38A (Fluid Dynamics II)

The radial strain draws fluid inward, axial strain expels it away from the midplane, and

v (r)

supplies swirl. In cylindrical coordinates

r^{- 1} \partial_{r} (r (- α r)) + \partial_{z} (2 α z) = - 2 α + 2 α = 0

, so the flow is incompressible.

b

The only vorticity component is

ω = (1/ r) (r v)'

. Its steady equation is

- α r ω' = 2 α ω + ν (ω'' + r^{- 1} ω')

, solved by

ω = ω_{0} e^{- α r^{2} / (2 ν)}

. Regularity at zero gives

v (r) = \frac{ω _{0} ν}{α r} (1 - e^{- α r^{2} / (2 ν)}) .

Its maximum satisfies

e^{s} = 1 + 2 s

s = α r^{* 2} / (2 ν)

, so

r^{*} = O (ν / α)

Adding inward side flux and outward end-cap flux gives the net inward angular-momentum advection

A = \frac{8 π ρ z _{0} ω _{0} ν ^{2}}{α} [1 - (1 + s_{0}) e^{- s_{0}}], s_{0} = \frac{α r _{0}^{2}}{2 ν} .

It is positive because

e^{s_{0}} > 1 + s_{0}

, and tends to the stated value.

Integrating the viscous traction

2 ρ ν e_{r θ}

over the side and end surfaces gives

T = - \frac{8 π ρ z _{0} ω _{0} ν ^{2}}{α} [1 - (1 + s_{0}) e^{- s_{0}}] = - A .

It is always negative. In steady state the cylinder stores no angular momentum, so the divergence theorem for angular-momentum balance requires viscous torque to cancel net advective influx.

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