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

This preview shows page 1. Sign up to view the full content.

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?
This is the end of the preview. Sign up to access the rest of the document.

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 ﬂow (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 ﬁnite distance from one another ( i.e. , in 2D). 4. Consider the geometry and ﬂow described by Kundu & Cohen, Chapter 4, Problem 11. Imagine that at t = 0, the upper plate was at rest, and the ﬂuid had no motion. Describe what happens to the ﬂow if at t = 0 + , the upper plate moves with constant velocity, U . Use the Navier-Stokes equation, Eq. 4.45. 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online