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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