1099 
with the above we shall assume an ordinary irrotational current 
to exist outside the boundary layer. 
From the calculation we see that an reversion of the direction of 
the flow may take place in the boundary layer, when the outside 
motion is retarded. In this case a counter current will be formed 
behind the body and the current coming from the front leaves 
the surface at a certain point. The place of this point depends on 
the form of the body but not on R. The thickness of the layer, in 
which these currents take place proves to be proportional toV rd/ U, 
where d is a dimension of the body; the relative thickness is 
1 a 1 
therefore of the order: 7 Der In these two points 
there is a qualitative agreement with the exact method of PRANDTL; 
there is however no quantitative agreement. 
For the details of the calculation see $10. The distribution of the 
vorticity has been represented schematically in the above mentioned 
fig. 15; fig. 17 gives a sketch of the image of the absolute flow; 
fig. 18 of that of the relative flow. 
$ 8. The method of Pranptu. 
The method of PranNprL and his collaborators is the only method 
of calculation in which equation (4) is not reduced artificially to a 
linear equation, but where directly a solution of the quadratic 
equation is sought for *). A detailed discussion of this method cannot 
be given here; a few remarks only may find place: 
a. The method has been worked out for the two-dimensional 
and for the axial-symmetrical three-dimensional flow, for high values 
of PR, so that the boundary layer is thin. 
6. Because of this last circumstance PRrANDTL simplifies Eurer’s 
equation to: 
dv, Ov, Ov, 1 dp 07», 
s— + 2», — => ——-—-4 pv : 
oy o dx Oy? 
where the w-axis has been taken parallel to the surface of the 
body, and the y-axis perpendicular to it. The pressure p is given 
by the state outside the boundary layer and can be treated in this 
(24) 
1) L. Pranptt, Ueber Flüssigkeitsbewegung bei sehr kleiner Reibung, Verhandl. 
d. III. internat. Mathematiker-Kongresses, Heidelberg 1904, p. 484. 
H. Brastus, Zeitschr. f. Math. u. Phys. 56, p. 1, 1908 (dissert. Göttingen 1907) 
en ibidem 58, p. 225, 1910. 
E. Botrze, Dissert. Göttingen 1908. 
K. Hiemenz, Dinglers Polytechn. Journal 826, p. 321, 1911. 
FG he 
