 Solution

ID: past-exam-of-the-advanced-fluid-dynamics-course-of-the-university-of-oxford/2024/1/c/solution

Take outward radial force as positive and keep the displaced element's angular rate fixed at

Ω_{0} = Ω (R_{0})

. Its imbalance is

Δ f_{R} = ρ [(R_{0} + ξ_{R}) Ω_{0}^{2} - Φ^{'} (R_{0} + ξ_{R})] .

Use the equilibrium relation from part b,

Φ^{'} (R) = R Ω (R)^{2}

Δ f_{R} = - ρ R_{0} \frac{d Ω ^{2}}{d R}_{R_{0}} ξ_{R} + O (ξ_{R}^{2}), ∣Δ f_{R} ∣ ≃ ρ R_{0} \frac{d Ω ^{2}}{d R} ξ_{R} .

The absolute values are needed for a magnitude. As printed, the proposed magnitude has a signed right side; it would be negative for outward displacement in a Keplerian disk. For decreasing

Ω^{2}

, the signed imbalance above points outward and therefore amplifies that fixed-angular-rate displacement. This calculation fixes the angular rate, rather than the specific angular momentum; the latter is the conserved quantity for an uncoupled inviscid radial displacement and leads to the Rayleigh centrifugal stability criterion.

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