### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mechanics of Materials
- Mechanical Engineering
- Applied Mathematics

### Cite this

*International Journal of Non-Linear Mechanics*,

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

}

*International Journal of Non-Linear Mechanics*, vol. 42, no. 9, pp. 1084-1091. https://doi.org/10.1016/j.ijnonlinmec.2007.06.003

**Analysis of stagnation point flow toward a stretching sheet.** / Paullet, Joseph; Weidman, Patrick.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Analysis of stagnation point flow toward a stretching sheet

AU - Paullet, Joseph

AU - Weidman, Patrick

PY - 2007/11/1

Y1 - 2007/11/1

N2 - 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 + b2 = 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 < bc ≈ 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.

AB - 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 + b2 = 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 < bc ≈ 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.

UR - http://www.scopus.com/inward/record.url?scp=35648972784&partnerID=8YFLogxK

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U2 - 10.1016/j.ijnonlinmec.2007.06.003

DO - 10.1016/j.ijnonlinmec.2007.06.003

M3 - Article

AN - SCOPUS:35648972784

VL - 42

SP - 1084

EP - 1091

JO - International Journal of Non-Linear Mechanics

JF - International Journal of Non-Linear Mechanics

SN - 0020-7462

IS - 9

ER -