== Boundary Layer for Flow Past a Wedge ==

The Blausius boundary layer velocity solution is a special case of a larger class of problems for [http://en.wikipedia.org/wiki/Blasius_boundary_layer flow over a wedge], as shown in the following figure. The angle <math>\beta \pi</math> represents the wedge angle.

[[Image(wedge_bl.png)]]

The general solution is called the Falkner-Skan boundary layer solution, which starts with a recognition that the free-stream velocity will accelerate for non-zero values of <math>\beta</math>:

<math>u_{e}(x)= U_{0} \left( x/L \right) ^{m}</math>

where <math> L </math> is a characteristic length and ''m'' is a dimensionless constant that depends on <math>\beta</math>:

<math>
{\beta} = \frac{2m}{m + 1}
</math>

The condition ''m = 0'' gives zero flow acceleration corresponding to the Blausius solution for flat-plate flow.

We then define a similarity variable <math>\eta</math> that combines the local streamwise and cross-flow coordinates ''x'' and ''y'' (defined relative to the surface of the wedge):

<math>
{\eta} = y \sqrt{\frac{U_{0}(m+1)}{2{\nu}L}}\left(\frac{x}{L}\right)^{\frac{m-1}{2}}
</math>

Then, defining a function ''f'' that relates to the streamwise and cross-flow velocities, a single ordinary differential equation ensues from boundary layer momentum and mass conservation:

<math>
\frac{\mathrm{d}^3 f}{\mathrm{d} \eta ^3}+f\frac{\mathrm{d}^2 f}{\mathrm{d} \eta^2}+ \beta \left[1-\left(\frac{\mathrm{d}f}{\mathrm{d}\eta}\right)^2 \right]=0
</math>

This non-linear equation is not amenable to an exact solution (even for the Blausius solution <math>\beta=0</math>, which eliminates the last term on the right side). The [http://demonstrations.wolfram.com/NumericalSolutionOfTheFalknerSkanEquationForVariousWedgeAngl/ following Mathematic CDF file] solves the equation numerically and provides the streamwise velocity normalized by the local freestream velocity as a function of <math>\eta</math>.


=== CDF Tool [[FootNote("Numerical Solution of the Falkner-Skan Equation for Various Wedge Angles" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/NumericalSolutionOfTheFalknerSkanEquationForVariousWedgeAngl/)]] ===
[[File(NumericalSolutionOfTheFalknerSkanEquationForVariousWedgeAngl.cdf)]]