worksheet3 - Fluid Dynamics 3 2011/12 Sheet 3 Homework to...

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Unformatted text preview: Fluid Dynamics 3 2011/12 Sheet 3 Homework to be handed in on Friday 4th November: Q 1,3,5,7 1. Consider an axisymmetric, incompressible flow in cylindrical polar coordinates ( r, , z ). This means the velocity field is given by u = u r r + u z z . Assume that the vector potential has the form A = ( r, z ) r , where is called the Stokes streamfunction . Note that the flow is not confined to a plane, but is axisymmetric, hence the Stokes streamfunction is an object slightly different from the ordinary stream function, used to describe planar flow. (i) Compute the components u r and u z of the velocity field and confirm that u = 0 using cylindrical polars. (ii) Show that = const along streamlines. (iii) Calculate for a uniform stream along the z-axis: u = U z . 2. For a streamfunction in a two-dimensional flow with u = y and v = x , show that (i) the streamlines are given by =constant; (ii) | u | = | | which implies that the flow is faster where the streamlines are closer; (iii) the volume flux crossing any curve from...
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worksheet3 - Fluid Dynamics 3 2011/12 Sheet 3 Homework to...

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