# Analysis of stagnation point flow toward a stretching sheet

Joseph Paullet, Patrick Weidman

Research output: Contribution to journalArticle

63 Citations (Scopus)

### 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 + 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.

Original language English (US) 1084-1091 8 International Journal of Non-Linear Mechanics 42 9 https://doi.org/10.1016/j.ijnonlinmec.2007.06.003 Published - Nov 1 2007

### Fingerprint

Stagnation Point Flow
Stretching Sheet
Monotonic
Stretching
Branch
Linearly
Boundary Value Problem
Dual Solutions
Strain Rate
Asymptotic Behavior of Solutions
Oblique
Stretch
Numerical Analysis
Heat Transfer
Stability Analysis
Horizontal
Unstable
Fluid
Boundary value problems
Numerical analysis

### All Science Journal Classification (ASJC) codes

• Mechanics of Materials
• Mechanical Engineering
• Applied Mathematics

### Cite this

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

In: International Journal of Non-Linear Mechanics, Vol. 42, No. 9, 01.11.2007, p. 1084-1091.

Research output: Contribution to journalArticle

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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.

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