# Blasius boundary layer

Blasius boundary layer

A Blasius boundary layer, in physics and fluid mechanics, 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

:$frac\left\{U^\left\{2\left\{L\right\}approx ufrac\left\{U\right\}\left\{delta^\left\{2$,

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

:$\left\{partial uoverpartial x\right\}+\left\{partial voverpartial y\right\}=0$

:$u\left\{partial u over partial x\right\}+v\left\{partial u over partial y\right\}=\left\{ u\right\}\left\{partial^2 uover partial y^2\right\}$

(note that the x-independence of $U$ has been accounted for in the boundary-layer equations) admit a similarity solution. In the system of partial differential equations written above it is assumed that a fixed solid body wall is parallel to the x-direction whereas the y-direction is normal with respect to the fixed wall. $u$ and $v$ denote here the x- and y-components of the fluid velocity vector. Furthermore, from the scaling argument it is apparent that the boundary layer grows with the downstream coordinate $x$, e.g.

:$delta\left(x\right)approx left\left(frac\left\{ u x\right\}\left\{U\right\} ight\right)^\left\{1/2\right\}.$

This suggests adopting the similarity variable

:$eta=frac\left\{y\right\}\left\{delta\left(x\right)\right\}=yleft\left( frac\left\{U\right\}\left\{ u x\right\} ight\right)^\left\{1/2\right\}$and writing

:$u=U f \text{'}\left(eta\right).$ It proves convenient to work with the stream function $psi$, in which case

:$psi=\left( u U x\right)^\left\{1/2\right\} f\left(eta\right)$

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

:$f"\text{'} + frac\left\{1\right\}\left\{2\right\}f f" =0$

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

:$sigma_\left\{xy\right\} = frac\left\{f" \left(0\right) ho U^\left\{2\right\}sqrt\left\{ u\left\{sqrt\left\{Ux.$

can then be computed. The numerical solution gives $f" \left(0\right) approx 0.332$.

Falkner-Skan boundary layer

A generalisation of the Blasius boundary layer that considers outer flows of the form $U=cx^\left\{m\right\}$results in a boundary-layer equation of the form:$u\left\{partial u over partial x\right\}+v\left\{partial u over partial y\right\}=c^\left\{2\right\}m x^\left\{2m-1\right\}+\left\{ u\right\}\left\{partial^2 uover partial y^2\right\}.$Under these circumstances the appropriate similarity variable becomes

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

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

$psi=U\left(x\right)delta\left(x\right)f\left(eta\right) = c x^m delta\left(x\right)f\left(eta\right)$

This results in the Falkner-Skan equation

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

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

References

*Schlichting, H. (2004), "Boundary-Layer Theory", Springer. ISBN 3-540-66270-7

*Pozrikidis, C. (1998), "Introduction to Theoretical and Computational Fluid Dynamics", Oxford. ISBN 0-19-509320-8

*Blasius, H. (1908), "Grenzschichten in Flussigkeiten mit kleiner Reibung", Z. Math. Phys. vol 56, pp. 1-37. http://naca.larc.nasa.gov/reports/1950/naca-tm-1256 (English translation)

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