PS1 - ∂ Ω show that the velocity must be zero everywhere...

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MATH 505: Mathematical Fluid Dynamics Problem Set 1 due Friday, September 14, 2007 1. Consider an outward radial flow in R 3 , given in spherical coordinates by ~u = ( q/r α , 0 , 0), where q and α are positive constants. (a) For what values of α is ∇ · ~u = 0? What happens at the origin? (b) Let S be any smooth surface surrounding the origin. For the values of α in (a), find the constant rate at which the fluid flows through S (the mass flow rate), assuming that the density ρ is constant. 2. Given an incompressible velocity ~u field in R 2 , define a streamfunction ψ ( r, θ, t ) for the polar coordinate system. 3. Show that if a fluid is rotating around a given axis as a solid body with angular velocity f , then the vorticity ω = 2 f . 4. Consider an incompressible irrotational flow of a fluid confined in a rigid square box in R 2 (side length L ). Given the boundary condition on the velocity ~u = 0 on the sides of the box
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Unformatted text preview: ∂ Ω, show that the velocity must be zero everywhere inside the box. Note that you may show this at least two ways: either by explicitly solving for the incompressible irrotational velocity field, or by invoking certain theorems for solutions to particular partial differential equations. 5. Beginning with Euler’s equation for an incompressible fluid, take the curl and thus obtain an equation for the vorticity ω = ∇ × ~u . Under what conditions will ω be conserved in the Lagrangian frame? 6. Make an estimate of the Reynolds number characterizing the flow for: a) Your car travelling at 55 mph. b) The water coming out of your kitchen faucet. c) A red blood cell (diameter 1 μ m) moving through your artery at 2 cm/s, assuming that blood is a Newtonian (Navier-Stokes) fluid. Version 1.1...
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