Basics of Fluid Mechanics, 2014a

10.3. POTENTIAL FLOW FUNCTIONS INVENTORY 347
integral around area (in two dimensional flow) of the velocity along the path. The
circulation is denoted as Γ and defined as
∮
Γ= U s ds (10.100)
Where the velocity U s represents the velocity component in the direction of the path.
The symbol
∮
indicating that the integral in over a close path.
x
Mathematically to obtain the integral the velocity
component in the direction of the path has to be chosen
and it can be defined as
∮
Γ= U · ̂ds
(10.101)
C
Substituting the definition potential function into equation
(10.101) provides
∮
Γ=
C
∇φ ·
cds
{}}{
ŝs ds (10.102)
And using some mathematical manipulations yields
y
U s
U
ds
C
Fig. -10.9. Circulation path
to illustrate varies calculations.
∮
Γ=
C
∇φ · bs
{}}{
dφ
ds
∮C
ds = dφ (10.103)
The integration of equation (10.103) results in
∮
Γ= dφ = φ 2 (starting point) − φ 1 (starting point) (10.104)
C
Unless the potential function is dual or multi value, the difference between the two
points is zero. In fact this what is expected from the close path integral. However, in a
free vortex situation the situation is different. The integral in that case is the integral
around a circular path which is
∮
Γ=
U · i {}}{
rdθds=
∮
c1
r rdθ= c 1 2 π (10.105)
In this case the circulation, Γ is not vanishing. In this example, the potential function
φ is a multiple value as potential function the potential function with a single value.
Example 10.5:
Calculate the circulation of the source on the path of the circle around the origin with
radius a for a source of a given strength.