Homework Set #3
1. In 2-D flows, show that streamlines (along which ψ is constant), and lines along which the velocity
potential Φ is constant are orthogonal to each other.
Stream function C1
Stream function C2
Velocity potential Lines
Hint: the solution is in the text book, in Chapter II.
2. In the figure above, show that the difference between the stream function values C1-C2 equals the volume
rate of flow (per unit width in a direction perpendicular to the plane of the paper) of the fluid that flows between
these two streamlines.
Hint: the solution is in the text, in Chapter II.
3. Given the stream function for a doublet:
r
θ
π
κ
ψ
sin
2
−=
find the velocity potential
φ
associated with this doublet.
Hint: Take derivatives of the stream function with respect to r and θ to find radial and tangential components of
velocity (
) vand
θ
r
v
. Equate these to corresponding derivatives of the velocity potential
φ
. Integrate
θ
φφ
∂
∂
∂
∂
and
r
to get
φ
. The answer will be
r
θ
π
κ
φ
cos
2
=.
4. Using the MATLAB script given in a recent handout as a starting point, construct contours of the following
stream function, which corresponds to the superposition of a uniform flow and a doublet:
−=
∞2
2
1r
R
yu
ψ
You may use any numerical value for the velocity u∞ and R (I would non-dimensionalize the stream function by
u∞ * R and use unity). Visually verify that this represents flow over a circular cylinder of radius R.
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