- The basic equation for this method is obtained by integrating the x direction momentum equation (boundary layer momentum equation) with respect to y from the wall (at y = 0) to a distance
which is assumed to be outside the boundary layer. Using this notation, we can rewrite the Karman momentum integral equation as
 | (30.1) |
- The effect of pressure gradient is described by the second term on the left hand side. For pressure gradient surfaces in external flow or for the developing sections in internal flow, this term contributes to the pressure gradient.
- We assume a velocity profile which is a polynomial of
. 
being a form of similarity variable , implies that with the growth of boundary layer as distance x varies from the leading edge, the velocity profile 
remains geometrically similar.
- We choose a velocity profile in the form
 | (30.2) |
In order to determine the constants 
we shall prescribe the following boundary conditions
 | (30.3a) |
 | (30.3b) |
at
 | (30.3c) |
at
 | (30.3d) |
These requirements will yield 
respectively
Finally, we obtain the following values for the coefficients in Eq. (30.2),
and the velocity profile becomes
 | (30.4) |
- For flow over a flat plate,
and the governing Eq. (30.1) reduces to
 | (30.5) |
The wall shear stress is given by
- Substituting the values of
and 
in Eq. (30.5) we get,
 | (30.6) |
where C1 is any arbitrary unknown constant.
- The condition at the leading edge (
) yields 
Finally we obtain,
 | (30.7) |
 | (30.8) |
- This is the value of boundary layer thickness on a flat plate. Although, the method is an approximate one, the result is found to be reasonably accurate. The value is slightly lower than the exact solution of laminar flow over a flat plate . As such, the accuracy depends on the order of the velocity profile. We could have have used a fourth order polynomial instead --
- In addition to the boundary conditions in Eq. (30.3), we shall require another boundary condition at
- This yields the constants as
. Finally the velocity profile will be
Subsequently, for a fourth order profile the growth of boundary layer is given by
 | (30.10) |
