A Blasius boundary layer describes the steady two-dimensional boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow $U$. Within the boundary layer the usual balance between viscosity and convective inertia is struck, resulting in the scaling argument

$\backslash frac\{U^\{2\}\}\{L\}\backslash approx\; \backslash nu\backslash frac\{U\}\{\backslash delta^\{2\}\}$,

where $\backslash delta$ is the boundary-layer thickness and $\backslash nu$ is the fluid viscosity. However the semi-infinite plate has no natural length scale $L$ and so the steady, two-dimensional boundary-layer equations

$\{\backslash partial\; u\backslash over\backslash partial\; x\}+\{\backslash partial\; v\backslash over\backslash partial\; y\}=0$

$u\{\backslash partial\; u\; \backslash over\; \backslash partial\; x\}+v\{\backslash partial\; u\; \backslash over\; \backslash partial\; y\}=\{\backslash nu\}\{\backslash partial^2\; u\backslash over\; \backslash partial\; y^2\}$

(note that the x-independence of $U$ has been accounted for in the boundary-layer equations) admit a similarity solution. Furthermore, from the scaling argument it is apparent that the boundary layer grows with the downstream coordinate x, e.g.

$$

\delta(x)\approx \left( \frac{\nu x}{U} \right)^{1/2}.

This suggests adopting the similarity variable

$\backslash eta=\backslash frac\{y\}\{\backslash delta(x)\}=y\backslash left(\; \backslash frac\{U\}\{\backslash nu\; x\}\; \backslash right)^\{1/2\}$

and writing

$u=U\; f\; \text{'}(\backslash eta).$

It proves convenient to work with the streamfunction, in which case

$\backslash psi=(\backslash nu\; U\; x)^\{1/2\}\; f(\backslash eta)$

and on differentiating, to find the velocities, and substituting into the boundary-layer equation we obtain the Blasius equation

$$

f' + \frac{1}{2}f f =0

subject to $f=f\text{'}=0$ on $\backslash eta=0$ and $f\text{'}\backslash rightarrow\; 1$ as $\backslash eta\backslash rightarrow\; \backslash infty$. This non-linear ODE must be solved numerically, with the shooting method proving an effective choice. The shear stress on the plate

$\backslash sigma\_\{xy\}\; =\; \backslash frac\{f\text{'}\text{'}\; (0)\; \backslash rho\; U^\{2\}\backslash sqrt\{\backslash nu\}\}\{\backslash sqrt\{Ux\}\}.$

can then be computed. The numerical solution gives $f\text{'}\text{'}\; (0)\; \backslash approx\; 0.332$.

=Falkner-Skan boundary layer=

A generalisation of the Blasius boundary layer which considers outer flows of the form $U=cx^\{m\}$ results in a boundary-layer equation of the form

$$

u{\partial u \over \partial x} + v{\partial u \over \partial y} = c^{2}m x^{2m-1} + {\nu}{\partial^2 u\over \partial y^2}. Under these circumstances the appropriate similarity variable becomes

$\backslash eta=\backslash frac\{y\}\{\backslash delta(x)\}=\backslash frac\{\backslash sqrt\{c\}y\}\{\backslash sqrt\{\backslash nu\}x^\{(1-m)/2\}\},$

and, as in the Blasius boundary layer, it is convenient to use a stream function

$\backslash psi=U(x)\backslash delta(x)f(\backslash eta)\; =\; c\; x^m\; \backslash delta(x)f(\backslash eta)$

This results in the Falkner-Skan equation

$f$'+\frac{1}{2}(m+1)f f - m f'^{2} + m =0

(note that $m=0$ produces the Blasius equation).

References

---

• Pozrikidis, C. (1998), 'Introduction to theoretical and computational fluid dynamics', Oxford. ISBN 0195093208

• Blasius, H. (1908), 'Grenzschichten in Flussigkeiten mit kleiner Reibung', Z. Math. Phys. vol 56, pp. 1-37.

Fluid dynamics