Order of Magnitude Analysis
- Let us examine what happens to the u velocity as we go across the boundary layer.
At the wall the u velocity is zero [ with respect to the wall and absolute zero for a stationary wall (which is normally implied if not stated otherwise)].
The value of u on the inviscid side, that is on the free stream side beyond the boundary layer is U.
For the case of external flow over a flat plate, this U is equal to 
.
- Based on the above, we can identify the following scales for the boundary layer variables:
| Variable | Dimensional scale | Non-dimensional scale |
 |  |  |
 |  |  |
 |  |  |
The symbol 
describes a value much smaller than 1.
- Now we analyse equations 28.4 - 28.6, and look at the order of magnitude of each individual term
Eq 28.6 - the continuity equation
One general rule of incompressible fluid mechanics is that we are not allowed to drop any term from the continuity equation.
- From the scales of boundary layer variables, the derivative
is of the order 1.
- The second term in the continuity equation
should also be of the order 1.The reason being 
has to be of the order 
because 
becomes 
at its maximum.
Eq 28.4 - x direction momentum equation
- Inertia terms are of the order 1.
-
is of the order 1
-
is of the order 
.
However after multiplication with 1/Re, the sum of the two second order derivatives should produce at least one term which is of the same order of magnitude as the inertia terms. This is possible only if the Reynolds number (Re) is of the order of 
.
- It follows from that
will not exceed the order of 1 so as to be in balance with the remaining term.
- Finally, Eqs (28.4), (28.5) and (28.6) can be rewritten as
 | (28.6) |
 | |
As a consequence of the order of magnitude analysis, 
can be dropped from the x direction momentum equation, because on multiplication with 
it assumes the smallest order of magnitude.
Eq 28.5 - y direction momentum equation.
- All the terms of this equation are of a smaller magnitude than those of Eq. (28.4).
This equation can only be balanced if 
is of the same order of magnitude as other terms.
- Thus they momentum equation reduces to
- This means that the pressure across the boundary layer does not change. The pressure is impressed on the boundary layer, and its value is determined by hydrodynamic considerations.
- This also implies that the pressure p is only a function of x. The pressure forces on a body are solely determined by the inviscid flow outside the boundary layer.
- The application of Eq. (28.4) at the outer edge of boundary layer gives
 | (28.8a) |
In dimensional form, this can be written as
 |
(28.8b)
|
On integrating Eq ( 28.8b) the well known Bernoulli's equation is obtained
 a constant |
(28.9)
|
- Finally, it can be said that by the order of magnitude analysis, the Navier-Stokes equations are simplified into equations given below.
 | (28.10) |
- These are known as Prandtl's boundary-layer equations. The available boundary conditions are:
| Solid surface |  | |
| or |  | (28.13) |
Outer edge of boundary-layer
| or |  | (28.14) |
- The unknown pressure p in the x-momentum equation can be determined from Bernoulli's Eq. (28.9), if the inviscid velocity distribution U(x) is also known.
We solve the Prandtl boundary layer equations for 
and 
with U obtained from the outer inviscid flow analysis. The equations are solved by commencing at the leading edge of the body and moving downstream to the desired location
- it allows the no-slip boundary condition to be satisfied which constitutes a significant improvement over the potential flow analysis while solving real fluid flow problems.
- The Prandtl boundary layer equations are thus a simplification of the Navier-Stokes equations.
