Chapter 4 • Differential Relations for a Fluid Particle
285
Further show that the angular velocity about the z axis in such a flow would be given by
2ω z =
1 ∂
1 ∂
(rυθ ) −
(υr )
r ∂r
r ∂θ
Finally show that φ as defined satisfies Laplace’s equation in polar coordinates for incompressible flow.
Solution: All of these things are quite true and easy to show from their definitions. Ans.
4.59 Consider the two-dimensional incompressible velocity potential φ = xy + x2 – y2.
(a) Is it true that ∇2φ = 0, and, if so, what does this mean? (b) If it exists, find the stream
function ψ(x, y) of this flow. (c) Find the equation of the streamline which passes through
(x, y) = (2, 1).
Solution: (a) First check that ∇2φ = 0, which means that incompressible continuity is
satisfied.
∇ 2φ =
∂ 2φ ∂ 2φ
+
= 0 + 2 − 2 = 0 Yes
∂ x 2 ∂ y2
(b) Now use φ to find u and v and then integrate to find ψ.
u=
y2
∂φ
∂ψ
= y + 2x =
, hence ψ =
+ 2 xy + f ( x )
2
∂x
∂y
df
x2
∂φ
∂ψ
v=
= x − 2y = −
= −2 y − , hence f ( x ) = − + const
dx
2
∂y
∂x
The final stream function is thus ψ =
1 2
( y − x 2 ) + 2 xy + const
2
Ans. (b)
(c) The streamline which passes through (x, y) = (2, 1) is found by setting ψ = a constant:
At ( x, y) = (2, 1), ψ =
1 2
3
5
(1 − 2 2 ) + 2(2)(1) = − + 4 =
2
2
2
Thus the proper streamline is ψ =
1 2
5
( y − x 2 ) + 2 xy =
2
2
Ans. (c)