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Laplace transform and new homotopy perturbation methods are adopted to study Blasius’ viscous flow equation analytically. The solutions approximated by the proposed method are shown to be precise as compared to the corresponding results obtained by Howarth’s numerical method. A high accuracy of the new method is evident.

One of the well-known equations arising in fluid mechanics and boundary layer approach is Blasius’ differential equation. Blasius [

Boundary layer flow over a flat plate is governed by the continuity and the Navier-Stokes equations. For a two-dimensional, steady-state, incompressible flow with zero pressure gradient over a flat plate, governing equations are simplified to

Blasius evaluated

To illustrate the basic ideas of this method, let us consider the following nonlinear differential equation

Consider the nonlinear Blasius ordinary differential equation (

Clearly, we have from (

Substituting (

Comparison between the Howarth and LTNHPM methods for

Howarth [ | |||
---|---|---|---|

0 | 0.00000 | 0.00000 | 0.00000 |

0.2 | 0.00664 | 0.00664 | 0.00664 |

0.4 | 0.02656 | 0.02656 | 0.02656 |

0.6 | 0.05974 | 0.05973 | 0.05973 |

0.8 | 0.10611 | 0.10611 | 0.10611 |

1 | 0.16557 | 0.16557 | 0.16557 |

1.2 | 0.23795 | 0.23795 | 0.23795 |

1.4 | 0.32298 | 0.32298 | 0.32298 |

1.6 | 0.42032 | 0.42032 | 0.42032 |

1.8 | 0.52952 | 0.52952 | 0.52952 |

2 | 0.65003 | 0.65002 | 0.65002 |

2.2 | 0.78120 | 0.78119 | 0.78119 |

2.4 | 0.92230 | 0.92228 | 0.92228 |

2.6 | 1.07252 | 1.07250 | 1.07250 |

2.8 | 1.23099 | 1.23098 | 1.23098 |

3 | 1.39682 | 1.39681 | 1.39681 |

3.2 | 1.56911 | 1.56909 | 1.56909 |

3.4 | 1.74696 | 1.74694 | 1.74695 |

3.6 | 1.92954 | 1.92951 | 1.92952 |

3.8 | 2.11605 | 2.11596 | 2.11602 |

4 | 2.30576 | 2.30550 | 2.30575 |

4.2 | 2.49806 | 2.49720 | 2.49805 |

4.4 | 2.69238 | 2.68965 | 2.69242 |

4.6 | 2.88826 | 2.88002 | 2.88859 |

4.8 | 3.08534 | 3.06157 | 3.08718 |

5 | 3.28239 | 3.21785 | 3.29272 |

Comparison between the Howarth and LTNHPM methods for

Howarth [ | |||
---|---|---|---|

0 | 0.00000 | 0.00000 | 0.00000 |

0.2 | 0.06641 | 0.06641 | 0.06641 |

0.4 | 0.13277 | 0.13276 | 0.13276 |

0.6 | 0.19894 | 0.19894 | 0.19894 |

0.8 | 0.26471 | 0.26471 | 0.26471 |

1 | 0.32979 | 0.32978 | 0.32978 |

1.2 | 0.39378 | 0.39378 | 0.39378 |

1.4 | 0.45627 | 0.45626 | 0.45626 |

1.6 | 0.51676 | 0.51676 | 0.51676 |

1.8 | 0.57477 | 0.57476 | 0.57476 |

2 | 0.62977 | 0.62977 | 0.62977 |

2.2 | 0.68132 | 0.68131 | 0.68131 |

2.4 | 0.72899 | 0.72898 | 0.72898 |

2.6 | 0.77246 | 0.77245 | 0.77245 |

2.8 | 0.81152 | 0.81151 | 0.81151 |

3 | 0.84605 | 0.84604 | 0.84604 |

3.2 | 0.87609 | 0.87607 | 0.87608 |

3.4 | 0.90177 | 0.90173 | 0.90176 |

3.6 | 0.92333 | 0.92321 | 0.92333 |

3.8 | 0.94112 | 0.94067 | 0.94112 |

4 | 0.95552 | 0.95396 | 0.95553 |

4.2 | 0.96696 | 0.96192 | 0.96704 |

4.4 | 0.97587 | 0.96047 | 0.97639 |

4.6 | 0.98269 | 0.93806 | 0.98564 |

4.8 | 0.98779 | 0.86475 | 1.00322 |

5 | 0.99155 | 0.66723 | 1.06671 |

The comparison of answers obtained by LTNHPM and Howarth’s results for

The comparison of answers obtained by LTNHPM and Howarth’s results for

Table

In this paper, the combined Laplace transform and homotopy perturbation methods are employed to give numerical solutions of the classical Blasius flat-plate flow in fluid mechanics. To illustrate the accuracy and efficiency of the proposed procedure, various different examples in the interval