##
**A novel numerical technique for two-dimensional laminar flow between two moving porous walls.**
*(English)*
Zbl 1195.76387

Summary: We investigate the steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain that is bounded by two permeable surfaces. The governing fourth-order nonlinear differential equation is solved by applying the spectral-homotopy analysis method and a novel successive linearisation method. Semianalytical results are obtained and the convergence rate of the solution series was compared with numerical approximations and with earlier results where the homotopy analysis and homotopy perturbation methods were used. We show that both the spectral-homotopy analysis method and successive linearisation method are computationally efficient and accurate in finding solutions of nonlinear boundary value problems.

### MSC:

76S05 | Flows in porous media; filtration; seepage |

### Software:

Matlab
PDFBibTeX
XMLCite

\textit{Z. G. Makukula} et al., Math. Probl. Eng. 2010, Article ID 528956, 15 p. (2010; Zbl 1195.76387)

### References:

[1] | E. C. Dauenhauer and J. Majdalani, “Exact self-similarity solution of the Navier-Stokes equations for a porous channel with orthogonally moving walls,” Physics of Fluids, vol. 15, no. 6, pp. 1485-1495, 2003. · Zbl 1186.76126 · doi:10.1063/1.1567719 |

[2] | S. Dinarvand, A. Doosthoseini, E. Doosthoseini, and M. M. Rashidi, “Comparison of HAM and HPM methods for Berman’s model of two-dimensional viscous flow in porous channel with wall suction or injection,” Advances in Theoretical and Applied Mechanics, vol. 1, no. 7, pp. 337-347, 2008. · Zbl 1276.76051 |

[3] | J. Majdalani, “The oscillatory channel flow with arbitrary wall injection,” Zeitschrift für Angewandte Mathematik und Physik, vol. 52, no. 1, pp. 33-61, 2001. · Zbl 1172.76368 · doi:10.1007/PL00001539 |

[4] | J. Majdalani and T.-S. Roh, “The oscillatory channel flow with large wall injection,” Proceedings of the Royal Society of London. Series A, vol. 456, no. 1999, pp. 1625-1657, 2000. · Zbl 1030.76012 · doi:10.1098/rspa.2000.0579 |

[5] | J. Majdalani and W. K. van Moorhem, “Multiple-scales solution to the acoustic boundary layer in solid rocket motors,” Journal of Propulsion and Power, vol. 13, no. 2, pp. 186-193, 1997. |

[6] | J. Majdalani and C. Zhou, “Moderate-to-large injection and suction driven channel flows with expanding or contracting walls,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 83, no. 3, pp. 181-196, 2003. · Zbl 1116.76348 · doi:10.1002/zamm.200310018 |

[7] | L. Oxarango, P. Schmitz, and M. Quintard, “Laminar flow in channels with wall suction or injection: a new model to study multi-channel filtration systems,” Chemical Engineering Science, vol. 59, no. 5, pp. 1039-1051, 2004. · doi:10.1016/j.ces.2003.10.027 |

[8] | A. S. Berman, “Laminar flow in channels with porous walls,” Journal of Applied Physics, vol. 24, pp. 1232-1235, 1953. · Zbl 0050.41101 · doi:10.1063/1.1721476 |

[9] | J. F. Brady, “Flow development in a porous channel and tube,” Physics of Fluids, vol. 27, no. 5, pp. 1061-1076, 1984. |

[10] | S. M. Cox, “Two-dimensional flow of a viscous fluid in a channel with porous walls,” Journal of Fluid Mechanics, vol. 227, pp. 1-33, 1991. · Zbl 0721.76080 · doi:10.1017/S0022112091000010 |

[11] | S. P. Hastings, C. Lu, and A. D. MacGillivray, “A boundary value problem with multiple solutions from the theory of laminar flow,” SIAM Journal on Mathematical Analysis, vol. 23, no. 1, pp. 201-208, 1992. · Zbl 0749.34014 · doi:10.1137/0523010 |

[12] | T. A. Jankowski and J. Majdalani, “Symmetric solutions for the oscillatory channel flow with arbitrary suction,” Journal of Sound and Vibration, vol. 294, no. 4, pp. 880-893, 2006. · doi:10.1016/j.jsv.2005.12.035 |

[13] | T. A. Jankowski and J. Majdalani, “Laminar flow in a porous channel with large wall suction and a weakly oscillatory pressure,” Physics of Fluids, vol. 14, no. 3, pp. 1101-1110, 2002. · doi:10.1063/1.1445419 |

[14] | C. Lu, “On the asymptotic solution of laminar channel flow with large suction,” SIAM Journal on Mathematical Analysis, vol. 28, no. 5, pp. 1113-1134, 1997. · Zbl 0886.34053 · doi:10.1137/S0036141096297704 |

[15] | R. M. Terrill, “Laminar flow in a uniformly porous channel,” The Aeronautical Quarterly, vol. 15, pp. 299-310, 1964. |

[16] | S. Uchida and H. Aoki, “Unsteady flows in a semi-infinite contracting or expanding pipe,” Journal of Fluid Mechanics, vol. 82, no. 2, pp. 371-387, 1977. · Zbl 0367.76100 · doi:10.1017/S0022112077000718 |

[17] | C. Zhou and J. Majdalani, “Improved mean-flow solution for slab rocket motors with regressing walls,” Journal of Propulsion and Power, vol. 18, no. 3, pp. 703-711, 2002. |

[18] | G. Adomian, “Nonlinear stochastic differential equations,” Journal of Mathematical Analysis and Applications, vol. 55, no. 2, pp. 441-452, 1976. · Zbl 0351.60053 · doi:10.1016/0022-247X(76)90174-8 |

[19] | G. Adomian, “A review of the decomposition method and some recent results for nonlinear equations,” Computers & Mathematics with Applications, vol. 21, no. 5, pp. 101-127, 1991. · Zbl 0758.35003 · doi:10.1016/0898-1221(91)90018-Y |

[20] | J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7 |

[21] | J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87-88, 2006. · Zbl 1195.65207 · doi:10.1016/j.physleta.2005.10.005 |

[22] | S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, 1992. |

[23] | S. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. · Zbl 1051.76001 |

[24] | H. Xu, Z. L. Lin, S. J. Liao, J. Z. Wu, and J. Majdalani, “Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls,” Physics of Fluids, vol. 22, Article ID 053601, 18 pages, 2010. · Zbl 1190.76132 · doi:10.1063/1.3392770 |

[25] | M. B. Zaturska, P. G. Drazin, and W. H. H. Banks, “On the flow of a viscous fluid driven along a channel by suction at porous walls,” Fluid Dynamics Research, vol. 4, no. 3, pp. 151-178, 1988. |

[26] | S. S. Motsa, P. Sibanda, and S. Shateyi, “A new spectral-homotopy analysis method for solving a nonlinear second order BVP,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2293-2302, 2010. · Zbl 1222.65090 · doi:10.1016/j.cnsns.2009.09.019 |

[27] | S. S. Motsa, P. Sibanda, F. G. Awad, and S. Shateyi, “A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem,” Computers and Fluids, vol. 39, no. 7, pp. 1219-1225, 2010. · Zbl 1242.76363 · doi:10.1016/j.compfluid.2010.03.004 |

[28] | Z. Makukula, S. S. Motsa, and P. Sibanda, “On a new solution for the viscoelastic squeezing flow between two parallel plates,” Journal of Advanced Research in Applied Mathematics, vol. 2, no. 4, pp. 31-38, 2010. |

[29] | S. S. Motsa and P. Sibanda, “A new algorithm for solving singular IVPs of Lane-Emden type,” in Proceedings of the 4th International Conference on Applied Mathematics, Simulation, Modelling, WSEAS International Conferences, pp. 176-180, Corfu Island, Greece, July 2010. · Zbl 1343.65087 |

[30] | S. Dinarvand and M. M. Rashidi, “A reliable treatment of a homotopy analysis method for two-dimensional viscous flow in a rectangular domain bounded by two moving porous walls,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1502-1512, 2010. · Zbl 1189.35249 · doi:10.1016/j.nonrwa.2009.03.006 |

[31] | J. Majdalani, C. Zhou, and C. A. Dawson, “Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,” Journal of Biomechanics, vol. 35, no. 10, pp. 1399-1403, 2002. · doi:10.1016/S0021-9290(02)00186-0 |

[32] | C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer, New York, NY, USA, 1988. · Zbl 0658.76001 |

[33] | L. N. Trefethen, Spectral Methods in MATLAB, vol. 10 of Software, Environments, and Tools, SIAM, Philadelphia, Pa, USA, 2000. · Zbl 0953.68643 · doi:10.1137/1.9780898719598 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.