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# PS4 - MATH 505 Mathematical Fluid Dynamics Problem Set 4...

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MATH 505: Mathematical Fluid Dynamics Problem Set 4 due Friday, November 16, 2007 1. Consider a U-shaped tube filled with two different fluids, of equal volumes and densities ρ 1 and ρ 2 > ρ 1 . At t = 0 the two fluids are arranged so that each one occupies half of the “U”, thus they each rise to the same height. Find the equilibrium arrangement of the fluids. 2. Consider steady Poiseuille flow through a pipe of radius R 1 , using cylindrical coordi- nates. Assume that the pressure is given by p ( z ) = Cz , that u z = u z ( r ), and that u r = u θ = 0. Additionally, assume a solid inner cylinder concentric to the first, with radius R 2 < R 1 , such that the fluid flows axially between the two cylinders. Find u z ( r ). Using a Mathematica .nb file (old example still available on the webpage), plot the solution for several values of R 1 with fixed R 2 = 1 (and fixed values of the other constants). Print out your results, and include them with your PS. 3. In class we solved the problem of axisymmetric incompressible Stokes flow past a sphere (radius a ), and by assuming the streamfunction could be written as ψ ( r, θ
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