##
**On comparison of the solutions for an axisymmetric flow.**
*(English)*
Zbl 1173.76039

Summary: Recently, P. D. Ariel [Comput. Math. Appl. 54, No. 7–8, 1169–1183 (2007; Zbl 1138.76030)] explored the axially stretching flow of a viscous fluid in the presence of a velocity slip. He computed the solutions by noniterative technique, the homotopy perturbation method (HPM), and the perturbation and asymptotic methods (for small and large values of the slip parameter, respectively). Through comparison among these solutions, he claimed that HPM solution is the best solution showing close agreement with an exact solution. Here, we recomputed the flow problem considered in Ariel’s work for the series solution by homotopy analysis method (HAM). It is found that HAM solution is identical with the presented exact solution in Ariel’s work. Furthermore, the HAM solution is better than the HPM solution.

### MSC:

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

76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |

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

### Citations:

Zbl 1138.76030
PDFBibTeX
XMLCite

\textit{T. Hayat} et al., Numer. Methods Partial Differ. Equations 25, No. 5, 1204--1211 (2009; Zbl 1173.76039)

Full Text:
DOI

### References:

[1] | Sakiadis, Boundary-layer behaviour on continuous solid surfaces, Am Inst Chem Eng J 7 pp 26– (1961) · doi:10.1002/aic.690070108 |

[2] | Crane, Flow past a stretching plate, Math Phys 21 pp 645– (1970) |

[3] | Bataller, Effects of heat source/sink, radiation and work done by deformation on flow and heat transfer of a viscoelastic fluid over a stretching sheet, Comput Math Appl 53 pp 305– (2007) · Zbl 1138.80003 |

[4] | Cortell, MHD flow and mass transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet with chemically reactive species, Chem Eng Process 46 pp 721– (2007) |

[5] | Cortell, A note on magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Appl Math Comput 168 pp 557– (2005) · Zbl 1081.76059 |

[6] | Ariel, On computation of the three-dimensional flow past a stretching sheet, Appl Math Comput 187 pp 1244– (2007) · Zbl 1114.76056 |

[7] | Ariel, Homotopy perturbation method and axisymmetric flow over a stretching sheet, Int J Nonlinear Sci Numer Simul 7 pp 399– (2006) · Zbl 06942218 · doi:10.1515/IJNSNS.2006.7.4.399 |

[8] | Ariel, The flow of an elastico-viscous fluid past a stretching sheet with partial slip, Acta Mech 187 pp 29– (2006) · Zbl 1103.76010 |

[9] | Hayat, The influence of thermal radiation on MHD flow of a second grade fluid, Int J Heat Mass Transfer 50 pp 931– (2007) · Zbl 1124.80325 |

[10] | Liao, On the analytic solution of magnetohydrodynamic flows of non-Newonian fluids over a stretching sheet, J Fluid Mech 488 pp 189– (2003) · Zbl 1063.76671 |

[11] | Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J Heat Mass Transfer 48 pp 2529– (2005) · Zbl 1189.76142 |

[12] | Sajid, Non-similar solution for the axisymmetric flow of a third-grade fluid over a radially stretching sheet, Acta Mech 189 pp 193– (2007) · Zbl 1117.76006 |

[13] | Sajid, Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet, Int J Heat Mass Transfer 50 pp 1723– (2007) · Zbl 1140.76042 |

[14] | Hayat, On analytic solution for generalized three-dimensional MHD flow over a porous stretching sheet, Phys Lett A 370 pp 243– (2007) · Zbl 1209.76024 |

[15] | Ariel, Axisymmetric flow due to a stretching sheet with partial slip, Comput Math Appl 54 pp 1169– (2007) · Zbl 1138.76030 |

[16] | Liao, Beyond perturbation: Introduction to homotopy analysis method (2003) · Zbl 1051.76001 · doi:10.1201/9780203491164 |

[17] | S. J. Liao, On the proposed homotopy analysis technique for nonlinear problems and its applications, PhD Thesis, Shanghai Jiao Tong University, China, 1992. |

[18] | Liao, On the homotopy analysis method for nonlinear problems, Appl Math Comput 147 pp 499– (2004) · Zbl 1086.35005 |

[19] | Yang, On the explicit, purely analytic solution of Von Kármán swirling viscous flow, Comm Non linear Sci Numer Simm 11 pp 83– (2006) |

[20] | Liao, An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude, Int J Nonlinear Mech 38 pp 1173– (2003) · Zbl 1348.74225 |

[21] | Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys Lett A 360 pp 109– (2006) · Zbl 1236.80010 |

[22] | S. Abbasbandy, Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method, Chaos, Solitons & Fractals 36 (2008), 581-591. · Zbl 1139.76013 |

[23] | Abbasbandy, Solitary wave solutions to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method, Non linear Dyn 52 pp 35– (2008) · Zbl 1173.35646 · doi:10.1007/s11071-007-9255-9 |

[24] | Hayat, The influence of variable viscosity and viscous dissipation on the non-Newtonian flow: An analytical solution, Comm Non linear Sci Numer Simm 12 pp 300– (2006) |

[25] | Hayat, On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Phys Lett A 361 pp 316– (2007) · Zbl 1170.76307 |

[26] | Hayat, Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field, J Math Anal Appl 298 pp 225– (2004) · Zbl 1067.35074 |

[27] | Abbas, MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel, Theor Comput Fluid Dyn 20 pp 229– (2006) · Zbl 1109.76065 |

[28] | Sajid, On the analytic solution of the steady flow of a fourth grade fluid, Phys Lett A 355 pp 18– (2006) |

[29] | Sajid, Non-similar series solution for boundary layer flow of a third-order fluid over a stretching sheet, Appl Math Comput 189 pp 1576– (2007) · Zbl 1120.76004 |

[30] | Hayat, Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid, Acta Mech 191 pp 219– (2007) · Zbl 1117.76069 |

[31] | Sajid, The application of homotopy analysis method to thin film flows of a third order fluid, Chaos Solutions Fractals 31 pp 506– (2008) · Zbl 1146.76588 |

[32] | Sajid, Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet, Int Comm Heat Mass Transfer 35 pp 347– (2008) |

[33] | Hayat, Series solution for the upper-convected Maxwell fluid over a porous stretching plate, Phys Lett A 358 pp 396– (2006) · Zbl 1142.76511 |

[34] | Hayat, Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet, Int J Heat Mass Transfer 50 pp 75– (2007) · Zbl 1104.80006 |

[35] | Sajid, Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet, Comm Non linear Sci Numer Simm 13 pp 2193– (2008) |

[36] | Sajid, Comparison of the HAM and HPM solutions of thin film flow of non-Newtonian fluids on a moving belt, Non linear Dyn 50 pp 27– (2007) · Zbl 1181.76031 · doi:10.1007/s11071-006-9140-y |

[37] | Abbasbandy, Homotopy analysis method for heat radiation equations, Int Comm Heat Mass Transfer 34 pp 380– (2007) |

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.