In 2-D hydrodynamics extensive use is made of the functions of a
complex variable. The independent variables x,y are combined
into the complex variable z=x+iy.The dependent variables 
and 
are combined into a variable known as complex potential
w
The potential of an infinite row of doublets in a uniform flow can
be obtained by combining the potential of uniform flow and the
potentials of a series of doublets
(when source and sink are located at the same point). We know
that the potential of a uniform flow with 
The potential of a doublet at the orgin is
in which 
is a constant which represents the strength
of the doublet.
A series doublets can be added symmetrily according the origin along y axis with equal distance S, the potential is
So the potential w can be written as
when 
,
After x is replaced by 
, we have
Take logarithm of (6) and differentiating it, we obtain
Substituting (7) into (4),
where
Employing the above equation, we have
Then we got the expression for the functions and 
Setting 
, we obtain the function of the curve of the body,
(1) y=0 is the trival solution, the x axis, which goes through the body. We don't consider this condition.
(2) Let 
, the equation remains the same, the function is
symmetric to the y axis;
Let 
, the equation remains the same, the function is
symmetric to the x axis;
Let 
, the equation remains the same,
the function is the symmetric about the origin.
(13) respensents the curve of the surface.
At point B,
so,
when 
is sufficently small,
At point A,
First let 
(infintely small)
When 
, 
, 
(15) can be written as:
So,
From (14) and (17) we obtain:
The velocity is obtained
Let
From Bernoulli's equation, we have
Let 
, and D is the diameter of the cylinder,then
Let 
, 
and S is the distance between two
cylinders, L the horizonal distance from the the front of the cylinder
to the point we consider.
The maximum concures at the same horizonal line with the center of the cylinder. The mininum concures at the the interdistance between two cylinders.
