### Abstract

There has been much recent interest in the stagnation point flow of a fluid toward a stretching sheet. Investigations that may include oblique stagnation flow and heat transfer to a horizontal plate all involve the same boundary value problem (BVP):f^{‴} + ff^{″} - (f^{′})^{2} + b^{2} = 0,f (0) = 0, f^{′} (0) = 1, f^{′} (∞) = b .Here b is the ratio of the stagnation flow strain rate to the stretch rate of the sheet. Through numerical analysis of the problem, several authors have conjectured the existence of a solution for all values of b > 0. In this note we present numerical evidence that a second solution exists for 0 < b < b_{c} ≈ 0.16906. Further we present a mathematical proof that for all b > 0 there exists a monotonic solution to the BVP and if b > 1, this solution is unique. If b < 1 it can be shown that any further solutions cannot be monotonic and the second solution found here numerically is non-monotonic. The asymptotic behavior of solutions near b = 0 and 1 is also presented. Finally, a stability analysis is performed to show that solutions on the upper branch of the dual solutions are linearly stable, while those on the lower branch are linearly unstable.

Original language | English (US) |
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Pages (from-to) | 1084-1091 |

Number of pages | 8 |

Journal | International Journal of Non-Linear Mechanics |

Volume | 42 |

Issue number | 9 |

DOIs | |

State | Published - Nov 1 2007 |

### All Science Journal Classification (ASJC) codes

- Mechanics of Materials
- Mechanical Engineering
- Applied Mathematics

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## Cite this

*International Journal of Non-Linear Mechanics*,

*42*(9), 1084-1091. https://doi.org/10.1016/j.ijnonlinmec.2007.06.003