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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, θ, z ). This means the velocity field is given by u = u r ˆ r + u z ˆ z . Assume that the vector potential has the form A = Ψ( r, z ) r ˆ θ , where Ψ 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 Ψ = const along streamlines. (iii) Calculate Ψ for a uniform stream along the ˆ z-axis: u = U ˆ z . 2. For a streamfunction in a two-dimensional flow with u = ∂ψ ∂y and v = − ∂ψ ∂x , show that (i) the streamlines are given by ψ =constant; (ii) | u | = |∇ ψ | which implies that the flow is faster where the streamlines are closer; (iii) the volume flux crossing any curve from...
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This note was uploaded on 01/20/2012 for the course MATH 33200 taught by Professor Eggers during the Fall '11 term at University of Bristol.
- Fall '11
- Polar Coordinates