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worksheet6 - Fluid Dynamics 3 2011/12 Sheet 6 Homework to...

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Fluid Dynamics 3 2011/12 Sheet 6 Homework to be handed in 25th November: questions 2,3,4 1. (i) Show that in spherical polars, the potential of a uniform stream and of a dipole are (a) φ = Ur cos θ , and (b) φ = - μ cos θ r 2 , respectively. (ii) By using the expression for the Laplacian in spherical coordinates, confirm that each of these potentials satisfy Laplace’s equation. (iii) Calculate u in spherical coordinates for each of the potentials. (iv) Verify that φ = Ur cos θ + A cos θ r 2 satisfies the boundary condition n · ∇ φ = 0 on the surface of a sphere r = R for a suitably chosen A . 2. A sphere is moving with velocity U through an ideal, incompressible fluid, which is at rest at infinity. (i) Compute φ = ( A · ∇ ) 1 r , where A is a constant vector, and show that 4 φ = 0. (ii) Placing x = 0 at the instantaneous centre of the sphere, compute the normal velocity u n u · n on the surface. Determine A . (iii) Using Bernoulli’s equation for unsteady flow, find the pressure on the surface of the sphere. Compare to what we found in the lecture for a sphere in a uniform stream.
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