ps3 - Explain why viscosity causes the vortex to decay in...

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APPH4200 Physics of Fluids: Homework 3 1. Show that for an axisymmetric, two-dimensional flow with vorticity distribution ω = (0 , 0 ( r )) in cylindrical polar coordinates ( r,φ,z ) (where 2-dimensional means no motion in the z direction, and no variation of the flow with respect to z ), the vorticity equation for an incompressible, constant density fluid in an unbounded domain reduces to ∂ω ∂t = ν r ∂r ( r ∂ω ∂r ) . If there is a concentrated line vortex along the z-axis at time t=0 in an otherwise irrotational fluid, show by substitution (that is, you don’t have to derive it, just plug it into the equation) that the vorticity distribution at any subsequent time is ω = Φ 4 πνt e - r 2 / 4 νt , where Φ is a constant proportional to the initial circulation. What is the corresponding velocity distribution?
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Unformatted text preview: Explain why viscosity causes the vortex to decay in time even though it was shown in class and in the book that the net viscous force vanishes in a point vortex ow (K&C, pp. 131-134). 2. Kundu & Cohen, Chapter 5, Problem 3. 3. Kundu & Cohen, Chapter 5, Problem 6. Just think of a vortex ring as a dipole of opposite-signed point vortices at a nite distance from one another ( i.e. , in 2D). 4. Consider the geometry and ow described by Kundu & Cohen, Chapter 4, Problem 11. Imagine that at t = 0, the upper plate was at rest, and the uid had no motion. Describe what happens to the ow if at t = 0 + , the upper plate moves with constant velocity, U . Use the Navier-Stokes equation, Eq. 4.45. 1...
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