Implicit differential equation arising in the steady flow of a Sisko fluid.

*(English)*Zbl 1160.76002Summary: Consideration is given to the thin film flow of a Sisko fluid on a moving belt. Using the implicit function theorem, an existence result for the solution of the resulting nonlinear differential equation is established. Also, the homotopy analysis method is used to obtain approximate analytical solution of the problem for non-integer values of the power index. The numerical results are presented graphically, and the salient features of the solution are discussed for various values of power index parameter. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.

##### MSC:

76A05 | Non-Newtonian fluids |

PDF
BibTeX
XML
Cite

\textit{F. T. Akyildiz} et al., Appl. Math. Comput. 210, No. 1, 189--196 (2009; Zbl 1160.76002)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Tanner, R.I., Engineering rheology, (1992), Oxford University Press Oxford · Zbl 0533.76004 |

[2] | Vradis, C.V.; Hammad, J.K., Strongly coupled block-implicit solution technique for non-Newtonian convective heat transfer problems, Numerical heat transfer, 33, 79-97, (1998) |

[3] | Alexandrou, A.N.; Mcgilvreay, T.M.; Burgos, G., Steady hershel – bulkley fluid flow in three-dimensional expansions, Journal of non-Newtonian fluid mechanics, 100, 77-96, (2001) · Zbl 1023.76004 |

[4] | Lefton, L.; Wei, D., A penalty method for approximations of the stationary power-law Stokes problem, Electronic journal of differential equations, 7, 1-12, (2001) · Zbl 0972.65096 |

[5] | Siddiqui, A.M.; Ahmed, M.; Ghori, Q.K., Thin film flow of non-Newtonian fluids on a moving belt, Chaos, solitons & fractals, 33, 1006-1016, (2007) · Zbl 1129.76009 |

[6] | Asghar, S.; Hayat, T.; Kara, A.H., Exact solutions of thin film flows, Nonlinear dynamics, 50, 229-233, (2007) · Zbl 1193.76022 |

[7] | Sajid, M.; Hayat, T.; Asghar, S., Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt, Nonlinear dynamics, 50, 27-35, (2007) · Zbl 1181.76031 |

[8] | Courant, R., Differential and integral calculus, vol. 2, (1988), Wiley · Zbl 0635.26001 |

[9] | Birkhoff, G.; Rota, G., Ordinary differential equation, (1989), Wiley New York · Zbl 0183.35601 |

[10] | Sanchez, D., Ordinary differential equations and stability theory: an introduction, (1979), Dover Publications New York |

[11] | S.J. Liao, On the proposed homotopy analysis techniques for nonlinear problems and its application, Ph.D. dissertation, Shanghai Jiao Tong University, 1992. |

[12] | Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton |

[13] | Liao, S.J., An explicit, totally analytic approximation of blasiusâ€™ viscous flow problems, International journal of non-linear mechanics, 34, 759-778, (1999) · Zbl 1342.74180 |

[14] | Liao, S.J., On the homotopy analysis method for nonlinear problems, Applied mathematics and computation, 147, 499-513, (2004) · Zbl 1086.35005 |

[15] | Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Studies in applied mathematics, 119, 254-297, (2007) |

[16] | S.J. Liao, Notes on the homotopy analysis method: some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation, online. · Zbl 1221.65126 |

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.