenzschicht[J]. J Appl Math Mech (ZAMM), 1921,1(4): 252-268.
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