To know about the distribution of wind velocity from a fan, we studied firstly on the relation between the various forms of wing and the wind velocities at near points from the wing.
The distribution of wind velocity jetting from the nozzle which is assumed as a point, is expressed as follows :
U = KU_oz^<-1>exp{-((2√<2>)/(3ck))((r/z)^<3/2>)}
<tt>
where, r is the radial distance from the center axis
of jet flow,
z is the distance from the originalpoint along
the center axis,
c and k are the coefficients of the diffusion
of jet flow and the mixing length respectively,
K is constant.
</tt>
Jetting out into the uniform stream in the side direction, it is ;

U = KU_oz^<-1>exp{-((2√<2>)/(3ck))〔(<{x^2+<(y-y_<U-max>)>^2}>^<1/2>)/z〕^<3/2>}

y_<U-max> = (V_o)/(2KU_o)z^2

where, x and y are the axes of coordinate on the r-plane,
V_o is the constant wind velocity along y axis.
Then applicating them to the jet flow from the fan limited wing radius, the wind velocities at any pointare decided as a sum of the following three velocities.
Into the rest air

U_1 = ∫^P_Q Kz^<-1>f(n)exp{-(2√<2>)/(3ck)<((n-r)/z)>^<3/2>}dn
U_2 = ∫^Q_R Kz^<-1>f(n)exp{-(2√<2>)/(3ck)<((r-n)/z)>^<3/2>}dn
U_3 = ∫^S_H Kz^<-1>f(n)exp{-(2√<2>)/(3ck)<((r+n)/z)>^<3/2>}dn
Into the air stream above ;

U_1 = ∫^P_Q Kz^<-1>f(n)exp{-(2√<2>)/(3ck)<〔<(n-{x^2+<(y-y_n)>^2}>^<1/2>)/z〕>^<3/2>}dn
U_2 = ∫^Q_R Kz^<-1>f(n)exp{-(2√<2>)/(3ck)<〔<({x^3+<(y-y_n)>^3}>^<1/3>-n)/z〕>^<3/3>}dn
U_3 = ∫^S_H Kz^<-1>f(n)exp{-(2√<2>)/(3ck)<〔<({x^2+<(y-y_n)>^2}>^<1/2>+n)/z〕>^<3/2>}dn
y_n = (V_o)/(2kf(n))z^2
where, n is the auxiliary variable of r and f(n) is a function of the initial wind velocity.
The integrating ranges are decided according with the length of wing and the radius of center boss.
These expression corresponded to the experimental data through a home fan counting c=0.22 and k=0.16, but these coefficients must be consulted by many practical uses of Helicopter and Speed Sprayer etc.