SI Xin-yi, SI Xin-hui, ZHENG Lian-cun, ZHANG Xin-xin. Homotopy Analysis Solution for Micropolar Fluid Flow Through Porous Channel With Expanding or Contracting Walls of Different Permeabilities[J]. Applied Mathematics and Mechanics, 2011, 32(7): 807-820. doi: 10.3879/j.issn.1000-0887.2011.07.005
Citation: SI Xin-yi, SI Xin-hui, ZHENG Lian-cun, ZHANG Xin-xin. Homotopy Analysis Solution for Micropolar Fluid Flow Through Porous Channel With Expanding or Contracting Walls of Different Permeabilities[J]. Applied Mathematics and Mechanics, 2011, 32(7): 807-820. doi: 10.3879/j.issn.1000-0887.2011.07.005

Homotopy Analysis Solution for Micropolar Fluid Flow Through Porous Channel With Expanding or Contracting Walls of Different Permeabilities

doi: 10.3879/j.issn.1000-0887.2011.07.005
  • Received Date: 2010-12-06
  • Rev Recd Date: 2011-04-20
  • Publish Date: 2011-07-15
  • The flow of amicropolar fluid through a porous channel with expanding or contracting walls of different permeability was investigated. Two cases were considered in which the opposing walls undergoeither uniformor nonuniform motion. In the first case, homotopy analysis method (HAM) was employed to obtain the expressions for velocity and micro-rotation fields. Graphs were sketched for some values of the param eters. The first conclusion can be made that expan sion ratio and different perm eability have miportant effects on the dynamic characteristics of the fluid. Following Xu. smodel, the second and more general case is that the wall expansion ratiovaries with time. Under this assumption, the govern ing equations were transformed in to non linear partial differential equations that also are solved analytically using HAM procedure. In the process, both algebraic and exponen tialmodels were considered to describe the evolution of a (t) from the initial a0 to a final state a1. As a result, it is found that the tmie-dependent solutions approach very rapidly to the steady state behavior. The second important conclusion can be made that the time-dependent variation of the wall expansion ratio plays a secondary role which maybe justifiably ignored.
  • loading
  • [1]
    Berman A S. Laminar flow in channels with porous walls[J]. Journal of Applied Physics, 1953, 24(9): 1232-1235. doi: 10.1063/1.1721476
    [2]
    Terrill R M, Thomas P W. Laminar flow in a uniformly porous pipe[J]. Applied Science Research, 1969, 21(1): 37-67. doi: 10.1007/BF00411596
    [3]
    Terrill R M. On some exponentially small terms arising in flow through a porous pipe[J]. The Quarterly Journal of Mechanics and Applied Mathematics, 1973, 26(3): 347-354. doi: 10.1093/qjmam/26.3.347
    [4]
    Terrill R M. Laminar flow in a uniformly porous channel[J]. The Aeronautical Quarterly, 1964, 15: 299-310.
    [5]
    Robinson W A. The existence of multiple solutions for the laminar flow in a uniformly porous channel with suction at both walls[J]. Journal of Engineering Mathematics, 1976, 10(1): 23-40. doi: 10.1007/BF01535424
    [6]
    Terrill R M. Laminar flow through parallel and uniformly porous walls of different permeability[J].ZAMP, 1965, 16: 470-482. doi: 10.1007/BF01593923
    [7]
    Terrill R M, Shrestha G M. Laminar flow through parallel and uniformly porous walls of different permeability[J]. Zeitschrift fur Angewandte Mathematik und Physik, 1965, 16: 470-482. doi: 10.1007/BF01593923
    [8]
    Uchida S, Aoki H. Unsteady flows in a semi-infinite contracting or expanding pipe[J]. Journal of Fluid Mechanics, 1977,82(2):371-387. doi: 10.1017/S0022112077000718
    [9]
    Ohki Morimatsu. Unsteady flows in a porous, elastic, circular tube—part 1: the wall contracting or expanding in an axial direction[J]. Bulletin of the JSME, 1980, 23(179): 679-686. doi: 10.1299/jsme1958.23.679
    [10]
    Goto M, Uchida S. Unsteady flow in a semi-infinite expanding pipe with injection through wall[J]. Journal of the Japan Society for Aeronautical and Space Science, 1990, 33(9): 14-27.
    [11]
    Bujurke N M, Pai N P, Jayaraman G. Computer extended series solution for unsteady flow in a contracting or expanding pipe[J]. IMA Journal of Applied Mathematics, 1998,60(2): 151-165. doi: 10.1093/imamat/60.2.151
    [12]
    Majdalani J, Zhou C, Dawson C D. Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability[J]. Journal of Biomechanics, 2002,35(10): 1399-1403. doi: 10.1016/S0021-9290(02)00186-0
    [13]
    Dauenhauer C E, Majdalani J. Exact self-similarity solution of the Navier-Stokes equations for a porous channel with orthogonally moving walls[J]. Physics of Fluids, 2003, 15(6): 1485-1495. doi: 10.1063/1.1567719
    [14]
    Majdalani J, Zhou C. Moderate-to-large injection and suction driven channel flows with expanding or contracting walls[J]. Zeitschrift fur Angewandte Mathematik und Mechanik, 2003,83(3): 181-196. doi: 10.1002/zamm.200310018
    [15]
    Asghar S, Mushtaq M, Hayat T. Flow in a slowly deforming channel with weak permeability:an analytical approach[J]. Nonlinear Analysis:Real World Applications, 2010, 11(1): 555-561. doi: 10.1016/j.nonrwa.2009.01.049
    [16]
    Si X H, Zheng L C, Zhang X X, Chao Y. Homotopy analysis solutions for the asymmetric laminar flow in a porous channel with expanding or contracting walls[J]. Acta Mechanica Sinica, 2011, 27(2): 208-214. doi: 10.1007/s10409-011-0430-3
    [17]
    Si X H, Zheng L C, Zhang X X, Chao Y. The flow of a micropolar fluid through a porous channel with expanding or contracting walls[J]. Central European Journal of Physics, 2011, 9(3): 825-834.
    [18]
    司新辉, 郑连存, 张欣欣, 晁莹. 半渗透涨缩管道内微极性流动解析求解[J].应用数学和力学, 2010, 31(9): 1027-1035.(SI Xin-hui, ZHENG Lian-cun, ZHANG Xin-xin, CHAO Ying. Analytic solution to the micropolar-fluid flow through a semi-porous channel with an expanding or contracting wall[J]. Applied Mathematics and Mechanics(English Edition), 2010, 31(9): 1073-1080.)
    [19]
    Xu H, Lin Z L, Liao S J, Wu J Z, Majdalani J. Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonlly moving walls[J]. Physics of Fluids, 2010, 22(5): 053601. doi: 10.1063/1.3392770
    [20]
    Makukula Z G, Precious Sibanda Motsa S S. A novel numerical technique for two dimensional laminar flow between two moving porous walls[J]. Mathematical Problems in Engineering, 2010. ID 528956. doi: 10.1155/2010/528956.
    [21]
    Si X H, Zheng L C, Zhang X X, Si X Y, Yang J H. flow of a viscoelastic through a porous channel with expanding or contracting walls[J]. Chinese Physics Letters, 2011, 28(4): 044702. doi: 10.1088/0256-307X/28/4/044702
    [22]
    Eringen A C. Theory of micropolar fluids[J].Journal of Mathematics & Mechanics, 1966, 16(1): 1-18.
    [23]
    Eringen A C. Theory of Thermomicropolar fluids[J]. Journal of Mathematical Analysis and Applications, 1972, 38(2): 480-496. doi: 10.1016/0022-247X(72)90106-0
    [24]
    Ariman T, Turk M A, Sylvester N D. Microcontinuum fluid mechanics—a review[J]. International Journal of Engineering Science, 1973, 11(8): 905-930. doi: 10.1016/0020-7225(73)90038-4
    [25]
    Ariman T, Turk M A, Sylvester N D. Application of Microcontinuum fluid mechanics—a review[J]. International Journal of Engineering, 1974, 12(4): 273-293. doi: 10.1016/0020-7225(74)90059-7
    [26]
    Eringen A C. Microcontinuum Field Theories Ⅱ:Fluent Media[M]. New York: Springer, 2001.
    [27]
    Subhadra Ramachandran P, Mathur M N, Ojha S K. Heat transfer in boundary layer flow of a micropolar fluid past a curved surface with suction and injection[J]. International Journal of Engineering Science, 1979, 17(5): 625-639. doi: 10.1016/0020-7225(79)90131-9
    [28]
    Takhar H S, Bhargava R, Agrawal R S, Balaji A V S. Finite element solution of micropolar fluid flow and heat transfer between two porous discs[J].International Journal of Engineering Science, 2000, 38(17): 1907-1922. doi: 10.1016/S0020-7225(00)00019-7
    [29]
    Kelson N A, Farrell T W. Micropolar fluid flow over a porous stretching sheet with strong suction or injection[J].International Communications in Heat and Mass Transfer, 2001, 28(4): 479-488. doi: 10.1016/S0735-1933(01)00252-4
    [30]
    Muhammad Ashraf, Anwar Kamal M, Syed K S. Numerical study of asymmetric laminar flow of a micropolar fluid in a porous channel[J].Computers & Fluids, 2009, 38(10): 1895-1902.
    [31]
    Muhammad Ashraf, Anwar Kamal M, Syed K S. Numerical simulation of flow of a micropolar fluid between a porous disk and a non-porous disk[J]. Applied Mathematical Modelling, 2009, 33(4): 1933-1943. doi: 10.1016/j.apm.2008.05.002
    [32]
    Liao S J. Beyond Perturbation: Introduction to Homotopy Analysis Method[M]. Boca, Raton: Chapman Hall/CRC Press, 2003.
    [33]
    Liao S J. On the homotopy analysis method for nonlinear problems[J]. Applied Mathematics and Computation, 2004, 147(2): 499-513. doi: 10.1016/S0096-3003(02)00790-7
    [34]
    Hayat T, Khan M. Homotopy solution for a generalized second grade fluid past a porous plate[J].Nonlinear Dynamics, 2005, 42(4): 395-405. doi: 10.1007/s11071-005-7346-z
    [35]
    Hayat T, Khan M, Asghar S. Magnetohydrodynamic flow of an oldroyd 6-constant fluid[J].Applied Mathematics and Computation, 2004, 155(2): 417-225. doi: 10.1016/S0096-3003(03)00787-2
    [36]
    Hayat T, Khan M, Siddiqui A M, Asghar S. Transient flows of a second grade fluid[J].International Journal of Non-Linear Mechanics, 2004, 39(10): 1621-1633. doi: 10.1016/j.ijnonlinmec.2002.12.001
    [37]
    Abbas Z, Sajid M, Hayat T. MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel[J].Theoretical and Computational Fluid Dynamics, 2006, 20(4): 229-238.
    [38]
    Sajid M, Hayat T, Asghar S. On the analytic solution of the steady flow of a fourth grade fluid[J]. Physics Letters A, 2006, 355(1): 18-26. doi: 10.1016/j.physleta.2006.01.092
    [39]
    Sajid M, Abbas Z, Hayat T. Homotopy analysis for boundary layer flow of a micropolar fluid through a porous channel[J].Applied Mathematical Modelling, 2009, 33(11): 4120-4125. doi: 10.1016/j.apm.2009.02.006
    [40]
    Srinivasacharya D, Ramana Murthy J V, Venugopalam D. Unsteady stokes flow of micropolar fluid between two parallel porous plates[J]. International Journal of Engineering Science, 2001, 39(14): 1557-1563. doi: 10.1016/S0020-7225(01)00027-1
    [41]
    Rees D A S, Pop I. Free convection boundary layer flow of a micropolar fluid from a vertical flat plate[J].IMA Journal of Applied Mathematics, 1998, 61(2): 179-197. doi: 10.1093/imamat/61.2.179
    [42]
    Guram G S, Smith A C. Stagnation flows of micropolar fluids with strong and weak interactions[J].Computers & Mathematics With Applications, 1980, 6(2): 213-233.
    [43]
    Liao S J. An optimal homotopy-analysis approach for strongly nonlinear differential equations[J].Communications in Nonlinear Science and Numerical Simulation, 2010, 15(8): 2003-2016. doi: 10.1016/j.cnsns.2009.09.002
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (1570) PDF downloads(654) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return