Basics of Fluid Mechanics, 2014a

10.3. POTENTIAL FLOW FUNCTIONS INVENTORY 351
Equation (10.123), when noticing that the cos θ coth(−x) =− coth(x), can be written
as
( ) 2 πφ
−2 r 0 r cos θ coth = r 2 2
+ r 0 (10.124)
Q
In Cartesian coordinates equation (10.124) can be written as
r cos θ
{}}{
−2 r 0 x
(
coth − 2 πφ )
= x 2 + y 2 2
+ r 0
Q
(10.125)
Equation (10.125) can be rearranged by the left hand side to right as and moving r 0
2
to left side result in
r cos θ
−r 2 {}}{
0 =2r 0 x
( 2 πφ
coth
Q
Add to both sides r 0 2 coth 2 2 πφ
Q 0
transfers equation (10.126)
r 0 2 coth 2 2 πφ
Q 0
− r 0 2 = r 0 2 coth 2 2 πφ
Q 0
The hyperbolic identity 6 can be written as
r 0 2 csch 2 2 πφ
Q 0
= r 0 2 coth 2 2 πφ
Q 0
r cos θ
{}}{
+2r 0 x
r cos θ
{}}{
+2r 0 x
)
+ x 2 + y 2 (10.126)
( 2 πφ
coth
Q
( 2 πφ
coth
Q
)
+ x 2 + y 2
(10.127)
)
+ x 2 + y 2 (10.128)
End Caution: mathematical details
It can be noticed that first three term on the right hand side are actually quadratic
and can be written as
(
r 2 0 csch 2 2 πφ
= r 0 coth 2 πφ ) 2
+ x + y 2 (10.129)
Q 0 Q 0
equation (10.129) represents a circle with a radius r 0 csch 2 πφ
( ) 2 πφ
and a center at ±r 0 coth .
Q 0
Q 0
The potential lines depicted on Figure 10.11.
For the drawing purposes equation (10.129) is transformed into a dimensionless
form as
(
coth 2 πφ + x ) 2 ( ) 2 y
+ = csch 2 2 πφ
(10.130)
Q 0 r 0 r 0 Q 0