 Solution

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

R

has radial acceleration

- R Ω (R)^{2} e_{R}

. Neglecting fluid and magnetic forces gives

- ρR Ω (R)^{2} = - ρ \frac{d Φ}{d R}, \frac{d Φ}{d R} = R Ω (R)^{2} .

In a local rotating frame with

Ω_{0} = Ω (R_{0})

, the fluid is instantaneously at rest at the reference radius

R_{0}

+ ρ R_{0} Ω_{0}^{2}

then balances the inward gravitational force. Away from

R_{0}

, the fluid's nonzero rotating-frame velocity must also be retained; the entire differentially rotating disk is not at rest in that one frame.

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