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worksheet7 - Fluid Dynamics 3 2011/12 Sheet 7 Homework to...

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Fluid Dynamics 3 2011/12 Sheet 7 Homework to be handed in 2nd December: questions 2,4,5. 1. z bubble θ R When a large bubble rises in water, it assumes the shape of a spherical cap as shown in the Figure. At its lower end it forms an irregular edge. Inside, the pressure is constant, which we take to be zero. (i) Consider a coordinate system in which the bubble is stationary. Show that the speed q along the surface of the bubble satisfies q 2 = 2 gR (1 cos θ ) . (ii) Assuming that the flow near the tip of the spherical cap bubble is the same as that around a sphere in uniform flow, show that q = 3 U sin θ/ 2, where U is the rise speed of the bubble. (iii) Show that the rise speed is U = 2 radicalbig gR/ 3 . 2. In the lecture, we showed that the potential of the flow around a cylinder of radius R is φ = U parenleftbigg r + R 2 r parenrightbigg cos θ. (i) Show that the stream function is ψ = U parenleftbigg r R 2 r parenrightbigg sin θ.
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