## Transcribed Text

1) Secondary flows: [50 marks] In some situations where the flow is externally bounded by
walls (as in flow through pipes, or inside a tea cup), apart from the primary flow, there exists
a secondary flow of typically much smaller magnitude. These secondary flows are usually a
result of some imbalance in the forces that makeup the primary flow. They can, however, create
interesting flow phenomena that might seem counterintuitive if you never knew the existence
of the secondary flows.
(i)
For making tea, let's suppose you take tea leaves in a cup, add hot water and let
it settle. You would see tea leaves evenly settled at the bottom of the cup. Now,
stir the cup with a spoon and leave it. You will see water circulating, and
surprisingly, the tea leaves settle at the bottom but pile up in the middle like a
cone. Explain why this happens. [20 marks]
(Hint: This is not due to a low-pressure
vortex core, where radial velocity V =
C/r. The primary flow here is a 'solid
body rotation' = r, so you would
expect the tea leaves to be thrown to the
side walls, but of course this is not
happening. Use Euler's equations, and
Before stirring
After stirring
note that at the bottom wall the wall-
friction slows down V, which creates a force imbalance.)
(ii)
The same principle can be applied to another problem of that of steering a ship
in a meandering river. Those who steer a
ship under these conditions know that
they should stay close to the outer bank,
else they will hit the ground. Why is the
water shallow at the inner bank? [20
A-A
marks] (Hint: It has to do with movement
of the sand - the same as tea leaves in the
above problem.)
(iii)
In a pipe flow with a 90° bend, a similar effect gives rise to secondary flows.
Show the direction of the secondary flow in the pipe cross-section after the
bend. [10 marks]
2) Surface tension: [30 marks] An interesting property of fluid interfaces is "surface tension'
- which is the energy required to stretch the fluid interface by one unit area. Watch the
following video, and try to understand the concept:
Now, answer the following questions using the concepts from the videos or from elsewhere.
(i)
What is the maximum mass of a 10cm long straw that can be supported on the water
surface? [10 marks]
(ii)
What is the pressure inside a 1 micron water bubble, and a 100 micron bubble? This
will tell you that if you connected a smaller bubble to a
larger bubble with a tube, the larger bubble will get larger
and the smaller would get smaller! [10 marks]
(iii)
Explain why 'tears of wine' appear. See the adjacent
picture. [10 marks]
3) Bernoulli's equation for compressible flows: [20 marks] We have derived and used
Bernoulli's equation multiple times in this subject, which was for incompressible flows. For
compressible flows, however, we would need to modify Bernoulli's equation. Derive the
following compressible form of Bernoulli's equation,
const.
Start from the Euler equation we derived in the class, and use the adiabatic relation:
p =
constant.

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