On the exact analytical and numerical solutions of nano boundary-layer fluid flows.

*(English)*Zbl 1253.76089Summary: The nonlinear boundary value problem describing the nanoboundary-layer flow with linear Navier boundary condition is investigated theoretically and numerically in this paper. The \(G'/G\)-expansion method is applied to search for the all possible exact solutions, and its results are then validated by the Chebyshev pseudospectral differentiation matrix (ChPDM) approach which has been recently introduced and successfully used. This numerical technique is firstly applied and, on comparing with the other recent work, it is found that the results are very accurate and effective to deal with the current problem. It is then used to examine and validate the present analytical analysis. Although the \(G'/G\)-expansion method has been used widely to solve nonlinear wave equations, its application for nonlinear boundary value problems has not been discussed yet, and the present paper may be the first to address this point. It is clarified that the exact solutions obtained via the \(G'/G\)-expansion method cannot be obtained by using some of the other methods. In addition, the domain of the physical parameters involved in the current boundary value problem is also discussed. Furthermore, the convex, vicinity of zero, and asymptotic solutions are deduced.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

##### Keywords:

nanoboundary-layer flow; \(G'/G\)-expansion method; Chebyshev pseudospectral differentiation matrix
PDF
BibTeX
XML
Cite

\textit{E. H. Aly} and \textit{A. Ebaid}, Abstr. Appl. Anal. 2012, Article ID 415431, 22 p. (2012; Zbl 1253.76089)

Full Text:
DOI

**OpenURL**

##### References:

[1] | J.-H. He, Y.-Q. Wan, and L. Xu, “Nano-effects, quantum-like properties in electrospun nanofibers,” Chaos, Solitons and Fractals, vol. 33, no. 1, pp. 26-37, 2007. |

[2] | L. Prandtl, “Über Flüssigkeitsbewegung bei sehr kleiner Reibung,” in Proceedings of the 3rd International Mathematical Congress, 1904. · JFM 36.0800.02 |

[3] | M. Gad-el-Hak, “The fluid mechanics of macrodevices-the Freeman scholar lecture,” Journal of Fluids Engineering, vol. 121, pp. 5-33, 1999. |

[4] | M. T. Matthews and J. M. Hill, “Nano boundary layer equation with nonlinear Navier boundary condition,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 381-400, 2007. · Zbl 1207.76050 |

[5] | M. T. Matthews and J. M. Hill, “Micro/nano thermal boundary layer equations with slip-creep-jump boundary conditions,” IMA Journal of Applied Mathematics, vol. 72, no. 6, pp. 894-911, 2007. · Zbl 1388.76055 |

[6] | M. T. Matthews and J. M. Hill, “A note on the boundary layer equations with linear slip boundary condition,” Applied Mathematics Letters, vol. 21, no. 8, pp. 810-813, 2008. · Zbl 1148.76019 |

[7] | J. Cheng, S. Liao, R. N. Mohapatra, and K. Vajravelu, “Series solutions of nano boundary layer flows by means of the homotopy analysis method,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 233-245, 2008. · Zbl 1135.76016 |

[8] | C. Y. Wang, “Analysis of viscous flow due to a stretching sheet with surface slip and suction,” Nonlinear Analysis. Real World Applications, vol. 10, no. 1, pp. 375-380, 2009. · Zbl 1154.76330 |

[9] | F. Talay Akyildiz, H. Bellout, K. Vajravelu, and R. A. Van Gorder, “Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces,” Nonlinear Analysis. Real World Applications, vol. 12, no. 6, pp. 2919-2930, 2011. · Zbl 1231.35155 |

[10] | M. M. Rashidi and E. Erfani, “The modified differential transform method for investigating nano boundary-layers over stretching surfaces,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21, pp. 864-883, 2011. |

[11] | R. A. Van Gorder, E. Sweet, and K. Vajravelu, “Nano boundary layers over stretching surfaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 6, pp. 1494-1500, 2010. · Zbl 1221.76024 |

[12] | C. Y. Wang, “Flow due to a stretching boundary with partial slip-an exact solution of the Navier-Stokes equations,” Chemical Engineering Science, vol. 57, pp. 3745-3747, 2002. |

[13] | E. J. Parkes and B. R. Duffy, “An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations,” Computer Physics Communications, vol. 98, pp. 288-300, 1996. · Zbl 0948.76595 |

[14] | S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69-74, 2001. · Zbl 0972.35062 |

[15] | A. Ebaid and E. H. Aly, “Exact solutions for the transformed reduced Ostrovsky equation via the F-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions,” Wave Motion, vol. 49, pp. 296-308, 2012. · Zbl 1360.35039 |

[16] | J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448 |

[17] | X.-H. Wu and J.-H. He, “EXP-function method and its application to nonlinear equations,” Chaos, Solitons and Fractals, vol. 38, no. 3, pp. 903-910, 2008. · Zbl 1153.35384 |

[18] | A. Ebaid, “Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method,” Physics Letters A, vol. 365, no. 3, pp. 213-219, 2007. · Zbl 1203.35213 |

[19] | A. Ebaid, “Application of the exp-function method for solving some evolution equations with nonlinear terms of any orders,” Zeitschrift für Naturforschung, vol. 65, pp. 1039-1044, 2010. |

[20] | C. Chun, “New solitary wave solutions to nonlinear evolution equations by the Exp-function method,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2107-2110, 2011. · Zbl 1219.35224 |

[21] | A. Ebaid, “Generalization of He’s exp-function method and new exact solutions for Burgers equation,” Zeitschrift für Naturforschung, vol. 64, pp. 604-608, 2009. |

[22] | M. Wang, X. Li, and J. Zhang, “The G\(^{\prime}\)/G-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417-423, 2008. · Zbl 1217.76023 |

[23] | R. Abazari, “Application of G\(^{\prime}\)/G-expansion method to travelling wave solutions of three nonlinear evolution equation,” Computers & Fluids, vol. 39, no. 10, pp. 1957-1963, 2010. · Zbl 1245.76096 |

[24] | S. Yu, “A generalized G\(^{\prime}\)/G-expansion method and its application to the MKdV equation,” International Journal of Nonlinear Science, vol. 8, no. 3, pp. 374-378, 2009. · Zbl 1192.35161 |

[25] | X. Fan, S. Yang, and D. Zhao, “Travelling wave solutions for the Gilson-Pickering equation by using the simplified G\(^{\prime}\)/G-expansion method,” International Journal of Nonlinear Science, vol. 8, no. 3, pp. 368-373, 2009. · Zbl 1192.35149 |

[26] | E. H. Aly, M. Benlahsen, and M. Guedda, “Similarity solutions of a MHD boundary-layer flow past a continuous moving surface,” International Journal of Engineering Science, vol. 45, no. 2-8, pp. 486-503, 2007. · Zbl 1213.76234 |

[27] | M. Guedda, E. H. Aly, and A. Ouahsine, “Analytical and ChPDM analysis of MHD mixed convection over a vertical flat plate embedded in a porous medium filled with water at 4^\circ C,” Applied Mathematical Modelling, vol. 35, no. 10, pp. 5182-5197, 2011. · Zbl 1228.76190 |

[28] | E. H. Aly, N. T. El-Dabe, and A. S. Al-Bareda, “ChPDM analysis for MHD flow of viscoelastic fluid through porous media,” Journal of Applied Sciences Research. In press. |

[29] | L. J. Crane, “Flow past a stretching plate,” Zeitschrift für Angewandte Mathematik und Physik, vol. 21, pp. 645-647, 1970. |

[30] | H. I. Andersson, “Slip flow past a stretching surface,” Acta Mechanica, vol. 158, pp. 121-125, 2002. · Zbl 1013.76020 |

[31] | P. S. Gupta and A. S. Gupta, “Heat and mass transfer on a stretching sheet with suction or blowing,” The Canadian Journal of Chemical Engineering, vol. 55, pp. 744-746, 1977. |

[32] | T. Fang, S. Yao, J. Zhang, and A. Aziz, “Viscous flow over a shrinking sheet with a second order slip flow model,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1831-1842, 2010. · Zbl 1222.76028 |

[33] | B. Brighi and J.-D. Hoernel, “Similarity solutions for high frequency excitation of liquid metal in an antisymmetric magnetic field,” in Self-Similar Solutions of Nonlinear PDE, vol. 74, pp. 41-57, Polish Academy of Science, Warsaw, Poland, 2006. · Zbl 1120.34009 |

[34] | E. H. Aly, L. Elliott, and D. B. Ingham, “Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium,” European Journal of Mechanics B, vol. 22, no. 6, pp. 529-543, 2003. · Zbl 1033.76055 |

[35] | A. Ebaid, “An improvement on the exp-function method when balancing the highest order linear and nonlinear terms,” Journal of Mathematical Analysis and Applications, vol. 392, pp. 1-5, 2012. · Zbl 1239.35008 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.