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:
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
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
. This reduces the boundary-layer equations to one ordinary differential equation.
Let us introduce a function
Then, we have from continuity equation:
The boundary-layer equations can be written as follows:
The above equation can be solved for a user-set value of parameter
using the shooting technique. The limiting case
is flow over a flat plate (Blasius problem). Using the following definitions of
, 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.