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One of the earliest and, until recently, most widely used approximate methods for the solution of the boundary layer equation is that developed by Pohlhausen. This method is based on the momentum equation of Kármán, which is obtained by integrating the boundary layer equation (4.77) across the layer, as shown in section 5.1. In the case of steady flow over an impermeable surface, the momentum equation (5.93) reduces to

(99) 
where
is the skin friction,
is the displacement thickness, and
is the momentum thickness of the boundary layer. The boundary conditions for the boundary layer equations (4.77) and (4.78) are

(100) 

(101) 
From the boundary layer equations (4.77) and (4.78) and their derivatives with respect to y, a set of conditions on u can be derived with the aid of the boundary conditions (6.100) and (6.101). These conditions are

(102) 

(103) 

(104) 

(105) 
and at
we have

(106) 
In the Pohlhausen's method, and similar approximate methods, a form for the velocity profile u(x,y) is sought which satisfies the momentum equation (6.99) and some of the boundary condition (6.102) to (6.106). It is hoped that this form will approximate to the exact profile, which satisfies all the conditions (6.102) to (6.106) as well as (6.99). The form assumed is

(107) 
where
is the effective total thickness of the boundary layer. The function
may also depend on x through certain coefficients which are chosen so as to satisfy some of the conditions (6.102) to (6.106). Altough strictly the conditions at infinity are only approached asymptotically, it is assumed that these conditions can be transfered from infinity to
without appreciable error. Thus equations (6.102) to (6.106) become

(109) 

(111) 

(112) 
For the assumed velocity profile (6.107), we have that

(113) 

(114) 

(115) 
If the form assumed for
involves m unknown coefficients, these can be specified using m of the boundary conditions (6.108) to (6.112) and the remaining unknown
can be determined from the momentum equation (6.99). Also, if only the conditions (6.108), (6.109) and (6.110) are considered,
and
are functions of
alone. In this case, substitution of relations (6.113) to (6.115) into (6.99) leads to an equation of the form

(116) 
Since
,
we can write the equation above in the form

(117) 
or simply

(118) 
where

(119) 

(120) 
Pohlhausen used a family of velocity profiles given by the quartic polynomial

(121) 
chosen to satisfy the boundary conditions
With the chosen velocity profile (6.121), we can evaluate the functions
and
,
and then integrate numerically equation (6.118) to obtain
as a function of x and through equation
.
By the substitution of
and
in the assumed velocity profile ,
we obtain the velocity profile for any x and y.
Next: Stagnation Point flow.
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Previous: Energy Integral.
Karl P Burr
20030312