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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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