Basics of Fluid Mechanics, 2014a

340 CHAPTER 10. POTENTIAL FLOW
φ = −2 yx 9.0 ψ = x 2 − y 2
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
-9.0 -8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
-1.0
-2.0
-3.0
-4.0
-5.0
-6.0
-7.0
-8.0
-9.0
Fig. -10.4. Stream lines and potential lines are drawn as drawn for two dimensional flow. The
green to green–turquoise color are the potential lines. Note that opposing quadrants (first
and third quadrants) have the same colors. The constant is larger as the color approaches
the turquoise color. Note there is no constant equal to zero while for the stream lines the
constant can be zero. The stream line are described by the orange to blue lines. The orange
lines describe positive constant while the purple lines to blue describe negative constants. The
crimson line are for zero constants. 5
where f(y) is arbitrary function of y. Utilizing the other relationship ((10.66)) leads
∂φ
∂y
= −2 x +
df(y)
dy
= − ∂ψ
∂x = −4 x3 (10.71)
Therefore
df(y)
=2x − 4 x 3 (10.72)
dy
After the integration the function φ is
φ = ( 2 x − 4 x 3) y + c
(10.III.c)
The results are shown in Figure
End Solution