Lecture F12 Mud: Stream Function, Potential Function
(25 respondents)
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1. The book claims “�2 = �1 represents mass flow perpendicular to the page”.
Isn’t 2-D flow in the plane of the page? (1 student)
It actually says “mass flow per unit depth perpendicular to the page”. The “perpen-
dicular” refers to the unit depth, not to the flow.
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2. What’s the difference between � and �? (2 students)
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� is the most general type of stream function, and gives the mass flux components:
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��/�y = �u −��/� x = �v
For low speed flows where � is a constant, it is convenient to absorb the constant �
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factor into the stream function by defining � = �/�. We now have:
��/�y = u −��/�x = v
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It’s important to remember that � can be used only for low speed flows, while � has
no such restriction.
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The difference �2 −�1 gives the mass flow between two streamlines, while �2 −�1
gives the volume flow between two streamlines (but only in low-speed flow).
3. Where does �(x, y) = ln �x2 + y2 come from? (1 student)
I just made it up. The neat thing about a stream function is that no matter how
complicated �(x, y) might get, the resulting u(x, y) and v(x, y) that you get from it
satisfy the mass continuity equation
ρ
V � �u/�x + �v/�y = 0� ·
and hence represent a physically possible flow.
4. Is there are graphical relation between streamlines and stream function
lines? (1 student)
Lines of constant stream function are the same as streamlines.
5. I don’t understand how the funnel-shaped � and the spiral-shaped ∂ repre-
sent the same vortex flow? (1 student)
Both functions produce the same u(x, y) and v(x, y), and hence they both represent
the same physical flowfield.
6. Will be dealing mostly with 2-D flows? (1 student)
The approach in Unified Fluids is to present concepts in the simplest way possible, so
that the understanding isn’t lost in unnecessary complexity. So we use the smallest
number of spatial dimensions in the examples and applications. Sometimes 1-D, usually
2-D, and occasionally 3-D if it’s unavoidable.
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