9804

Boundary-Layer Flow Past a Semi-Infinite Wedge: The Falkner-Skan Problem

In 1904 Prandtl showed that the effects of viscosity at a high Reynolds number could be represented by approximating the Navier–Stokes equations with the celebrated boundary-layer equations, which for two-dimensional steady flow reduce to:
Here, and are the coordinates parallel and perpendicular to the body surface, respectively.
For a semi-infinite wedge with an angle of taper , one can prove that far from the wedge the potential flow is given by
, where or , and is a scale factor.
The above boundary-layer equations admit a self-similar solution such that the velocity profiles at different distances can be made congruent with suitable scale factors for and . This reduces the boundary-layer equations to one ordinary differential equation.
Let us introduce a function such that: where .
Then, we have from continuity equation: .
The boundary-layer equations can be written as follows:
with and
The above equation can be solved for a user-set value of parameter when and using the shooting technique. The limiting case is flow over a flat plate (Blasius problem). Using the following definitions of and , one can obtain the particle trajectories (streamlines) for different starting points. This Demonstration plots such trajectories and also shows the growth of the boundary-layer (red curve) in a separate plot for any value of the wedge angle. The evolution of the -velocity component and its congruent properties with the growth of the boundary layer is also shown.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

References
[1] L. G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport, Cambridge: Cambridge University Press, 2007.
[2] A. D. Polyanin, A. M. Kutepov, A. V. Vyazmin, and D. A. Kazenin, Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Boca Raton, FL: CRC Press, 2002.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+