The Falkner–Skan separation parameter

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Numbers

parameterisation

separation parameter

$\beta_s$:

-0.198837735046677546889047131815

$m_s$:

-0.0904285622706291073513665724347

Definition

The Falkner-Skan separation parameter $\beta_s$ is the Hartree pressure-gradient value for which the boundary-value problem $f''' + f f'' + \beta(1-(f')^2)=0$, $f(0)=f'(0)=0$ and $f'(\infty)=1$ has zero wall shear, $f''(0)=0$ [5] [4].

Parameters

parameterisation

— parameterisation (either beta or m)

Formulas

(1)

The two parameterisations are related by $m_s=\beta_s/(2-\beta_s)$ and $\beta_s=2m_s/(1+m_s)$.

Comments

(2)

The row labelled $\beta_s$ is the parameter used in the Falkner-Skan equation in this table. The row labelled $m_s$ is included because many boundary-layer sources quote the same separation point as the exponent in the external velocity law $u_e(x)=U_0(x/L)^m$. The two rows are one separation point written in two invertible parameterisations.

(3)

At $\beta=\beta_s$ the forward-flow and reversed-flow branches meet. For $0>\beta>\beta_s$, Stewartson found a second branch with $f''(0)<0$ [3].

Programs

(P1)

Python

from mpmath import mp
mp.dps = 50
# The attached generate.py solves for beta_s and converts it to m_s.

References

[1]

V. M. Falkner and S. W. Skan, Solutions of the boundary-layer equations, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 12 (1931), 865-896. (doi)

[2]

D. R. Hartree, On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer, Mathematical Proceedings of the Cambridge Philosophical Society 33 (1937), 223-239. (doi)

[3]

K. Stewartson, Further solutions of the Falkner-Skan equation, Mathematical Proceedings of the Cambridge Philosophical Society 50 (1954), 454-465. (doi)

[4]

E. R. Belden, Z. A. Dickman, S. J. Weinstein, A. D. Archibee, E. Burroughs and N. S. Barlow, Asymptotic Approximant for the Falkner-Skan Boundary-Layer equation, arXiv preprint, 2019. (arXiv)

Links

[5]

Wikipedia: Falkner-Skan boundary layer

[6]

Richard Fitzpatrick: Self-Similar Boundary Layers

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: yes

How they were obtained:

The attached generator computes $\beta_s$ by imposing $f''(0)=0$ and solving for the value of $\beta$ for which the finite-interval solution has $f'(\eta_{\max})=1$.

It requires agreement between independent working precisions and truncation lengths, converts $m_s$ from the same computed $\beta_s$, and checks the result against a collocation boundary-value solve and the printed value $\beta_s=-0.198837735$ in [4].