Basics of Fluid Mechanics, 2014a

350 CHAPTER 10. POTENTIAL FLOW
The sink is at the same distance but
at the negative side of the x coordinate
and hence it can be represented by the potential
function
Q 1 = − Q ( )
0
2 π ln rA
(10.116)
r 0
The description is depicted on Figure
10.10. The distances, r A and r B are defined
from the points A and B respectively.
The potential of the source and the sink is
φ = Q 0
2 π (ln r A − ln r B ) (10.117)
A
θ B
r 0
y
r B r
O θ
r 0
R
r A
B θ A
Fig. -10.10. Combination of the Source and
Sink located at a distance r 0 from the origin on
the x coordinate. The source is on the right.
In this case, it is more convenient to represent the situation utilizing the cylindrical
coordinates. The Law of Cosines for the right triangle (OBR) this cases reads
In the same manner it applied to the left triangle as
r B 2 = r 2 + r 0 2 − 2 rr 0 cosθ (10.118)
r A 2 = r 2 + r 0 2 +2rr 0 cosθ (10.119)
Therefore, equation (10.117) can be written as
⎛
r 2 2
⎞
+ r 0
φ = − Q 0 1
2 π 2 ln ⎜ 2 rr 0 cos θ +1
⎟
⎝ r 2 2
+ r 0
2 rr 0 cos θ − 1 ⎠ (10.120)
It can be shown that the following the identity exist
Caution: mathematical details which can be skipped
x
coth −1 (ξ) = 1 ( ) ξ +1
2 ln ξ − 1
(10.121)
where ξ is a dummy variable. Hence, substituting into equation (10.120) the identity
of equation (10.121) results in
φ = − Q (
0
r 2 )
2
+ r 0
2 π coth−1 (10.122)
2 rr 0 cos θ
The several following stages are more of a mathematical nature which provide
minimal contribution to physical understanding but are provide to interested reader.
The manipulations are easier with an implicit solution and thus
(
coth − 2 πφ )
= r2 2
+ r 0
(10.123)
Q 2 rr 0 cos θ