Basics of Fluid Mechanics, 2014a

354 CHAPTER 10. POTENTIAL FLOW
Equation (10.139) describes circles with center on the y coordinates at y = r 0 cot 2 πψ .
Q 0
It can be noticed that these circles are orthogonal to the the circle that represents the
the potential lines. For the drawing it is convenient to write equation (10.139) in
dimensionless form as
( ) 2 ( x y
+ − cot 2 πψ ) 2 (
= csc 2 πψ ) 2
(10.140)
r 0 r 0 Q 0 Q 0
Dipole Flow
It was found that when the distance between the sink and source shrinks to zero
a new possibility is created which provides benefits to new understanding. The new
combination is referred to as the dipole. Even though, the construction of source/sink
to a single location (as the radius is reduced to zero) the new “creature” has direction
as opposed to the scalar characteristics of source and sink. First the potential function
and stream function will be presented. The potential function is
lim φ = − Q (
0 1 r 2
r 0→0 2 π 2 ln + r 2 )
0 − 2 rr 0 cos θ
r 2 (10.141)
+ r 02 +2rr 0 cos θ
To determine the value of the quantity in equation (10.141) the L’Hpital’s rule will be
used. First the appropriate form will be derived so the technique can be used.
Caution: mathematical details which can be skipped
Multiplying and dividing equation (10.141) by 2 r 0 yields
2 nd part
1 st part
{ }} { { ( }} {
lim φ = Q 0 2 r 0 1 r 2 + r 2 )
0 − 2 rr 0 cos θ
ln
r 0 →0 2 π
}{{}
22 r 0 r 2 + r 02 +2rr 0 cos θ
4
(10.142)
Equation (10.142) has two parts. The first part, (Q 0 2 r 0 )/2 π, which is a function of
Q 0 and r 0 and the second part which is a function of r 0 . While reducing r 0 to zero,
the flow increases in such way that the combination of Q 0 r 0 is constant. Hence, the
second part has to be examined and arranged for this purpose.
( r 2 + r 2 )
0 − 2 rr 0 cos θ
ln
r
lim
2 + r 02 +2rr 0 cos θ
(10.143)
r 0 →0
4 r 0
It can be noticed that the ratio in the natural logarithm approach one r 0 → 0. The
L’Hopital’s rule can be applied because the situation of nature of 0/0. The numerator
can be found using a short cut 8
8 In general the derivative ln f(ξ) is done by derivative of the natural logarithm with fraction inside.
g(ξ)
The general form of this derivative is
d f(ξ)
ln
dξ g(ξ) = g(ξ) d
f(ξ) dξ
„ « f(ξ)
g(ξ)