Basics of Fluid Mechanics, 2014a

344 CHAPTER 10. POTENTIAL FLOW
The volumetric flow rate (two dimensional)
˙Q denotes the flow rate out
or in to control volume into the source or
sink. The flow rate is shown in Figure 10.7
is constant for every potential line. The
flow rate can be determined by
˙Q =2πrU r (10.81)
Where ˙Q is the volumetric flow rate, r
is distance from the origin and U r is the
velocity pointing out or into the origin
depending whether origin has source or
sink. The relationship between the potential
function to velocity dictates that
9.0
7.0
5.0
3.0
1.0
-9.0 -7.0 -5.0 -3.0 -1.0 1.0 3.0 5.0 7.0 9.0
-2.0
-4.0
-6.0
-8.0
ψ = const
ψ =0
φ = const
Fig. -10.7. Streamlines and Potential lines due
to Source or sink.
∇φ = U = U ̂r =
˙Q
(10.82)
2 πr̂r
Explicitly writing the gradient in cylindrical coordinate results as
∂φ
∂r̂r + 1 ∂φ
r ∂θ ̂θ + ∂φ
∂z ẑ =
˙Q
2 πr̂r +0̂θ +0ẑ (10.83)
Equation (10.83) the gradient components must satisfy the following
The integration of equation results in
∂φ
∂r =
˙Q
2 πr̂r
∂φ
∂z = ∂φ
(10.84)
∂θ =0
φ − φ 0 =
˙Q
2 πr ln r (10.85)
r 0
where r 0 is the radius at a known point and φ 0 is the potential at that point. The stream
function can be obtained by similar equations that were used or Cartesian coordinates.
In the same fashion it can be written that
dψ = U·ŝdl (10.86)
Where in this case dl = rdθ (the shortest distance between two adjoining stream lines
is perpendicular to both lines) and hence equation (10.87) is
dψ = U· rdθ̂r̂r̂r =
˙Q ˙Q
rdθ=
2 πr 2 π
dθ (10.87)