Basics of Fluid Mechanics, 2014a

10.3. POTENTIAL FLOW FUNCTIONS INVENTORY 345
Note that the direction of U and ̂r̂r̂r is identical. The integration of equation (10.87)
yields
ψ − ψ 0 =
˙Q
2 πr (θ − θ 0) (10.88)
It traditionally chosen that the stream function ψ 0 is zero at θ =0. This operation is
possible because the integration constant and the arbitrary reference.
In the case of the sink rather than the source, the velocity is in the opposite
direction. Hence the flow rate is negative and the same equations obtained.
φ − φ 0 = −
˙Q
2 πr ln r (10.89)
r 0
Free Vortex Flow
As opposed to the radial flow direction (which
was discussed under the source and sink) the flow
in the tangential direction is referred to as the free
vortex flow. Another typical name for this kind
of flow is the potential vortex flow. The flow is
circulating the origin or another point. The velocity
is only a function of the distance from the radius
as
ψ − ψ 0 = −
˙Q
2 πr (θ − θ 0) (10.90)
U θ = f(r) (10.91)
7.0
5.0
3.0
1.0
φ = const
U
ŝ = −̂θ
1.0 3.0 5.0 7.0
ψ = const
And in vector notation the flow is
U = ̂θ f(r) (10.92)
Fig. -10.8. Two dimensional Vortex
free flow. In the diagram exhibits part
the circle to explain the stream lines
and potential lines.
The fundamental aspect of the potential flow is that this flow must be irrotational flow.
The gradient of the potential in cylindrical coordinates is
U = ∇φ = ∂φ
∂r ̂r + 1 ∂φ
r ∂θ ̂θ (10.93)
Hence, equation (10.93) dictates that
φ =0
1 ∂φ
r ∂θ = f(r)
(10.94)
∂φ
∂r =0