Basics of Fluid Mechanics, 2014a

10.3. POTENTIAL FLOW FUNCTIONS INVENTORY 355
at
End Caution: mathematical details
lim
r 0 →0
✟2 r ✟✯ 0
0 − 2 r cos θ
r 2 + ✚r ✚❃0 2 0 − ✘ ✘✘✘ ✘✿ −
✟2 r ✟✯ 0
0 +2r cos θ
2 rr 0
0 cos θ r 2 + ✚r ✚❃0 2 0 + ✘ 2 rr ✘✘✘ ✘✿ 0
0 cos θ
= − cos θ
4
r
(10.144)
Combining the first and part with the second part results in
φ = − Q 0 r 0
π
cos θ
r
(10.145)
After the potential function was established the attention can be turned into the
stream function. To establish the stream function, the continuity equation in cylindrical
is used which is
∇·U = 1 ( ∂rUr
+ ∂U )
θ
r ∂r ∂θ
The transformation of equations (10.46) and (10.48) to cylindrical coordinates results
in
U r = 1 r
∂ψ
∂θ
(10.146a)
U θ = − ∂ψ
∂r
(10.146b)
The relationship for the potential function of the cylindrical coordinates was determined
before an appear the relationship (10.66) and (10.67) in cylindrical coordinates to be
U r = ∂φ
∂r
and
(10.147a)
The internal derivative is done by the quotient rule and using the prime notation as
0 “ ”′ “ ”′ 1
„
ln f(ξ)
«′
= g(ξ) f(ξ) g(ξ) − g(ξ) f(ξ)
B
C
@ “ ”
g(ξ) f(ξ)
2 A
g(ξ)
by canceling the various parts (notice the color coding). First canceling the square (the red color)
and breaking to two fractions and in the first one canceling the numerator (green color) second one
canceling the denominator (cyan color), one can obtain
0
„
ln f(ξ)
«′
= ✟✟ g(ξ)
✟ ✟ “ ”′
f(ξ) g(ξ) −✟ ✟ “
1
”′ “ ”′ “ ”′
g(ξ) f(ξ)
g(ξ) f(ξ)
g(ξ) ✟f(ξ)
✟ B
C
@ “
✟g(ξ)
✟ ” A = −
2 ✄ g(ξ) f(ξ)